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Zekelman1, Tengfei Xue1,2, Chaoyi Zhang2, Yang Song3, Nikos +Makris1, Yogesh Rathi1, Weidong Cai2, Lauren J. O’Donnell1 +1 Harvard Medical School, MA, USA +2 The University of Sydney, NSW, Australia +3 The University of New South Wales, NSW, Australia +ABSTRACT +The structure and variability of the brain’s connections can +be investigated via prediction of non-imaging phenotypes +using neural networks. However, known neuroanatomical +relationships between input features are generally ignored in +network design. We propose TractGraphCNN, a novel, +anatomically informed graph CNN framework for machine +learning +tasks +using +diffusion MRI tractography. An +EdgeConv module aggregates features from anatomically +similar white matter connections indicated by graph edges, +and an attention module enables interpretation of predictive +white matter tracts. Results in a sex prediction testbed task +demonstrate strong performance of TractGraphCNN in two +large datasets (HCP and ABCD). Graphs informed by white +matter geometry demonstrate higher performance than +graphs informed by gray matter connectivity. Overall, the +bilateral cingulum and left middle longitudinal fasciculus +are consistently highly predictive of sex. This work shows +the +potential of incorporating anatomical information, +especially known anatomical similarities between input +features, to guide convolutions in neural networks. +Index Terms— Sex classification, white matter tracts, +graph CNN, neuroanatomy, tractography +1. INTRODUCTION +The +human +brain’s +white +matter +(WM) +fiber +tract +connections have important inter-individual variability, with +implications +for +understanding +neurodevelopment +and +disease +[1]. Recently, brain variability is studied by +predicting non-imaging phenotypes from high-dimensional +neuroimaging data using machine learning [2]. Many +aspects of such machine learning methods are active areas +of research (e.g. multiple modalities [3], comparison of +methodology [4], and interpretation [2], [4]). However, we +find relatively fewer studies of tailored network design that +can +leverage +neuroanatomical +knowledge. +Here +we +investigate deep neural networks informed by the anatomy +and geometry of the brain’s WM structure. +A few studies have aimed to develop dedicated neural +networks for analyses of the brain’s structural connections. +The BrainNETCNN [5] includes novel convolutional filters +that improve performance [6] by handling the topology of +connectivity +matrices +(where +each +row +or +column +corresponds to a gray matter (GM) region or node, and +entries or edges in the matrix indicate connectivity strengths +between GM regions). Other approaches apply graph +convolutional neural networks to connectivity matrices, e.g. +[7]. However, the above classes of methods are restricted to +the anatomical information contained in the row and column +structure of the connectivity matrix, and they cannot +leverage any additional anatomical information to inform +network convolutions. +We hypothesize that the performance of deep learning +can +be +enhanced by incorporating information about +anatomical neighborhoods of WM connections with similar +geometry +and +connectivity. +To +encode +neighborhood +relationships, we adopt the popular EdgeConv neural +network module originally designed for the Dynamic Graph +CNN (DGCNN) [8], and we use it to construct static graphs +informed +by +brain +anatomy. +Following +two +major +approaches to study the brain’s structural connectivity [9], +we investigate 1) white-matter-centric graphs (WMG) that +define neighborhoods according to fiber tract geometry, and +2) +gray-matter-centric +graphs +(GMG) +that +define +neighborhoods according to connected gray matter regions. +To focus our project, we choose a testbed problem of +sex prediction. While this problem is not straightforward +[10], [11], sex is known to be an important source of WM +variability +[12]. +Many +studies +have +investigated +sex +prediction +[6], +[13]–[16] +using +microstructure +and/or +connectivity +features +from quantitative diffusion MRI +(dMRI) +tractography +[9]. +Both +microstructure +(e.g. +fractional anisotropy, FA [6]) and connectivity (e.g. number +of streamlines, NoS [14]) features provide good prediction +performance. However, NoS is affected by intracranial +volume, a common confound in sex prediction [10]. In this +work +we +therefore +utilize +FA +and +a percentage of +streamlines that is normalized to reduce the effect of brain +size. We quantify these features using an anatomically +curated, atlas-based WM fiber cluster parcellation that is +1 + +consistent across datasets, acquisitions, and the human +lifespan [17]. Importantly for this study, a description of the +WM geometry and GM connectivity of each fiber cluster is +provided in the ORG atlas [17]. +In this study, we propose an anatomically-informed +graph CNN framework, called TractGraphCNN, to leverage +neuroanatomical knowledge for sex prediction based on +cluster-wise WM features from dMRI tractography. The +main contributions of this study are as follows. First, for the +first time, we model the anatomical relationship between +clusters as a graph, informed by WM geometry and GM +connectivity information. Second, we integrate EdgeConv +modules into our framework to extract features from +anatomically similar clusters to improve performance of sex +prediction. Finally, our framework is able to identify +important WM tracts for sex classification by leveraging an +attention +module. +We +evaluate +our +method +on +two +large-scale datasets of children and healthy young adults. +2. METHODS +Fig. 1 gives an overview of our proposed TractGraphCNN +method. First, WM features are extracted from dMRI +tractography data (Sec. 2.1), resulting in two features for +each cluster. Second, we build a graph to model the +relationship between clusters, indicated by WM geometry or +GM connectivity information (Sec. 2.2). Third, the built +graph is input to the proposed TractGraphCNN framework +(Sec. 2.3) for sex classification. The framework aggregates +information from connecting clusters in the graph via +EdgeConv modules. An attention module is adopted to +enable interpretation of important tracts that are predictive +for sex classification. +Fig. 1. (a) Overall pipeline of TractGraphCNN. (b) Network +structure of the attention module. +2.1 dMRI datasets and feature extraction +2.1.1 Adolescent Brain Cognitive Development (ABCD) +This study utilized dMRI data of 9342 young children (age +9-11) from the large-scale, multi-site ABCD dataset [18]. +We harmonized the dMRI data across 21 acquisition sites to +remove +scanner-specific +biases +while +preserving +inter-subject biological variability [19], [20]. Of all subjects, +4879 (52.2%) are males and 4463 (47.8%) are females. +7473 subjects (80%) are used for training the neural +network, while 1869 (20%) are used for testing. +2.1.2 Human Connectome Project (HCP) +We also conducted experiments on a dataset of 964 subjects +(age 22-37) from the Human Connectome Project, a large +multimodal dataset composed of healthy young adults [21]. +Of all subjects, 443 are male (45.6%) and 521 are female +(54.4%). 772 subjects (80%) are used for training and 192 +(20%) subjects are used for testing. +2.1.3 White matter fiber cluster features +Two-tensor unscented Kalman filter tractography (UKFt) +[22] via SlicerDMRI [23], [24] was applied to obtain whole +brain tractography from the dMRI data. Tractography was +parcellated with an anatomically curated cluster atlas and a +machine +learning +approach +that +has +been shown to +consistently identify WM tracts across the human lifespan +[17]. +For +each +subject, +953 +expert-curated +clusters +categorized into 75 WM tracts were obtained. Importantly, +cluster +IDs are assigned according to the atlas and +correspond across subjects (e.g. cluster #1 corresponds +across +all +subjects +and +datasets +studied). +Statistical +microstructure measurements were then computed for each +cluster. We adopted two measurements for the task: +fractional anisotropy (FA) and percentage of streamlines +(PoS). FA of the cluster is computed as the mean FA across +all streamline points within the cluster. The PoS of a cluster +is calculated as the number of streamlines of the cluster +divided by the total number of streamlines across all clusters +of the subject, to reduce the effect of brain size. For each +subject, this resulted in an input feature matrix of size +2x953. For absent clusters due to individual anatomical +variation, we set features to zero. Finally, a min-max +normalization was performed on the input feature matrix for +FA and PoS individually. +2.2 Anatomically informed graph construction +We propose to build graphs such that edges connect +neighboring fiber clusters with similar anatomy. Each +cluster +is represented as a node in the graph with +cluster-wise WM features as node features. +2.2.1 Fiber tract geometry informed graph +The first type of graph (WMG) proposed in our study is +based on WM tractography fiber geometric similarity, a +well-established concept in the field of fiber clustering [9]. +Specifically, we first compute the geometric distance +between each pair of fiber clusters in the ORG atlas, which +is measured as the mean of the pairwise fiber distances (the +popular mean closest point fiber distance is used [25]) +between the two fiber clusters. A low distance between two +clusters represents a high similarity in terms of WM +anatomy. Then, for each cluster, we choose the top k (k =20 +is used in our study following the default setting in +DGCNN) clusters with the lowest geometric distances as +neighbors, and edges are placed in between for graph +construction. +2 + +N×64 +N×64 +N×64 +N×64 +N×2 +EdgeConv +EdgeConv +EdgeConv +C +mlp [64] +mlp [64] +mlp [64] +Att. +Mod. +Conv1d +mlp [32] +Conv1d +mlp (1] +Conv1d +mlp [32] +团2.2.2 Cortical and subcortical connectivity informed graph +The second type of graph (GMG) proposed in our study is +based on GM regions to which the fiber clusters connect. +Specifically, for each cluster, we first identify its connected +Freesurfer GM regions. The ORG atlas provides the +percentage of streamlines from each cluster that intersect +each Freesurfer region [26]. We leverage this information to +identify the top two FreeSurfer regions most commonly +intersected +by +the +streamlines +of +each +cluster. +The +neighborhood of a cluster is then defined as the set of +clusters with at least one top Freesurfer region in common, +and edges are placed in between for graph construction. +2.3 Network architecture +The overall architecture of our TractGraphCNN framework +is shown in Fig. 1. TractGraphCNN extends the 1D CNN +model [15] for group classification using fiber cluster +features with two innovative improvements. First, we +replace the 1D convolutional layers in the original model +with EdgeConv layers [8] to utilize the information of +anatomically neighboring clusters (Sec. 2.3.1). Second, we +add a gated attention module [27] in the network that can +assess the importance of each cluster to enable result +interpretation (Sec. 2.3.2). +Fig. 2. Graphic illustration of the usage of EdgeConv to +leverage +fiber +cluster neighborhood information, with +comparison to the standard 1D convolutional layer. +2.3.1 1D CNN with EdgeConv +Fig. 2 illustrates the use of EdgeConv in TractGraphCNN to +aggregate +information +from +neighboring +graph +nodes +representing fiber clusters. EdgeConv was proposed in the +popular DGCNN method to capture the local geometric +structure of point clouds [8]. The basic idea of EdgeConv is +to use a learnable fully-connected layer to compute an edge +feature of two neighboring nodes xi and xj based on their +input features. Then, the output of EdgeConv is calculated +by aggregating the edge features with max-pooling. This +learning process enables dynamic update of graph structure +by recomputing distances of points in the feature space. In +our +application, +because +the +anatomical +relationships +between fiber clusters do not change, we maintain a static +graph structure across layers by using the same graph +structure across all EdgeConv layers without recomputing +distances between node features. In addition, after feature +extraction, we do not use the max pooling operation as in +traditional Graph CNNs, but instead we retain the flatten +operation in the 1D CNN model [15] to preserve the +information about cluster correspondence across subjects. +2.3.2 Interpretation of important tracts +For the purpose of interpretation, it is important to identify +important WM connections for the task of sex classification. +To achieve this, we improve our neural network by adding +an attention mechanism using the popular gated attention +module from [27]. The attention module (Fig. 1(b)) is +composed of two parallel fully-connected layers followed +by a sigmoid and tanh activation functions, a concatenation +operation, and another fully-connected layer followed by a +sigmoid function. The output is a 1-D attention map of size +973 with values between 0 and 1, indicating the importance +of the corresponding cluster to the classification task. Next, +we identify the most predictive anatomical WM tracts. We +first compute the mean importance of each cluster across all +testing subjects. Then we find the top T (T = 50 in our +experiment) clusters with the highest mean importance +values. Finally, we identify all tracts to which the top 50 +clusters belong, according to the ORG atlas [17]. +2.4 Implementation details +All experiments are performed on a NVIDIA RTX A4000 +GPU +using +Pytorch +(v1.12.1) +[28]. +For +the +overall +architecture, we use two EdgeConv layers and one 1-D +convolutional layer to extract features. The two EdgeConv +layers compute edge features with two fully-connected +layers (64 , 64). Shortcut connections are included to extract +multi-scale features and one 1-D convolutional layer (kernel +size=1, +output +channel=64) +to +aggregate +multi-scale +features, where we concatenate features from previous +layers to get a 64+64=128 dimension feature. After that, a +flatten operation and two fully-connected layers follow to +obtain the final classification results. Our overall network is +trained for 200 epochs with a learning rate of 1e-5. The +batchsize of training is 32 and Admax [29] is used for +optimization. Source code will be made available. +3. RESULTS AND DISCUSSION +Four metrics are adopted in our study to evaluate sex +classification performance: accuracy, precision, recall and +F1 score. For precision, recall and F1 score, the averaged +values of the two classes are calculated for evaluation. +3.1 Sex prediction performance +We compared the performance of our proposed method with +two methods: SVM +and 1D CNN [15]. In addition, an +ablation study was performed to investigate the performance +of our model without the attention module (TractGraph +CNN w/o Att. Mod.). The sex classification results of all +3 + +EdgeConv +1D Convtesting subjects from the ABCD and HCP datasets are +shown in Tables 1 and 2. +Table 1. Comparison of sex classification performance +across different methods in the ABCD dataset. +Methods +SVM +1-D +CNN +TractGraph CNN +w/o Att. Mod. +TractGraph +CNN +WMG +GMG +WMG +GMG +Acc +73.46 +82.77 +85.13 +84.59 +85.50 +84.80 +Precision 73.21 +82.83 +85.09 +84.58 +85.46 +84.77 +Recall +73.23 +82.88 +85.11 +84.52 +85.49 +84.79 +F1 +73.22 +82.77 +85.10 +84.55 +85.48 +84.78 +Table 2. Comparison of sex classification performance +across different methods in the HCP dataset. +Methods +SVM +1-D +CNN +TractGraph CNN +w/o Att. Mod. +TractGraph +CNN +WMG +GMG +WMG +GMG +Acc +90.673 93.229 93.750 94.271 94.792 93.229 +Precision 90.506 93.229 93.760 94.326 94.791 93.245 +Recall +91.414 93.234 93.760 94.254 94.791 93.259 +F1 +90.601 93.228 93.750 94.269 94.791 93.229 +Generally speaking, our TractGraphCNN model with +WMG shows the best performance across all compared +methods. This indicates the strong potential of anatomically +informed graphs to improve performance in deep learning +tasks related to the brain’s WM connections. Furthermore, +we note that in general the WMG outperformed the GMG. +This is likely because the neighborhoods constructed using +fiber +distances +were +able +to capture more localized +information. In comparison, many larger neighborhoods +were induced when considering FS parcels. This could be +seen in the neighborhood size, where in WMG each node +(cluster) had 20 edges, while the number of edges per node +in +GMG +ranged +from +3 +to +180. +Overall, +both +TractGraphCNN approaches had good performance, in +comparison with typical sex prediction accuracies across +different MRI modalities and datasets ranging from 80-90% +[10]. Note that a comparable recent study of sex prediction +from HCP structural connectivity data achieved 92.75% +accuracy [30]. Finally, despite the larger size of the ABCD +dataset, all methods had much higher accuracy in the HCP +dataset, likely related to the different neurodevelopmental +stages of the subjects in the two datasets [31]. +3.2 Interpretation of important tracts +Fig. 3 shows important tracts for the sex prediction task that +were consistently identified across both ABCD and HCP +experiments, for each graph type. We can observe that +widespread regions in the WM are predictive of the sex of +an individual. Three tracts (the bilateral cingulum and the +left +middle +longitudinal +fasciculus) +were +consistently +predictive of sex across both graph types (WMG and GMG) +and across both large datasets. Other interpretation results +varied across graph types, indicating that the different graph +structures helped the network focus on different informative +brain connections. This further suggests the potentially +complementary nature of the two investigated graphs, and +the potential for future investigations into simultaneously +leveraging multiple sources of anatomical information in +network construction. +Fig. 3. Interpretation results of important tracts common +across both ABCD and HCP datasets. +4. CONCLUSION +In this study, we proposed a novel anatomically informed +graph CNN framework, TractGraphCNN, for machine +learning using diffusion MRI tractography. The framework +incorporates EdgeConv modules to aggregate features from +white matter connections that are anatomically related, and +an attention module that enables the interpretation of white +matter tracts that are important for prediction. The results in +a +sex +prediction +testbed +task +demonstrated +strong +performance of TractGraphCNN in two large datasets. We +found that white-matter-centric graphs were most successful +overall. Across both datasets and both graph types, the +bilateral cingulum and left middle longitudinal fasciculus +were most predictive of the sex of an individual. Overall, +this work shows the potential of incorporating sources of +anatomical +information, +especially +known +anatomical +similarities between input features, to guide convolutions in +neural networks. +4 + +WMG +WMG +Left +Right +snouunbunoe +cnguumbunde +mkdd e longitudinal faec +corticospnal tacl +Tronta +halamo-frontal +Buperior lonattualinar +halamo- fronta. +thalamo-parietal +GMG +GMG +Left +Right +tinguium bunde +cngulum bundle +ndde long itudina faec +striato fronta +supernicia pareta-tempo +suoercaHt +Htempora5. COMPLIANCE WITH ETHICAL STANDARDS +This study uses public HCP imaging data and no ethical +approval was required. Approval was granted by the BWH +IRB for use of the public ABCD data. +6. ACKNOWLEDGMENTS +We acknowledge the following NIH grants: P41EB015902, +R01MH125860 and R01MH119222. +7. REFERENCES +[1] +S. J. Forkel, P. Friedrich, M. 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Bremer, “Modeling human brain connectomes using +structured +neural +networks,” +2019. +https://www.osti.gov/servlets/purl/1579625 +(accessed +Nov. +10, 2022). +[31] C. Lebel, M. Gee, R. Camicioli, M. Wieler, W. Martin, and C. +Beaulieu, “Diffusion tensor imaging of white matter tract +evolution over the lifespan,” Neuroimage, vol. 60, no. 1, pp. +340–352, Mar. 2012. +5 + diff --git a/1tAzT4oBgHgl3EQf8_4M/content/tmp_files/load_file.txt b/1tAzT4oBgHgl3EQf8_4M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65ce1d27b8326eae5bb838274959d661d0025571 --- /dev/null +++ b/1tAzT4oBgHgl3EQf8_4M/content/tmp_files/load_file.txt @@ -0,0 +1,537 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf,len=536 +page_content='TRACTGRAPHCNN: ANATOMICALLY INFORMED GRAPH CNN FOR CLASSIFICATION USING DIFFUSION MRI TRACTOGRAPHY Yuqian Chen1,2, Fan Zhang1, Leo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Zekelman1, Tengfei Xue1,2, Chaoyi Zhang2, Yang Song3, Nikos Makris1, Yogesh Rathi1, Weidong Cai2, Lauren J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' O’Donnell1 1 Harvard Medical School, MA, USA 2 The University of Sydney, NSW, Australia 3 The University of New South Wales, NSW, Australia ABSTRACT The structure and variability of the brain’s connections can be investigated via prediction of non-imaging phenotypes using neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' However, known neuroanatomical relationships between input features are generally ignored in network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We propose TractGraphCNN, a novel, anatomically informed graph CNN framework for machine learning tasks using diffusion MRI tractography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' An EdgeConv module aggregates features from anatomically similar white matter connections indicated by graph edges, and an attention module enables interpretation of predictive white matter tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Results in a sex prediction testbed task demonstrate strong performance of TractGraphCNN in two large datasets (HCP and ABCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Graphs informed by white matter geometry demonstrate higher performance than graphs informed by gray matter connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Overall, the bilateral cingulum and left middle longitudinal fasciculus are consistently highly predictive of sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' This work shows the potential of incorporating anatomical information, especially known anatomical similarities between input features, to guide convolutions in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Index Terms— Sex classification, white matter tracts, graph CNN, neuroanatomy, tractography 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' INTRODUCTION The human brain’s white matter (WM) fiber tract connections have important inter-individual variability, with implications for understanding neurodevelopment and disease [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Recently, brain variability is studied by predicting non-imaging phenotypes from high-dimensional neuroimaging data using machine learning [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Many aspects of such machine learning methods are active areas of research (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' multiple modalities [3], comparison of methodology [4], and interpretation [2], [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' However, we find relatively fewer studies of tailored network design that can leverage neuroanatomical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Here we investigate deep neural networks informed by the anatomy and geometry of the brain’s WM structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' A few studies have aimed to develop dedicated neural networks for analyses of the brain’s structural connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The BrainNETCNN [5] includes novel convolutional filters that improve performance [6] by handling the topology of connectivity matrices (where each row or column corresponds to a gray matter (GM) region or node, and entries or edges in the matrix indicate connectivity strengths between GM regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Other approaches apply graph convolutional neural networks to connectivity matrices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' However, the above classes of methods are restricted to the anatomical information contained in the row and column structure of the connectivity matrix, and they cannot leverage any additional anatomical information to inform network convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We hypothesize that the performance of deep learning can be enhanced by incorporating information about anatomical neighborhoods of WM connections with similar geometry and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' To encode neighborhood relationships, we adopt the popular EdgeConv neural network module originally designed for the Dynamic Graph CNN (DGCNN) [8], and we use it to construct static graphs informed by brain anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Following two major approaches to study the brain’s structural connectivity [9], we investigate 1) white-matter-centric graphs (WMG) that define neighborhoods according to fiber tract geometry, and 2) gray-matter-centric graphs (GMG) that define neighborhoods according to connected gray matter regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' To focus our project, we choose a testbed problem of sex prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' While this problem is not straightforward [10], [11], sex is known to be an important source of WM variability [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Many studies have investigated sex prediction [6], [13]–[16] using microstructure and/or connectivity features from quantitative diffusion MRI (dMRI) tractography [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Both microstructure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' fractional anisotropy, FA [6]) and connectivity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' number of streamlines, NoS [14]) features provide good prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' However, NoS is affected by intracranial volume, a common confound in sex prediction [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' In this work we therefore utilize FA and a percentage of streamlines that is normalized to reduce the effect of brain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We quantify these features using an anatomically curated, atlas-based WM fiber cluster parcellation that is 1 consistent across datasets, acquisitions, and the human lifespan [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Importantly for this study, a description of the WM geometry and GM connectivity of each fiber cluster is provided in the ORG atlas [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' In this study, we propose an anatomically-informed graph CNN framework, called TractGraphCNN, to leverage neuroanatomical knowledge for sex prediction based on cluster-wise WM features from dMRI tractography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The main contributions of this study are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' First, for the first time, we model the anatomical relationship between clusters as a graph, informed by WM geometry and GM connectivity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Second, we integrate EdgeConv modules into our framework to extract features from anatomically similar clusters to improve performance of sex prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Finally, our framework is able to identify important WM tracts for sex classification by leveraging an attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We evaluate our method on two large-scale datasets of children and healthy young adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' METHODS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 1 gives an overview of our proposed TractGraphCNN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' First, WM features are extracted from dMRI tractography data (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1), resulting in two features for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Second, we build a graph to model the relationship between clusters, indicated by WM geometry or GM connectivity information (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Third, the built graph is input to the proposed TractGraphCNN framework (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3) for sex classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The framework aggregates information from connecting clusters in the graph via EdgeConv modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' An attention module is adopted to enable interpretation of important tracts that are predictive for sex classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' (a) Overall pipeline of TractGraphCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' (b) Network structure of the attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1 dMRI datasets and feature extraction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1 Adolescent Brain Cognitive Development (ABCD) This study utilized dMRI data of 9342 young children (age 9-11) from the large-scale, multi-site ABCD dataset [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We harmonized the dMRI data across 21 acquisition sites to remove scanner-specific biases while preserving inter-subject biological variability [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Of all subjects, 4879 (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2%) are males and 4463 (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='8%) are females.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 7473 subjects (80%) are used for training the neural network, while 1869 (20%) are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2 Human Connectome Project (HCP) We also conducted experiments on a dataset of 964 subjects (age 22-37) from the Human Connectome Project, a large multimodal dataset composed of healthy young adults [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Of all subjects, 443 are male (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='6%) and 521 are female (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 772 subjects (80%) are used for training and 192 (20%) subjects are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3 White matter fiber cluster features Two-tensor unscented Kalman filter tractography (UKFt) [22] via SlicerDMRI [23], [24] was applied to obtain whole brain tractography from the dMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Tractography was parcellated with an anatomically curated cluster atlas and a machine learning approach that has been shown to consistently identify WM tracts across the human lifespan [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' For each subject, 953 expert-curated clusters categorized into 75 WM tracts were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Importantly, cluster IDs are assigned according to the atlas and correspond across subjects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' cluster #1 corresponds across all subjects and datasets studied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Statistical microstructure measurements were then computed for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We adopted two measurements for the task: fractional anisotropy (FA) and percentage of streamlines (PoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' FA of the cluster is computed as the mean FA across all streamline points within the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The PoS of a cluster is calculated as the number of streamlines of the cluster divided by the total number of streamlines across all clusters of the subject, to reduce the effect of brain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' For each subject, this resulted in an input feature matrix of size 2x953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' For absent clusters due to individual anatomical variation, we set features to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Finally, a min-max normalization was performed on the input feature matrix for FA and PoS individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2 Anatomically informed graph construction We propose to build graphs such that edges connect neighboring fiber clusters with similar anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Each cluster is represented as a node in the graph with cluster-wise WM features as node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1 Fiber tract geometry informed graph The first type of graph (WMG) proposed in our study is based on WM tractography fiber geometric similarity, a well-established concept in the field of fiber clustering [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Specifically, we first compute the geometric distance between each pair of fiber clusters in the ORG atlas, which is measured as the mean of the pairwise fiber distances (the popular mean closest point fiber distance is used [25]) between the two fiber clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' A low distance between two clusters represents a high similarity in terms of WM anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Then, for each cluster, we choose the top k (k =20 is used in our study following the default setting in DGCNN) clusters with the lowest geometric distances as neighbors, and edges are placed in between for graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2 N×64 N×64 N×64 N×64 N×2 EdgeConv EdgeConv EdgeConv C mlp [64] mlp [64] mlp [64] Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Conv1d mlp [32] Conv1d mlp (1] Conv1d mlp [32] 团2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2 Cortical and subcortical connectivity informed graph The second type of graph (GMG) proposed in our study is based on GM regions to which the fiber clusters connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Specifically, for each cluster, we first identify its connected Freesurfer GM regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The ORG atlas provides the percentage of streamlines from each cluster that intersect each Freesurfer region [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We leverage this information to identify the top two FreeSurfer regions most commonly intersected by the streamlines of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The neighborhood of a cluster is then defined as the set of clusters with at least one top Freesurfer region in common, and edges are placed in between for graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3 Network architecture The overall architecture of our TractGraphCNN framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' TractGraphCNN extends the 1D CNN model [15] for group classification using fiber cluster features with two innovative improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' First, we replace the 1D convolutional layers in the original model with EdgeConv layers [8] to utilize the information of anatomically neighboring clusters (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Second, we add a gated attention module [27] in the network that can assess the importance of each cluster to enable result interpretation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Graphic illustration of the usage of EdgeConv to leverage fiber cluster neighborhood information, with comparison to the standard 1D convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1 1D CNN with EdgeConv Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2 illustrates the use of EdgeConv in TractGraphCNN to aggregate information from neighboring graph nodes representing fiber clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' EdgeConv was proposed in the popular DGCNN method to capture the local geometric structure of point clouds [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The basic idea of EdgeConv is to use a learnable fully-connected layer to compute an edge feature of two neighboring nodes xi and xj based on their input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Then, the output of EdgeConv is calculated by aggregating the edge features with max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' This learning process enables dynamic update of graph structure by recomputing distances of points in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' In our application, because the anatomical relationships between fiber clusters do not change, we maintain a static graph structure across layers by using the same graph structure across all EdgeConv layers without recomputing distances between node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' In addition, after feature extraction, we do not use the max pooling operation as in traditional Graph CNNs, but instead we retain the flatten operation in the 1D CNN model [15] to preserve the information about cluster correspondence across subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2 Interpretation of important tracts For the purpose of interpretation, it is important to identify important WM connections for the task of sex classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' To achieve this, we improve our neural network by adding an attention mechanism using the popular gated attention module from [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The attention module (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 1(b)) is composed of two parallel fully-connected layers followed by a sigmoid and tanh activation functions, a concatenation operation, and another fully-connected layer followed by a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The output is a 1-D attention map of size 973 with values between 0 and 1, indicating the importance of the corresponding cluster to the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Next, we identify the most predictive anatomical WM tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We first compute the mean importance of each cluster across all testing subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Then we find the top T (T = 50 in our experiment) clusters with the highest mean importance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Finally, we identify all tracts to which the top 50 clusters belong, according to the ORG atlas [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='4 Implementation details All experiments are performed on a NVIDIA RTX A4000 GPU using Pytorch (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' For the overall architecture, we use two EdgeConv layers and one 1-D convolutional layer to extract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The two EdgeConv layers compute edge features with two fully-connected layers (64 , 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Shortcut connections are included to extract multi-scale features and one 1-D convolutional layer (kernel size=1, output channel=64) to aggregate multi-scale features, where we concatenate features from previous layers to get a 64+64=128 dimension feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' After that, a flatten operation and two fully-connected layers follow to obtain the final classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Our overall network is trained for 200 epochs with a learning rate of 1e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The batchsize of training is 32 and Admax [29] is used for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Source code will be made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' RESULTS AND DISCUSSION Four metrics are adopted in our study to evaluate sex classification performance: accuracy, precision, recall and F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' For precision, recall and F1 score, the averaged values of the two classes are calculated for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='1 Sex prediction performance We compared the performance of our proposed method with two methods: SVM and 1D CNN [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' In addition, an ablation study was performed to investigate the performance of our model without the attention module (TractGraph CNN w/o Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The sex classification results of all 3 EdgeConv 1D Convtesting subjects from the ABCD and HCP datasets are shown in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Comparison of sex classification performance across different methods in the ABCD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Methods SVM 1-D CNN TractGraph CNN w/o Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' TractGraph CNN WMG GMG WMG GMG Acc 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='46 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='77 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='13 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='59 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='50 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='80 Precision 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='21 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='83 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='09 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='58 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='46 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='77 Recall 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='23 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='88 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='11 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='52 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='49 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='79 F1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='22 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='77 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='10 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='55 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='48 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='78 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Comparison of sex classification performance across different methods in the HCP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Methods SVM 1-D CNN TractGraph CNN w/o Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' TractGraph CNN WMG GMG WMG GMG Acc 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='673 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='229 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='750 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='271 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='792 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='229 Precision 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='506 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='229 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='760 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='326 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='791 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='245 Recall 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='414 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='234 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='760 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='254 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='791 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='259 F1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='601 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='228 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='750 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='269 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='791 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='229 Generally speaking, our TractGraphCNN model with WMG shows the best performance across all compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' This indicates the strong potential of anatomically informed graphs to improve performance in deep learning tasks related to the brain’s WM connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Furthermore, we note that in general the WMG outperformed the GMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' This is likely because the neighborhoods constructed using fiber distances were able to capture more localized information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' In comparison, many larger neighborhoods were induced when considering FS parcels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' This could be seen in the neighborhood size, where in WMG each node (cluster) had 20 edges, while the number of edges per node in GMG ranged from 3 to 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Overall, both TractGraphCNN approaches had good performance, in comparison with typical sex prediction accuracies across different MRI modalities and datasets ranging from 80-90% [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Note that a comparable recent study of sex prediction from HCP structural connectivity data achieved 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='75% accuracy [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Finally, despite the larger size of the ABCD dataset, all methods had much higher accuracy in the HCP dataset, likely related to the different neurodevelopmental stages of the subjects in the two datasets [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content='2 Interpretation of important tracts Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 3 shows important tracts for the sex prediction task that were consistently identified across both ABCD and HCP experiments, for each graph type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We can observe that widespread regions in the WM are predictive of the sex of an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Three tracts (the bilateral cingulum and the left middle longitudinal fasciculus) were consistently predictive of sex across both graph types (WMG and GMG) and across both large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Other interpretation results varied across graph types, indicating that the different graph structures helped the network focus on different informative brain connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' This further suggests the potentially complementary nature of the two investigated graphs, and the potential for future investigations into simultaneously leveraging multiple sources of anatomical information in network construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Interpretation results of important tracts common across both ABCD and HCP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' CONCLUSION In this study, we proposed a novel anatomically informed graph CNN framework, TractGraphCNN, for machine learning using diffusion MRI tractography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The framework incorporates EdgeConv modules to aggregate features from white matter connections that are anatomically related, and an attention module that enables the interpretation of white matter tracts that are important for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' The results in a sex prediction testbed task demonstrated strong performance of TractGraphCNN in two large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' We found that white-matter-centric graphs were most successful overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Across both datasets and both graph types, the bilateral cingulum and left middle longitudinal fasciculus were most predictive of the sex of an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Overall, this work shows the potential of incorporating sources of anatomical information, especially known anatomical similarities between input features, to guide convolutions in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 4 WMG WMG Left Right snouunbunoe cnguumbunde mkdd e longitudinal faec corticospnal tacl Tronta halamo-frontal Buperior lonattualinar halamo- fronta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' thalamo-parietal GMG GMG Left Right tinguium bunde cngulum bundle ndde long itudina faec striato fronta supernicia pareta-tempo suoercaHt Htempora5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' COMPLIANCE WITH ETHICAL STANDARDS This study uses public HCP imaging data and no ethical approval was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' Approval was granted by the BWH IRB for use of the public ABCD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge the following NIH grants: P41EB015902, R01MH125860 and R01MH119222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} +page_content=' REFERENCES [1] S.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf'} diff --git a/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf b/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..047b3a07da4be64e875d0593f862afdb625098a9 --- /dev/null +++ b/29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4b441323350525b02d4927164bd145116d979a6e6323799c9322d57d3789a27a +size 10782147 diff --git a/69E5T4oBgHgl3EQfPw7T/content/tmp_files/2301.05508v1.pdf.txt b/69E5T4oBgHgl3EQfPw7T/content/tmp_files/2301.05508v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c744079414a1d57782abf523a832b3f2a80417e4 --- /dev/null +++ b/69E5T4oBgHgl3EQfPw7T/content/tmp_files/2301.05508v1.pdf.txt @@ -0,0 +1,938 @@ +arXiv:2301.05508v1 [cs.IR] 13 Jan 2023 +Do the Findings of Document and Passage +Retrieval Generalize to the Retrieval of +Responses for Dialogues? +Gustavo Penha and Claudia Hauff +TU Delft +{g.penha-1,c.hauff}@tudelft.nl +Abstract. A number of learned sparse and dense retrieval approaches +have recently been proposed and proven effective in tasks such as pas- +sage retrieval and document retrieval. In this paper we analyze with +a replicability study if the lessons learned generalize to the retrieval +of responses for dialogues, an important task for the increasingly pop- +ular field of conversational search. Unlike passage and document re- +trieval where documents are usually longer than queries, in response +ranking for dialogues the queries (dialogue contexts) are often longer +than the documents (responses). Additionally, dialogues have a particu- +lar structure, i.e. multiple utterances by different users. With these dif- +ferences in mind, we here evaluate how generalizable the following major +findings from previous works are: (F1) query expansion outperforms +a no-expansion baseline; (F2) document expansion outperforms a no- +expansion baseline; (F3) zero-shot dense retrieval underperforms sparse +baselines; (F4) dense retrieval outperforms sparse baselines; (F5) hard +negative sampling is better than random sampling for training dense +models. Our experiments1—based on three different information-seeking +dialogue datasets—reveal that four out of five findings (F2–F5) gener- +alize to our domain. +1 +Introduction +Conversational search is concerned with creating agents that satisfy an informa- +tion need by means of a mixed-initiative conversation through natural language +interaction, rather than through the traditional search engine results page. A +popular approach to conversational search is retrieval-based [3]: given an on- +going conversation and a large corpus of historic conversations, retrieve the +response that is best suited from the corpus [45,48,28,47,11]. Due to the ef- +fectiveness of heavily pre-trained transformer-based language models such as +BERT [4], they have become the predominant approach for conversation re- +sponse re-ranking [28,43,53,8,42]. +The most common evaluation procedure for conversation response re-ranking +consists of re-ranking a limited set of n candidate responses (including the +1 https://github.com/Guzpenha/transformer_rankers/tree/full_rank_retrieval_dialogues. + +2 +Gustavo Penha and Claudia Hauff +ground-truth response(s)), followed by measuring the number of relevant re- +sponses found in the first K positions—Recalln@K [52]. Since the entire collec- +tion of available responses is typically way bigger2 than such a set of candidates, +this setup is in fact a selection problem, where we have to choose the correct +response out of a few options. This evaluation overlooks the first-stage retrieval +step, which retrieves a set of n responses to be re-ranked. If the first-stage model, +e.g. BM25, fails to retrieve relevant responses, the entire pipeline fails. +Motivated by a lack of research on the first-stage retrieval step, we are inter- +ested in answering in our replicability study whether the considerable knowledge +obtained on document and passage retrieval tasks generalizes to the dialogue do- +main. Unlike document and passage retrieval where the documents are generally +longer than the queries, in response retrieval for dialogues the queries (dialogue +contexts) tend to be longer than the documents (responses). A second important +difference is the structure induced by the dialogue as seen in Table 1. +Table 1. Comparison between passage retrieval and response retrieval for dialogues. +In §3 we define the task of First-stage Retrieval for Dialogues. Colors symbolize the +information-seeker and information-provider. p+/r+ are the relevant passage/response. +Passage Retrieval +First-stage Retrieval for Dialogues +Input +Query q +Dialogue context U = {u1, u2, ..., uτ} +Example +q: what is theraderm used +for +u1: I want a firewall that will protect +me but more of that to monitor any +connection in or out of my mac [...] +u2: {url} allows you to map joypad +buttons to keyboard keys and [...] +u3: Do the diagonals for the analog +stick work correctly for you? [...] +Output +Ranked list of passages +Ranked list of responses +Example +p+: Thera-Derm Lotion +is used as a moisturizer +to treat or prevent dry, +rough, scaly, [...] +r+: In the ”Others” tab, try [...] +Given the differences between the domains, we verify empirically across three +information-seeking datasets and 1.7M queries, the generalizability of five find- +ings (F1 to F5) from the passage and document retrieval literature related to +state-of-the-art sparse and dense retrieval models. We are motivated in our se- +lection of these five findings by their impact in prior works (cf. §2). Our results +show that four out of five previous findings do indeed generalize to our domain: +2 While +for +most +benchmarks +[52] +we +have +only +10–100 +candi- +dates, +a +working +system +with +the +Reddit +data +from +PolyAI +https://github.com/PolyAI-LDN/conversational-datasets +would +need +to +retrieve from 3.7 billion responses. + +From Document and Passage Retrieval to Response Retrieval for Dialogues +3 +F1 ✗3 Dialogue context (i.e. query) expansion outperforms a no-expansion base- +line [1,18,49,21]. +F2 ✓ Response (i.e. document) expansion outperforms a no-expansion baseline +[25,21,19] if the expansion model is trained to generate the most recent context +(last utterance4 of the dialogue) instead of older context (all utterances). +F3 ✓ Dense retrieval in the zero-shot5 setting underperforms sparse baselines +[34,41] except when it goes through intermediate training on large amounts +of out-of-domain data. +F4 ✓ Dense retrieval with access to target data6 outperforms sparse baselines +[7,15,34] if an intermediate training step on out-of-domain data is performed +before the fine-tuning on target data. +F5 ✓ Harder negative sampling techniques lead to effectiveness gains [46,51] if +a denoising technique is used to reduce the number of false negative samples. +Our results indicate that most findings translate to the domain of retrieval +of responses for dialogues. A promising future direction is thus to start with +successful models from other domains—for which there are more datasets and +previous research—and study how to adapt and improve them for retrieval-based +conversational search. +2 +Related Work +In this section we first discuss current research in retrieval-based systems for +conversational search, followed by reviewing the major findings of (un)supervised +sparse and dense retrieval in the domains of passage and document retrieval. +2.1 +Ranking and Retrieval of Responses for Dialogues +Early neural models for response re-ranking were based on matching the repre- +sentations of the concatenated dialogue context and the representation of a re- +sponse in a single-turn manner with architectures such as CNN and LSTM [23,14]. +More complex neural architectures matching each utterance with the response +were also explored [54,9,22]. Heavily pre-trained language models such as BERT +were first shown to be effective by Nogueira and Cho [24] for re-ranking. Such +models quickly became a predominant approach for re-ranking in IR [21] and +were later shown to be effective for re-ranking responses in conversations [42,28]. +In contrast, the first-stage retrieval of responses for a dialogue received rela- +tively little attention [29]. Lan et al. [17] and Tao et al. [38] showed that BERT- +based dense retrieval models outperform BM25 for first-stage retrieval of re- +sponses for dialogues. A limitation of their work is that strong sparse retrieval +3 ✗ indicates that the finding does not hold in our domain whereas ✓ indicates that +it holds in our domain followed by the necessary condition or exception. +4 For example in Table 1 the last utterance is u3. +5 A zero-shot is a model that does not have access to target data, cf. Table 2. +6 Target data is data from the same distribution, i.e. dataset, of the evaluation dataset. + +4 +Gustavo Penha and Claudia Hauff +baselines that have shown to be effective in other retrieval tasks, e.g. BM25 with +dialogue context expansion [25] or BM25 with response expansion [49], were not +employed for dense retrieval. We do such comparisons here and test a total of five +major findings that have been not been evaluated before by previous literature +on the first-stage retrieval of responses for dialogues. +2.2 +Dense and Sparse Models for Passage and Document Retrieval +Context for F1 Retrieval models can be categorized into two dimensions: su- +pervised vs. unsupervised and dense vs. sparse representations [19]. An unsuper- +vised sparse representation model such as BM25 [35] represents each document +and query with a sparse vector with the dimension of the collection’s vocabu- +lary, having many zero weights due to non-occurring terms. Since the weights of +each term are entirely based on term statistics they are considered unsupervised +methods. Such approaches are prone to the vocabulary mismatch problem [6], +as semantic matches are not considered. A way to address such a problem is by +using query expansion methods. RM3 [1] is a competitive [49] query expansion +technique that uses pseudo-relevance feedback to add new terms to the queries +followed by another final retrieval step using the modified query. +Context for F2 A supervised sparse retrieval model can take advantage of +the effectiveness of transformer-based language models by changing the terms’ +weights from collection statistics to something that is learned. Document ex- +pansion with a learned model can be considered a learned sparse retrieval ap- +proach [19]. The core idea is to create pseudo documents that have expanded +terms and use them instead when doing retrieval. Doc2query [25] is a strong su- +pervised sparse retrieval baseline that uses a language model to predict queries +that might be issued to find a document. The predictions of this model are used +to create the augmented pseudo documents. +Context for F3 and F4 Supervised dense retrieval models7, such as ANCE [46] +and coCodenser [7], represent query and documents in a small fixed-length space, +for example of 768 dimensions. Dense retrieval models without access to target +data for training—known as the zero-shot scenario—have underperformed sparse +methods (F3). For example, the BEIR benchmark [41] showed that BM25 was +superior to dense retrieval from 9–18 (depending on the model) out of the 18 +datasets in the zero-shot scenario. In contrast, when having access to enough +supervision from target data, dense retrieval models have shown to consistently +outperform strong sparse baselines [7,15,34] (F4). +7 A distinction can also be made of cross-encoders and bi-encoders, where the first +encode the query and document jointly as opposed to separately [40]. Cross-encoders +are applied in a re-ranking step due to their inefficiency and thus are not our focus. + +From Document and Passage Retrieval to Response Retrieval for Dialogues +5 +Context for F5 In order to train neural ranking models, a small set of negative +(i.e. non-relevant) candidates are necessary as it is prohibitively expensive to +use every other document in the collection as negative sample for a query. A +limitation of randomly selecting negative samples is that they might be too easy +for the ranking model to discriminate from relevant ones, while for negative +documents that are harder the model might still struggle. For this reason hard +negative sampling has been shown to perform better than random sampling for +passage and document retrieval [46,36,51]. +3 +First-stage Retrieval for Dialogues +In this section we first describe the problem of first-stage retrieval of responses, +followed by the findings we want to replicate from sparse and dense approaches. +Problem Definition The task of first-stage retrieval of responses for dialogues, +concerns retrieving the best response out of the entire collection given the di- +alogue context. Formally, let D = {(Ui, Ri, Yi)}M +i=1 be a data set consisting of +M triplets: dialogue context, response candidates and response relevance labels. +The dialogue context Ui is composed of the previous utterances {u1, u2, ..., uτ} +at the turn τ of the dialogue. The candidate responses Ri = {r1, r2, ..., rn} are +either ground-truth responses r+ or negative sampled candidates r−, indicated +by the relevance labels Yi = {y1, y2, ..., yn}. In previous work, the number of +candidates is limited, typically n = 10 [29]. The findings we replicate here come +from passage and document retrieval tasks where there is no limit to the number +of documents or passages that have to be retrieved. Thus, in all of our first-stage +retrieval task experiments n is set to the size of the entire collection of responses +in the corpus. The number of ground-truth responses is one, the observed re- +sponse in the conversational data. The task is then to learn a ranking function +f(.) that is able to generate a ranked list from the entire corpus of responses Ri +based on their predicted relevance scores f(U, r). +F1: Unsupervised Sparse Retrieval We rely on classic retrieval methods, +for which the most commonly used baseline is BM25. One of the limitations of +sparse retrieval is the vocabulary mismatch problem. Expansion techniques are +able to overcome this problem by appending new words to the dialogue contexts +and responses. For this reason, we here translate a query expansion technique +to the dialogue domain and perform dialogue context expansion with RM3 [1], +a competitive unsupervised method that assumes that the top-ranked responses +by the sparse retrieval model are relevant. From these pseudo-relevant responses, +words are selected and an expanded dialogue context is created and subsequently +employed by the sparse retrieval method to rank the final list of responses. The +effectiveness of RM3 in the domain of dialogues is the first finding +that we validate. + +6 +Gustavo Penha and Claudia Hauff +F2: Learned Sparse Retrieval Alternatively, we can expand the responses in +the collection with a learned method. To do so we “translate” doc2query [25] into +our domain, yielding resp2ctxt. Formally, we fine-tune a generative transformer +model G for the task of generating the dialogue context Ui from the ground-truth +response r+ +i . This model is then used to generate expansions for all responses +in the collection, ri = concat(ri, G(ri)). These expansions are appended to the +responses and the collection is indexed again—the sparse retrieval method it- +self is not modified, i.e. we continue using BM25. This approach (which we coin +resp2ctxt) leads to two improvements: term re-weighting (adding terms that +already exist in the document) and dealing with the vocabulary mismatch prob- +lem (adding new terms). The effectiveness of doc2query in the domain of +dialogues is the second finding that we validate. +Unlike passage and document retrieval where the queries are smaller than the +documents, for the retrieval of responses for dialogues the queries are longer than +the documents8. This is a challenge for the generative model, since generating +larger pieces of text is a more difficult problem than smaller ones as there is +more room for errors. Motivated by this, we also explored a modified version of +resp2ctxt that aims to generate only the last utterance of the dialogue context: +resp2ctxtlu. This model is trained to generate uτ from r+ +i , instead of trying to +generate the whole utterance Ui = {u1, u2, ..., uτ}. The underlying premise is +that the most important utterance from the dialogue is the last one, and if it is +correctly generated by resp2ctxtlu, the sparse retrieval method will be able to +find the correct response from the collection. +F3: Zero-shot Dense Retrieval We rely on methods that learn to represent +the dialogue context and the responses separately in a dense embedding space. +Responses are then ranked by their similarity to the dialogue context. We rely +here on pre-trained language transformer models, such as BERT [4] and MP- +Net [37], to obtain such representations of the dialogue context and response. +This approach is generally referred to as a bi-encoder model [21] and is an ef- +fective family of models9. A zero-shot model is one that is not trained on the +target data. Target data is data from the same distribution, i.e. dataset, of the +evaluation dataset. +One way of improving the representations of a heavily pre-trained language +model for the zero-shot setting is to fine-tune it with intermediate data [33]. Such +intermediate data contains triplets of query, relevant document, and negative +document and can include multiple datasets. The advantage of adding this step +before employing the representations of the language model is to reduce the +8 For example, while the TREC-DL-2020 passage and document retrieval tasks the +queries have between 5–6 terms on average and the passages and documents have +over 50 and 1000 terms respectively, for the information-seeking dialogue datasets +used here the dialogue contexts (queries) have between 70 and 474 terms on average +depending on the dataset while the responses (documents) have between 11 and 71. +9 See for example the top models in terms of effectiveness from the MSMarco bench- +mark leaderboards https://microsoft.github.io/msmarco/. + +From Document and Passage Retrieval to Response Retrieval for Dialogues +7 +gap between the pre-training and the downstream task at hand [31,26,30]. In +Table 2 we clarify the relationship between pre-training, intermediate training +and fine-tuning. +Table 2. The different training stages and data, their purposes, examples of datasets, +and the type of dense model obtained after each stage. +Pre-training data +Intermediate data +Target data +Purpose Learn general represen- +tations +Learn sentence representations +for ranking +Learn representations for tar- +get distribution +Model is Zero-shot +Zero-shot +Fine-tuned +Example Wikipedia +MSMarco +MANtIS +The intermediate training step learns to represent pieces of text (query and +documents) by applying a mean pooling function over the transformer’s final +layer, which is then used to calculate the dot-product similarity. The loss function +employs multiple negative texts from the same batch to learn the representations +in a constrastive manner, also known as in-batch negative sampling. Such a +procedure learns better text representations than a naive approach that uses the +[CLS] token representation of BERT [33,2]. +The function f(U, r) is then dot(η(concat(U)), η(r)), where η is the repre- +sentation obtained by applying the mean pooling function over the last layer of +the transformer model, and concat(U) = u1 | [U] | u2 | [T ] | ... | uτ , where | +indicates the concatenation operation. The utterances from the context U are +concatenated with special separator tokens [U] and [T ] indicating end of utter- +ances and turns10. The effectiveness of a zero-shot bi-encoder model in +the domain of dialogues is the third finding we validate. +F4: Fine-tuned Dense Retrieval The standard procedure is to fine-tune +dense models with target data that comes from the same dataset that the model +will be evaluated. Since we do not have labeled negative responses, all the remain- +ing responses in the dataset can be thought of as non-relevant to the dialogue +context. Computing the probability of the correct response over all other re- +sponses in the dataset would give us P(r | U) = +P (U,r) +� +k P (U,rk). This computation is +prohibitively expensive, and the standard procedure is to approximate it using +a few negative samples. The negative sampling task is then as follows: given the +dialogue context U find challenging responses r− that are non-relevant for U. +Negative sampling can be seen as a retrieval task, where one can use a model to +retrieve negatives by applying a retrieval function to the collection of responses +using U as the query. +With such a dataset at hand, we continue the training—after the intermediate +step—in the same manner as done by the intermediate training step, with the +following cross-entropy loss function11 for a batch with size B: +10 The special tokens [U] and [T ] will not have any meaningful representation in the +zero-shot setting, but they can be learned on the fine-tuning step. +11 We refer to this loss as MultipleNegativesRankingLoss. + +8 +Gustavo Penha and Claudia Hauff +J (U, r, θ) = − 1 +B +�B +i=1 +� +f (Ui, ri) − log �B +j=1,j!=i ef(Ui,rj)� +, +where f(U, r) is the dot-product of the mean pooling of the last layer of the +transformer model. The effectiveness of a fine-tuned bi-encoder model +in the domain of dialogues is the fourth finding we validate here. +F5: Hard Negative Sampling A limitation of random samples is that they +might be too easy for the ranking model to discriminate from relevant ones, +while for negative documents that are hard the model might still struggle. For +this reason, another popular approach is to use a ranking model to retrieve +negative documents using the given query with a classic retrieval technique such +as BM25. This leads to finding negative documents that are closer to the query +in the sparse representation space, and thus they are harder negatives. Since +dense retrieval models have been outperforming sparse retrieval in a number of +cases with available training data, more complex negative sampling techniques +making use of dense retrieval have also been proposed [46,12]. The effectiveness +of hard negative sampling for a bi-encoder model in the domain of +dialogues is the fifth finding we validate here. +4 +Experimental Setup +In order to compare the different sparse and dense approaches we consider three +large-scale information-seeking conversation datasets12: MSDialog [32] contains +246K context-response pairs, built from 35.5K information seeking conversa- +tions from the Microsoft Answer community, a QA forum for several Microsoft +products; MANtIS [27] contains 1.3 million context-response pairs built from con- +versations of 14 Stack Exchange sites, such as askubuntu and travel; UDCDSTC8 [16] +contains 184k context-response pairs of disentangled Ubuntu IRC dialogues. +Implementation Details For BM25 and BM25+RM313 we rely on the pyserini +implementations [20]. In order to train resp2ctxt expansion methods we rely on +the Huggingface transformers library [44], using the t5-base model. We fine- +tune the T5 model for 2 epochs, with a learning rate of 2e-5, weight decay of 0.01, +and batch size of 5. When augmenting the responses with resp2ctxt we follow +docT5query [25] and append three different context predictions, using sampling +and keeping the top-10 highest probability vocabulary tokens. +For the zero-shot dense models, we rely on the SentenceTransformers [33] +model releases. The library uses Hugginface’s transformers for the pre-trained +models such as BERT [4] and MPNet [37]. For the bi-encoder models, we use the +12 MSDialog +is +available +at +https://ciir.cs.umass.edu/downloads/msdialog/; +MANtIS is available at https://guzpenha.github.io/MANtIS/; UDCDSTC8 is available +at https://github.com/dstc8-track2/NOESIS-II. +13 We perform hyperparameter tuning using grid search on the number of expansion +terms, number of expansion documents, and weight. + +From Document and Passage Retrieval to Response Retrieval for Dialogues +9 +pre-trained all-mpnet-base-v2 weights which were the most effective in our initial +experiments, compared with other pre-trained models14. When fine-tuning the +dense retrieval models, we rely on the MultipleNegativesRankingLoss, which ac- +cepts a number of hard negatives, and also uses the remaining in-batch random +negatives to train the model. We use a total of 10 negative samples for dialogue +context. We fine-tune the dense models for a total of 10k steps, and every 100 +steps we evaluate the models on a re-ranking task that selects the relevant re- +sponse out of 10 responses. We use the re-ranking validation MAP to select the +best model from the whole training to use in evaluation. We use a batch size of +5, with 10% of the training steps as warmup steps. The learning rate is 2e-5 and +the weight decay is 0.01. We use FAISS [13] to perform the similarity search. +Evaluation To evaluate the effectiveness of the retrieval systems we use R@K. +We thus evaluate the models’ capacity of finding the correct response out of +the whole possible set of responses15. We perform Students t-tests at the 0.95 +confidence level with Bonferroni correction to compare statistical significance of +methods. Comparisons are performed across the results for each dialogue context. +5 +Results +In this section, we discuss our empirical results along with the five major find- +ings from previous work (Section 1) in turn. Table 3 contains the main results +regarding F1 to F4. Table 5 contains the results for F5. +F1 ✗ Query expansion via RM3 leads to improvements over not using +query expansion [1,18,49,21]. BM25+RM3 (row 1b) does not improve over +BM25 (1a) on any of the three conversational datasets analyzed. We performed +thorough hyperparameter fine-tuning and no combination of the RM3 hyperpa- +rameters outperformed BM25. This indicates that F1 does not hold for +the task of response retrieval for dialogues. +A manual analysis of the new terms appended to a sample of 60 dialogue +contexts by one of the paper’s authors revealed that only 18% of them have at +least one relevant term added based on our best judgment. Unlike web search +where the query is often incomplete, under-specified, and ambiguous, in the +information-seeking datasets employed here the dialogue context (query) is quite +detailed and has more terms than the responses (documents). We hypothesize +that because the dialogue contexts are already quite descriptive, the task of +expansion is trickier in this domain and thus we observe many dialogues for +which the added terms are noisy. +14 The alternative models we considered are those listed in the model overview section +at https://www.sbert.net/docs/pretrained_models.html. +15 The standard evaluation metric in conversation response ranking [50,8,39] is recall +at position K with n candidates Rn@K. Since we are focused on the first-stage +retrieval we set n to be the entire collection of answers + +10 +Gustavo Penha and Claudia Hauff +Table 3. Results for the generalizability of F1–F4. Bold values indicate the high- +est recall for each type of approach. Superscripts indicate statistically significant im- +provements using Students t-test with Bonferroni correction. †=other methods from the +same group1=best from unsupervised sparse retrieval ; 2=best from supervised sparse +retrieval; 3=best from zero-shot dense retrieval. For example, in F3 † indicates that +row (3d) improves over rows (3a–c), 1 indicates that it improves over row (1a) and 2 +indicates it improves over row (2b). +MANtIS +MSDialog +UDCDSTC8 +R@1 +R@10 +R@1 +R@10 +R@1 +R@10 +(0) +Random +0.000 +0.000 +0.000 +0.001 +0.000 +0.001 +Unsupervised sparse +F1 +(1a) BM25 +0.133† 0.299† +0.064† +0.177† +0.027† 0.070† +(1b) BM25 + RM3 +0.073 +0.206 +0.035 +0.127 +0.011 +0.049 +Supervised sparse +F2 +(2a) BM25 + resp2ctxt +0.135 +0.309 +0.074 +0.208 +0.028 +0.067 +(2b) BM25 + resp2ctxtlu +0.147†1 0.325†1 0.0751 +0.2021 +0.029 +0.076 +Zero-shot dense (ModelIntermediateData) +F3 +(3a) ANCE600K−MSMarco 0.048 +0.111 +0.050 +0.124 +0.010 +0.028 +(3b) TAS-B400K−MSMarco 0.062 +0.143 +0.060 +0.157 +0.019 +0.050 +(3c) Bi-encoder215M−mul +0.138 +0.297 +0.108 +0.277 +0.023 +0.076 +(3d) Bi-encoder1.17B−mul +0.155†1 0.341†12 0.147†12 0.339†12 0.041† 0.097†12 +Fine-tuned dense (ModelNegativeSampler) +F4 +(4a) Bi-encoderRandom(0) +0.130 +0.307 +0.168123 0.387123 0.05012 0.128123 +F2 ✓ Document expansion via resp2ctxt leads to improvements over +no expansion [25,21,19]. We find that a naive approach to response expansion +improves marginally in two of the three datasets with BM25+resp2ctxt (2a) +outperforming BM25 (1a). However, the proposed modification of predicting only +the last utterance of the dialogue (resp2ctxtlu) performs better than predicting +the whole utterance, as shown by BM25+resp2ctxtlu’s (2b) higher recall values. +In the MANtIS dataset the R@10 goes from 0.309 when using the model trained to +predict the dialogue context to 0.325 when using the one trained to predict only +the last utterance of the dialogue context. We thus find that F2 generalizes +to response retrieval for dialogues, especially when predicting only +the last utterance of the context16. +In order to understand what the response expansion methods are doing +most—term re-weighting or adding novel terms—we present the percentage of +novel terms added by both methods in Table 4. The table shows that resp2ctxtlu +does more term re-weighting than adding new words when compared to resp2ctxt +16 As future work, more sophisticated techniques can be used to determine which parts +of the dialogue context should be predicted. + +From Document and Passage Retrieval to Response Retrieval for Dialogues +11 +Table 4. Statistics of the augmentations for resp2ctxt and resp2ctxtlu. New words are +the ones that did not exist in the document before. +MANtIS MSDialog UDCDSTC8 +Context avg length +474.12 +426.08 +76.95 +Response avg length +42.58 +71.38 +11.06 +Aug. avg length - resp2ctxt +494.23 +596.99 +202.3 +Aug. avg length - resp2ctxtlu +138.5 +135.29 +72.57 +% new words - resp2ctxt +71% +69% +71% +% new words - resp2ctxtlu +59% +37% +63% +(53% and 70% on average are new words respectively and thus 47% vs 30% are +changing the weights by adding existing words), generating overall smaller aug- +mentations (115.45 vs 431.17 on average respectively). +F3 ✓ Sparse retrieval outperforms zero-shot dense retrieval [34,41]. +Sparse retrieval models are more effective than the majority of zero-shot dense +models, as shown by the comparison of rows (1a–b), and (2a–b) with rows (3a– +c). However, a dense retrieval model that has gone through intermediate training +on large and diverse datasets including dialogues is more effective than a strong +sparse retrieval model, as we see by comparing row (3d) with row (2b) in Table 3. +For example, while the zero-shot dense retrieval models based only on the +MSMarco dataset (3a–b) perform on average 35% worse than the strong sparse +baseline (2b) in terms of R@10 for the MSDialog dataset, the zero-shot model +trained with 1.17B instances on diverse data (3d) is 68% better than the sparse +baseline (2b). When using a bigger amount of intermediate training data17, we +see that the zero-shot dense retrieval model (3d) is able to outperform the sparse +retrieval baseline by margins of 33% of R@10 on average across datasets. +We thus show that F3 only generalizes to response retrieval for +dialogues if we do not employ a large set of diverse intermediate data. +As expected, the closer the intermediate training data distribution is to the eval- +uation data, the better the dense retrieval model performs. The results indicate +that a good zero-shot retrieval model needs to go through intermediate training +on a large set of training data coming from multiple datasets to generalize well +to different domains and outperform strong sparse retrieval baselines. +F4 ✓ Dense models with access to target training data outperform +sparse models [7,15,34]. First, we see that fine-tuning the dense retrieval +model, which has gone through intermediate training already, with random +sampling—row (4a) in Table 3—achieves the best overall effectiveness in two +of the three datasets. This result shows that F4 generalizes to the task of +17 For +the +full +description +of +the +intermediate +data +see +https://huggingface.co/sentence-transformers/all-mpnet-base-v2. + +12 +Gustavo Penha and Claudia Hauff +response retrieval for dialogues when employing intermediate train- +ing18. Having access to the target data as opposed to only the intermediate +training data means that the representations learned by the model are closer to +the true distribution of the data. +We hypothesize that fine-tuning the bi-encoder for MANtIS (4a) is harmful +because the intermediate data contains Stack Exchange responses. In this way, +the set of dialogues of Stack Exchange that MANtIS encompasses might be serving +only to overfit the intermediate representations. As evidence for this hypothe- +sis, we found that (I) the learning curves flatten quickly (as opposed to other +datasets) and (II) fine-tuning another language model that does not have Stack +Exchange data (MSMarco) in their fine-tuning, bi-encoderbert−base (3c), improves +the effectiveness with statistical significance from 0.092 R@10 to 0.205 R@10. +F5 ✓ Hard negative sampling is better than random sampling for +training dense retrieval models [46,51]. Surprisingly we find that naively +using more effective models to select negative candidates is detrimental to the ef- +fectiveness of the dense retrieval model (see Hard negative sampling in Table 5). +We observe this phenomenon when using different language models, when switch- +ing intermediate training on or off for all datasets, and when using an alternative +contrastive loss [10] that does not employ in-batch negative sampling19. +After testing for a number of hypotheses that might explain why harder +negatives do not improve the effectiveness of the dense retrieval model, we found +that false negative samples increase significantly when using better negative +sampling methods. False negatives are responses that are potentially valid for +the context. Such relevant responses lead to unlearning relevant matches between +context and responses as they receive negative labels. See below an example of +a false negative sample retrieved by the bi-encoder model (row 3d of Table 3): +Dialogue context (U): hey... how long until dapper comes out? [U] 14 days [...] [U] i +thought it was coming out tonight +Correct response (r+): just kidding couple hours +False negative sample (r−): there is a possibility dapper will be delayed [...] mean- +while, dapper discussions should occur in ubuntu+1 +Denoising techniques try to solve this problem by reducing the number of false +negatives. We employ a simple approach that instead of using the top-ranked +responses as negative responses, we use the bottom responses of the top-ranked +responses as negatives20. This decreases the chances of obtaining false positives +and if k << |D| we will not obtain random samples. Our experiments in Table 5 +reveal that this denoising technique, row (3b), increases the effectiveness for +harder negative samples, beating all models from Table 3 for two of the three +18 Our experiments show that when we do not employ the intermediate training step the +fine-tuned dense model does not generalize well, with row (3d) performance dropping +to 0.172, 0.308 and 0.063 R@10 for MANtIS, MSDialog and UDCDSTC8 respectively. +19 The results are not shown here due to space limitations +20 For example, if we retrieve k = 100 responses, instead of using responses from top +positions 1–10, we use responses 91–100 from the bottom of the list. + +From Document and Passage Retrieval to Response Retrieval for Dialogues +13 +datasets. The results indicate that F5 generalizes to the task of response +retrieval for dialogues only when employing a denoising technique. +Table 5. Results for the generalizability of F5—with and without a denoising strategy +for hard negative sampling. Superscripts indicate statistically significant improvements +using Students t-test with Bonferroni correction . †=significance against the random +sampling baseline, ‡=significance against hard negative sampling without denoising. +MANtIS MSDialog UDCDSTC8 +R@10 +R@10 +R@10 +Baseline +(1) Bi-encoderRandom +0.307 +0.387 +0.128 +Hard negative sampling +(2a) Bi-encoderBM25 +0.271 +0.316 +0.087 +(2b) Bi-encoderBi−encoder +0.146 +0.306 +0.051 +Denoised hard negative sampling +(3a) Bi-encoderBM25 +0.257 +0.358‡ +0.121‡ +(3b) Bi-encoderBi−encoder +0.316†‡ 0.397†‡ +0.107‡ +6 +Conclusion +In this work, we tested if the knowledge obtained in dense and sparse retrieval +from experiments on the tasks of passage and document retrieval generalizes to +the first-stage retrieval of responses for dialogues. Our replicability study reveals +that while most findings do generalize to our domain, a simple translation of +the models is not always successful. A careful analysis of the domain in question +might reveal better ways to adapt techniques. +As future work, we believe an important direction is to evaluate learned +sparse methods that do weighting and expansion for both the queries and doc- +uments [5]—while resp2ctxt is able to both change the weights of the terms in +the response (by repeating existing terms) and expand terms (by adding novel +terms), it is not able to do weighting and expansion for the dialogue contexts. +Acknowledgements This research has been supported by NWO projects SearchX +(639.022.722) and NWO Aspasia (015.013.027). +References +1. Abdul-Jaleel, N., Allan, J., Croft, W.B., Diaz, F., Larkey, L., Li, X., Smucker, M.D., +Wade, C.: Umass at trec 2004: Novelty and hard. Computer Science Department +Faculty Publication Series p. 189 (2004) + +14 +Gustavo Penha and Claudia Hauff +2. 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In: +Proceedings of the 56th Annual Meeting of the Association for Computational +Linguistics (Volume 1: Long Papers). pp. 1118–1127 (2018) + diff --git a/69E5T4oBgHgl3EQfPw7T/content/tmp_files/load_file.txt b/69E5T4oBgHgl3EQfPw7T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2c960f136d0fcb59f8e1c516c8c82656e14a0de --- /dev/null +++ b/69E5T4oBgHgl3EQfPw7T/content/tmp_files/load_file.txt @@ -0,0 +1,918 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf,len=917 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='05508v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='IR] 13 Jan 2023 Do the Findings of Document and Passage Retrieval Generalize to the Retrieval of Responses for Dialogues?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Gustavo Penha and Claudia Hauff TU Delft {g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='penha-1,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='hauff}@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='nl Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A number of learned sparse and dense retrieval approaches have recently been proposed and proven effective in tasks such as pas- sage retrieval and document retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In this paper we analyze with a replicability study if the lessons learned generalize to the retrieval of responses for dialogues, an important task for the increasingly pop- ular field of conversational search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Unlike passage and document re- trieval where documents are usually longer than queries, in response ranking for dialogues the queries (dialogue contexts) are often longer than the documents (responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Additionally, dialogues have a particu- lar structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' multiple utterances by different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' With these dif- ferences in mind, we here evaluate how generalizable the following major findings from previous works are: (F1) query expansion outperforms a no-expansion baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' (F2) document expansion outperforms a no- expansion baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' (F3) zero-shot dense retrieval underperforms sparse baselines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' (F4) dense retrieval outperforms sparse baselines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' (F5) hard negative sampling is better than random sampling for training dense models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Our experiments1—based on three different information-seeking dialogue datasets—reveal that four out of five findings (F2–F5) gener- alize to our domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 1 Introduction Conversational search is concerned with creating agents that satisfy an informa- tion need by means of a mixed-initiative conversation through natural language interaction, rather than through the traditional search engine results page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A popular approach to conversational search is retrieval-based [3]: given an on- going conversation and a large corpus of historic conversations, retrieve the response that is best suited from the corpus [45,48,28,47,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Due to the ef- fectiveness of heavily pre-trained transformer-based language models such as BERT [4], they have become the predominant approach for conversation re- sponse re-ranking [28,43,53,8,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The most common evaluation procedure for conversation response re-ranking consists of re-ranking a limited set of n candidate responses (including the 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='com/Guzpenha/transformer_rankers/tree/full_rank_retrieval_dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 2 Gustavo Penha and Claudia Hauff ground-truth response(s)), followed by measuring the number of relevant re- sponses found in the first K positions—Recalln@K [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Since the entire collec- tion of available responses is typically way bigger2 than such a set of candidates, this setup is in fact a selection problem, where we have to choose the correct response out of a few options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This evaluation overlooks the first-stage retrieval step, which retrieves a set of n responses to be re-ranked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' If the first-stage model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' BM25, fails to retrieve relevant responses, the entire pipeline fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Motivated by a lack of research on the first-stage retrieval step, we are inter- ested in answering in our replicability study whether the considerable knowledge obtained on document and passage retrieval tasks generalizes to the dialogue do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Unlike document and passage retrieval where the documents are generally longer than the queries, in response retrieval for dialogues the queries (dialogue contexts) tend to be longer than the documents (responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A second important difference is the structure induced by the dialogue as seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Comparison between passage retrieval and response retrieval for dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In §3 we define the task of First-stage Retrieval for Dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Colors symbolize the information-seeker and information-provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' p+/r+ are the relevant passage/response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Passage Retrieval First-stage Retrieval for Dialogues Input Query q Dialogue context U = {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=', uτ} Example q: what is theraderm used for u1: I want a firewall that will protect me but more of that to monitor any connection in or out of my mac [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] u2: {url} allows you to map joypad buttons to keyboard keys and [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] u3: Do the diagonals for the analog stick work correctly for you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] Output Ranked list of passages Ranked list of responses Example p+: Thera-Derm Lotion is used as a moisturizer to treat or prevent dry, rough, scaly, [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] r+: In the ”Others” tab, try [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] Given the differences between the domains, we verify empirically across three information-seeking datasets and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='7M queries, the generalizability of five find- ings (F1 to F5) from the passage and document retrieval literature related to state-of-the-art sparse and dense retrieval models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We are motivated in our se- lection of these five findings by their impact in prior works (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' §2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Our results show that four out of five previous findings do indeed generalize to our domain: 2 While for most benchmarks [52] we have only 10–100 candi- dates, a working system with the Reddit data from PolyAI https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='com/PolyAI-LDN/conversational-datasets would need to retrieve from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='7 billion responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From Document and Passage Retrieval to Response Retrieval for Dialogues 3 F1 ✗3 Dialogue context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' query) expansion outperforms a no-expansion base- line [1,18,49,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F2 ✓ Response (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' document) expansion outperforms a no-expansion baseline [25,21,19] if the expansion model is trained to generate the most recent context (last utterance4 of the dialogue) instead of older context (all utterances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F3 ✓ Dense retrieval in the zero-shot5 setting underperforms sparse baselines [34,41] except when it goes through intermediate training on large amounts of out-of-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F4 ✓ Dense retrieval with access to target data6 outperforms sparse baselines [7,15,34] if an intermediate training step on out-of-domain data is performed before the fine-tuning on target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F5 ✓ Harder negative sampling techniques lead to effectiveness gains [46,51] if a denoising technique is used to reduce the number of false negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Our results indicate that most findings translate to the domain of retrieval of responses for dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A promising future direction is thus to start with successful models from other domains—for which there are more datasets and previous research—and study how to adapt and improve them for retrieval-based conversational search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 2 Related Work In this section we first discuss current research in retrieval-based systems for conversational search, followed by reviewing the major findings of (un)supervised sparse and dense retrieval in the domains of passage and document retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='1 Ranking and Retrieval of Responses for Dialogues Early neural models for response re-ranking were based on matching the repre- sentations of the concatenated dialogue context and the representation of a re- sponse in a single-turn manner with architectures such as CNN and LSTM [23,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' More complex neural architectures matching each utterance with the response were also explored [54,9,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Heavily pre-trained language models such as BERT were first shown to be effective by Nogueira and Cho [24] for re-ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Such models quickly became a predominant approach for re-ranking in IR [21] and were later shown to be effective for re-ranking responses in conversations [42,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In contrast, the first-stage retrieval of responses for a dialogue received rela- tively little attention [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' [17] and Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' [38] showed that BERT- based dense retrieval models outperform BM25 for first-stage retrieval of re- sponses for dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A limitation of their work is that strong sparse retrieval 3 ✗ indicates that the finding does not hold in our domain whereas ✓ indicates that it holds in our domain followed by the necessary condition or exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 4 For example in Table 1 the last utterance is u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 5 A zero-shot is a model that does not have access to target data, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 6 Target data is data from the same distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' dataset, of the evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 4 Gustavo Penha and Claudia Hauff baselines that have shown to be effective in other retrieval tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' BM25 with dialogue context expansion [25] or BM25 with response expansion [49], were not employed for dense retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We do such comparisons here and test a total of five major findings that have been not been evaluated before by previous literature on the first-stage retrieval of responses for dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='2 Dense and Sparse Models for Passage and Document Retrieval Context for F1 Retrieval models can be categorized into two dimensions: su- pervised vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' unsupervised and dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' sparse representations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' An unsuper- vised sparse representation model such as BM25 [35] represents each document and query with a sparse vector with the dimension of the collection’s vocabu- lary, having many zero weights due to non-occurring terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Since the weights of each term are entirely based on term statistics they are considered unsupervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Such approaches are prone to the vocabulary mismatch problem [6], as semantic matches are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A way to address such a problem is by using query expansion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' RM3 [1] is a competitive [49] query expansion technique that uses pseudo-relevance feedback to add new terms to the queries followed by another final retrieval step using the modified query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Context for F2 A supervised sparse retrieval model can take advantage of the effectiveness of transformer-based language models by changing the terms’ weights from collection statistics to something that is learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Document ex- pansion with a learned model can be considered a learned sparse retrieval ap- proach [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The core idea is to create pseudo documents that have expanded terms and use them instead when doing retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Doc2query [25] is a strong su- pervised sparse retrieval baseline that uses a language model to predict queries that might be issued to find a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The predictions of this model are used to create the augmented pseudo documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Context for F3 and F4 Supervised dense retrieval models7, such as ANCE [46] and coCodenser [7], represent query and documents in a small fixed-length space, for example of 768 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Dense retrieval models without access to target data for training—known as the zero-shot scenario—have underperformed sparse methods (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For example, the BEIR benchmark [41] showed that BM25 was superior to dense retrieval from 9–18 (depending on the model) out of the 18 datasets in the zero-shot scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In contrast, when having access to enough supervision from target data, dense retrieval models have shown to consistently outperform strong sparse baselines [7,15,34] (F4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 7 A distinction can also be made of cross-encoders and bi-encoders, where the first encode the query and document jointly as opposed to separately [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Cross-encoders are applied in a re-ranking step due to their inefficiency and thus are not our focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From Document and Passage Retrieval to Response Retrieval for Dialogues 5 Context for F5 In order to train neural ranking models, a small set of negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' non-relevant) candidates are necessary as it is prohibitively expensive to use every other document in the collection as negative sample for a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A limitation of randomly selecting negative samples is that they might be too easy for the ranking model to discriminate from relevant ones, while for negative documents that are harder the model might still struggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For this reason hard negative sampling has been shown to perform better than random sampling for passage and document retrieval [46,36,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 3 First-stage Retrieval for Dialogues In this section we first describe the problem of first-stage retrieval of responses, followed by the findings we want to replicate from sparse and dense approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Problem Definition The task of first-stage retrieval of responses for dialogues, concerns retrieving the best response out of the entire collection given the di- alogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Formally, let D = {(Ui, Ri, Yi)}M i=1 be a data set consisting of M triplets: dialogue context, response candidates and response relevance labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The dialogue context Ui is composed of the previous utterances {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=', uτ} at the turn τ of the dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The candidate responses Ri = {r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=', rn} are either ground-truth responses r+ or negative sampled candidates r−, indicated by the relevance labels Yi = {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=', yn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In previous work, the number of candidates is limited, typically n = 10 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The findings we replicate here come from passage and document retrieval tasks where there is no limit to the number of documents or passages that have to be retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Thus, in all of our first-stage retrieval task experiments n is set to the size of the entire collection of responses in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The number of ground-truth responses is one, the observed re- sponse in the conversational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The task is then to learn a ranking function f(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=') that is able to generate a ranked list from the entire corpus of responses Ri based on their predicted relevance scores f(U, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F1: Unsupervised Sparse Retrieval We rely on classic retrieval methods, for which the most commonly used baseline is BM25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' One of the limitations of sparse retrieval is the vocabulary mismatch problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Expansion techniques are able to overcome this problem by appending new words to the dialogue contexts and responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For this reason, we here translate a query expansion technique to the dialogue domain and perform dialogue context expansion with RM3 [1], a competitive unsupervised method that assumes that the top-ranked responses by the sparse retrieval model are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From these pseudo-relevant responses, words are selected and an expanded dialogue context is created and subsequently employed by the sparse retrieval method to rank the final list of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The effectiveness of RM3 in the domain of dialogues is the first finding that we validate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 6 Gustavo Penha and Claudia Hauff F2: Learned Sparse Retrieval Alternatively, we can expand the responses in the collection with a learned method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' To do so we “translate” doc2query [25] into our domain, yielding resp2ctxt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Formally, we fine-tune a generative transformer model G for the task of generating the dialogue context Ui from the ground-truth response r+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This model is then used to generate expansions for all responses in the collection, ri = concat(ri, G(ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' These expansions are appended to the responses and the collection is indexed again—the sparse retrieval method it- self is not modified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' we continue using BM25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This approach (which we coin resp2ctxt) leads to two improvements: term re-weighting (adding terms that already exist in the document) and dealing with the vocabulary mismatch prob- lem (adding new terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The effectiveness of doc2query in the domain of dialogues is the second finding that we validate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Unlike passage and document retrieval where the queries are smaller than the documents, for the retrieval of responses for dialogues the queries are longer than the documents8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This is a challenge for the generative model, since generating larger pieces of text is a more difficult problem than smaller ones as there is more room for errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Motivated by this, we also explored a modified version of resp2ctxt that aims to generate only the last utterance of the dialogue context: resp2ctxtlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This model is trained to generate uτ from r+ i , instead of trying to generate the whole utterance Ui = {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=', uτ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The underlying premise is that the most important utterance from the dialogue is the last one, and if it is correctly generated by resp2ctxtlu, the sparse retrieval method will be able to find the correct response from the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F3: Zero-shot Dense Retrieval We rely on methods that learn to represent the dialogue context and the responses separately in a dense embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Responses are then ranked by their similarity to the dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We rely here on pre-trained language transformer models, such as BERT [4] and MP- Net [37], to obtain such representations of the dialogue context and response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This approach is generally referred to as a bi-encoder model [21] and is an ef- fective family of models9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A zero-shot model is one that is not trained on the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Target data is data from the same distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' dataset, of the evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' One way of improving the representations of a heavily pre-trained language model for the zero-shot setting is to fine-tune it with intermediate data [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Such intermediate data contains triplets of query, relevant document, and negative document and can include multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The advantage of adding this step before employing the representations of the language model is to reduce the 8 For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' while the TREC-DL-2020 passage and document retrieval tasks the queries have between 5–6 terms on average and the passages and documents have over 50 and 1000 terms respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' for the information-seeking dialogue datasets used here the dialogue contexts (queries) have between 70 and 474 terms on average depending on the dataset while the responses (documents) have between 11 and 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 9 See for example the top models in terms of effectiveness from the MSMarco bench- mark leaderboards https://microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='io/msmarco/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From Document and Passage Retrieval to Response Retrieval for Dialogues 7 gap between the pre-training and the downstream task at hand [31,26,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In Table 2 we clarify the relationship between pre-training, intermediate training and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The different training stages and data, their purposes, examples of datasets, and the type of dense model obtained after each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Pre-training data Intermediate data Target data Purpose Learn general represen- tations Learn sentence representations for ranking Learn representations for tar- get distribution Model is Zero-shot Zero-shot Fine-tuned Example Wikipedia MSMarco MANtIS The intermediate training step learns to represent pieces of text (query and documents) by applying a mean pooling function over the transformer’s final layer, which is then used to calculate the dot-product similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The loss function employs multiple negative texts from the same batch to learn the representations in a constrastive manner, also known as in-batch negative sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Such a procedure learns better text representations than a naive approach that uses the [CLS] token representation of BERT [33,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The function f(U, r) is then dot(η(concat(U)), η(r)), where η is the repre- sentation obtained by applying the mean pooling function over the last layer of the transformer model, and concat(U) = u1 | [U] | u2 | [T ] | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' | uτ , where | indicates the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The utterances from the context U are concatenated with special separator tokens [U] and [T ] indicating end of utter- ances and turns10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The effectiveness of a zero-shot bi-encoder model in the domain of dialogues is the third finding we validate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F4: Fine-tuned Dense Retrieval The standard procedure is to fine-tune dense models with target data that comes from the same dataset that the model will be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Since we do not have labeled negative responses, all the remain- ing responses in the dataset can be thought of as non-relevant to the dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Computing the probability of the correct response over all other re- sponses in the dataset would give us P(r | U) = P (U,r) � k P (U,rk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This computation is prohibitively expensive, and the standard procedure is to approximate it using a few negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The negative sampling task is then as follows: given the dialogue context U find challenging responses r− that are non-relevant for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Negative sampling can be seen as a retrieval task, where one can use a model to retrieve negatives by applying a retrieval function to the collection of responses using U as the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' With such a dataset at hand, we continue the training—after the intermediate step—in the same manner as done by the intermediate training step, with the following cross-entropy loss function11 for a batch with size B: 10 The special tokens [U] and [T ] will not have any meaningful representation in the zero-shot setting, but they can be learned on the fine-tuning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 11 We refer to this loss as MultipleNegativesRankingLoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 8 Gustavo Penha and Claudia Hauff J (U, r, θ) = − 1 B �B i=1 � f (Ui, ri) − log �B j=1,j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='=i ef(Ui,rj)� , where f(U, r) is the dot-product of the mean pooling of the last layer of the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The effectiveness of a fine-tuned bi-encoder model in the domain of dialogues is the fourth finding we validate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F5: Hard Negative Sampling A limitation of random samples is that they might be too easy for the ranking model to discriminate from relevant ones, while for negative documents that are hard the model might still struggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For this reason, another popular approach is to use a ranking model to retrieve negative documents using the given query with a classic retrieval technique such as BM25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This leads to finding negative documents that are closer to the query in the sparse representation space, and thus they are harder negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Since dense retrieval models have been outperforming sparse retrieval in a number of cases with available training data, more complex negative sampling techniques making use of dense retrieval have also been proposed [46,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The effectiveness of hard negative sampling for a bi-encoder model in the domain of dialogues is the fifth finding we validate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 4 Experimental Setup In order to compare the different sparse and dense approaches we consider three large-scale information-seeking conversation datasets12: MSDialog [32] contains 246K context-response pairs, built from 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='5K information seeking conversa- tions from the Microsoft Answer community, a QA forum for several Microsoft products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' MANtIS [27] contains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='3 million context-response pairs built from con- versations of 14 Stack Exchange sites, such as askubuntu and travel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' UDCDSTC8 [16] contains 184k context-response pairs of disentangled Ubuntu IRC dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Implementation Details For BM25 and BM25+RM313 we rely on the pyserini implementations [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In order to train resp2ctxt expansion methods we rely on the Huggingface transformers library [44], using the t5-base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We fine- tune the T5 model for 2 epochs, with a learning rate of 2e-5, weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='01, and batch size of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' When augmenting the responses with resp2ctxt we follow docT5query [25] and append three different context predictions, using sampling and keeping the top-10 highest probability vocabulary tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For the zero-shot dense models, we rely on the SentenceTransformers [33] model releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The library uses Hugginface’s transformers for the pre-trained models such as BERT [4] and MPNet [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For the bi-encoder models, we use the 12 MSDialog is available at https://ciir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='umass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='edu/downloads/msdialog/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' MANtIS is available at https://guzpenha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='io/MANtIS/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' UDCDSTC8 is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='com/dstc8-track2/NOESIS-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 13 We perform hyperparameter tuning using grid search on the number of expansion terms, number of expansion documents, and weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From Document and Passage Retrieval to Response Retrieval for Dialogues 9 pre-trained all-mpnet-base-v2 weights which were the most effective in our initial experiments, compared with other pre-trained models14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' When fine-tuning the dense retrieval models, we rely on the MultipleNegativesRankingLoss, which ac- cepts a number of hard negatives, and also uses the remaining in-batch random negatives to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We use a total of 10 negative samples for dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We fine-tune the dense models for a total of 10k steps, and every 100 steps we evaluate the models on a re-ranking task that selects the relevant re- sponse out of 10 responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We use the re-ranking validation MAP to select the best model from the whole training to use in evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We use a batch size of 5, with 10% of the training steps as warmup steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The learning rate is 2e-5 and the weight decay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We use FAISS [13] to perform the similarity search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Evaluation To evaluate the effectiveness of the retrieval systems we use R@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We thus evaluate the models’ capacity of finding the correct response out of the whole possible set of responses15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We perform Students t-tests at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='95 confidence level with Bonferroni correction to compare statistical significance of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Comparisons are performed across the results for each dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 5 Results In this section, we discuss our empirical results along with the five major find- ings from previous work (Section 1) in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Table 3 contains the main results regarding F1 to F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Table 5 contains the results for F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F1 ✗ Query expansion via RM3 leads to improvements over not using query expansion [1,18,49,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' BM25+RM3 (row 1b) does not improve over BM25 (1a) on any of the three conversational datasets analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We performed thorough hyperparameter fine-tuning and no combination of the RM3 hyperpa- rameters outperformed BM25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This indicates that F1 does not hold for the task of response retrieval for dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A manual analysis of the new terms appended to a sample of 60 dialogue contexts by one of the paper’s authors revealed that only 18% of them have at least one relevant term added based on our best judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Unlike web search where the query is often incomplete, under-specified, and ambiguous, in the information-seeking datasets employed here the dialogue context (query) is quite detailed and has more terms than the responses (documents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We hypothesize that because the dialogue contexts are already quite descriptive, the task of expansion is trickier in this domain and thus we observe many dialogues for which the added terms are noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 14 The alternative models we considered are those listed in the model overview section at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='sbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='net/docs/pretrained_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 15 The standard evaluation metric in conversation response ranking [50,8,39] is recall at position K with n candidates Rn@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Since we are focused on the first-stage retrieval we set n to be the entire collection of answers 10 Gustavo Penha and Claudia Hauff Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Results for the generalizability of F1–F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Bold values indicate the high- est recall for each type of approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Superscripts indicate statistically significant im- provements using Students t-test with Bonferroni correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' †=other methods from the same group1=best from unsupervised sparse retrieval ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 2=best from supervised sparse retrieval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 3=best from zero-shot dense retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For example, in F3 † indicates that row (3d) improves over rows (3a–c), 1 indicates that it improves over row (1a) and 2 indicates it improves over row (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' MANtIS MSDialog UDCDSTC8 R@1 R@10 R@1 R@10 R@1 R@10 (0) Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='001 Unsupervised sparse F1 (1a) BM25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='133† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='299† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='064† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='177† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='027† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='050 (3c) Bi-encoder215M−mul 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='108 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+page_content='147†12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='339†12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='041† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='097†12 Fine-tuned dense (ModelNegativeSampler) F4 (4a) Bi-encoderRandom(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='168123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='387123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='05012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='128123 F2 ✓ Document expansion via resp2ctxt leads to improvements over no expansion [25,21,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We find that a naive approach to response expansion improves marginally in two of the three datasets with BM25+resp2ctxt (2a) outperforming BM25 (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' However, the proposed modification of predicting only the last utterance of the dialogue (resp2ctxtlu) performs better than predicting the whole utterance, as shown by BM25+resp2ctxtlu’s (2b) higher recall values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In the MANtIS dataset the R@10 goes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='309 when using the model trained to predict the dialogue context to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='325 when using the one trained to predict only the last utterance of the dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We thus find that F2 generalizes to response retrieval for dialogues, especially when predicting only the last utterance of the context16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In order to understand what the response expansion methods are doing most—term re-weighting or adding novel terms—we present the percentage of novel terms added by both methods in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The table shows that resp2ctxtlu does more term re-weighting than adding new words when compared to resp2ctxt 16 As future work, more sophisticated techniques can be used to determine which parts of the dialogue context should be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From Document and Passage Retrieval to Response Retrieval for Dialogues 11 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Statistics of the augmentations for resp2ctxt and resp2ctxtlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' New words are the ones that did not exist in the document before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' MANtIS MSDialog UDCDSTC8 Context avg length 474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='12 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='08 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='95 Response avg length 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='58 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='38 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='06 Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' avg length - resp2ctxt 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='23 596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='99 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='3 Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' avg length - resp2ctxtlu 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='5 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='29 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='57 % new words - resp2ctxt 71% 69% 71% % new words - resp2ctxtlu 59% 37% 63% (53% and 70% on average are new words respectively and thus 47% vs 30% are changing the weights by adding existing words), generating overall smaller aug- mentations (115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='45 vs 431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='17 on average respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F3 ✓ Sparse retrieval outperforms zero-shot dense retrieval [34,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Sparse retrieval models are more effective than the majority of zero-shot dense models, as shown by the comparison of rows (1a–b), and (2a–b) with rows (3a– c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' However, a dense retrieval model that has gone through intermediate training on large and diverse datasets including dialogues is more effective than a strong sparse retrieval model, as we see by comparing row (3d) with row (2b) in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' For example, while the zero-shot dense retrieval models based only on the MSMarco dataset (3a–b) perform on average 35% worse than the strong sparse baseline (2b) in terms of R@10 for the MSDialog dataset, the zero-shot model trained with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='17B instances on diverse data (3d) is 68% better than the sparse baseline (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' When using a bigger amount of intermediate training data17, we see that the zero-shot dense retrieval model (3d) is able to outperform the sparse retrieval baseline by margins of 33% of R@10 on average across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We thus show that F3 only generalizes to response retrieval for dialogues if we do not employ a large set of diverse intermediate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' As expected, the closer the intermediate training data distribution is to the eval- uation data, the better the dense retrieval model performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The results indicate that a good zero-shot retrieval model needs to go through intermediate training on a large set of training data coming from multiple datasets to generalize well to different domains and outperform strong sparse retrieval baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F4 ✓ Dense models with access to target training data outperform sparse models [7,15,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' First, we see that fine-tuning the dense retrieval model, which has gone through intermediate training already, with random sampling—row (4a) in Table 3—achieves the best overall effectiveness in two of the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This result shows that F4 generalizes to the task of 17 For the full description of the intermediate data see https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='co/sentence-transformers/all-mpnet-base-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 12 Gustavo Penha and Claudia Hauff response retrieval for dialogues when employing intermediate train- ing18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Having access to the target data as opposed to only the intermediate training data means that the representations learned by the model are closer to the true distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We hypothesize that fine-tuning the bi-encoder for MANtIS (4a) is harmful because the intermediate data contains Stack Exchange responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In this way, the set of dialogues of Stack Exchange that MANtIS encompasses might be serving only to overfit the intermediate representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' As evidence for this hypothe- sis, we found that (I) the learning curves flatten quickly (as opposed to other datasets) and (II) fine-tuning another language model that does not have Stack Exchange data (MSMarco) in their fine-tuning, bi-encoderbert−base (3c), improves the effectiveness with statistical significance from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='092 R@10 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='205 R@10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' F5 ✓ Hard negative sampling is better than random sampling for training dense retrieval models [46,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Surprisingly we find that naively using more effective models to select negative candidates is detrimental to the ef- fectiveness of the dense retrieval model (see Hard negative sampling in Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We observe this phenomenon when using different language models, when switch- ing intermediate training on or off for all datasets, and when using an alternative contrastive loss [10] that does not employ in-batch negative sampling19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' After testing for a number of hypotheses that might explain why harder negatives do not improve the effectiveness of the dense retrieval model, we found that false negative samples increase significantly when using better negative sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' False negatives are responses that are potentially valid for the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Such relevant responses lead to unlearning relevant matches between context and responses as they receive negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' See below an example of a false negative sample retrieved by the bi-encoder model (row 3d of Table 3): Dialogue context (U): hey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' how long until dapper comes out?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' [U] 14 days [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] [U] i thought it was coming out tonight Correct response (r+): just kidding couple hours False negative sample (r−): there is a possibility dapper will be delayed [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='] mean- while, dapper discussions should occur in ubuntu+1 Denoising techniques try to solve this problem by reducing the number of false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' We employ a simple approach that instead of using the top-ranked responses as negative responses, we use the bottom responses of the top-ranked responses as negatives20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' This decreases the chances of obtaining false positives and if k << |D| we will not obtain random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Our experiments in Table 5 reveal that this denoising technique, row (3b), increases the effectiveness for harder negative samples, beating all models from Table 3 for two of the three 18 Our experiments show that when we do not employ the intermediate training step the fine-tuned dense model does not generalize well, with row (3d) performance dropping to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='172, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='308 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='063 R@10 for MANtIS, MSDialog and UDCDSTC8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 19 The results are not shown here due to space limitations 20 For example, if we retrieve k = 100 responses, instead of using responses from top positions 1–10, we use responses 91–100 from the bottom of the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' From Document and Passage Retrieval to Response Retrieval for Dialogues 13 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' The results indicate that F5 generalizes to the task of response retrieval for dialogues only when employing a denoising technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Results for the generalizability of F5—with and without a denoising strategy for hard negative sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Superscripts indicate statistically significant improvements using Students t-test with Bonferroni correction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' †=significance against the random sampling baseline, ‡=significance against hard negative sampling without denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' MANtIS MSDialog UDCDSTC8 R@10 R@10 R@10 Baseline (1) Bi-encoderRandom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='128 Hard negative sampling (2a) Bi-encoderBM25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='087 (2b) Bi-encoderBi−encoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='051 Denoised hard negative sampling (3a) Bi-encoderBM25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='358‡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='121‡ (3b) Bi-encoderBi−encoder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='316†‡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='397†‡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='107‡ 6 Conclusion In this work, we tested if the knowledge obtained in dense and sparse retrieval from experiments on the tasks of passage and document retrieval generalizes to the first-stage retrieval of responses for dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Our replicability study reveals that while most findings do generalize to our domain, a simple translation of the models is not always successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' A careful analysis of the domain in question might reveal better ways to adapt techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' As future work, we believe an important direction is to evaluate learned sparse methods that do weighting and expansion for both the queries and doc- uments [5]—while resp2ctxt is able to both change the weights of the terms in the response (by repeating existing terms) and expand terms (by adding novel terms), it is not able to do weighting and expansion for the dialogue contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Acknowledgements This research has been supported by NWO projects SearchX (639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='722) and NWO Aspasia (015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content='027).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' Abdul-Jaleel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=', Allan, J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=': Multi- turn response selection for chatbots with deep attention matching network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' In: Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} +page_content=' 1118–1127 (2018)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E5T4oBgHgl3EQfPw7T/content/2301.05508v1.pdf'} diff --git a/6tAyT4oBgHgl3EQfpvgE/vector_store/index.pkl b/6tAyT4oBgHgl3EQfpvgE/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..71e50a92a5aeeb59f4e19c3b4b234b4e9d283474 --- /dev/null +++ b/6tAyT4oBgHgl3EQfpvgE/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f4e9d2fd6378cc1990f3c5bebf7987956e00e87d840dc992ea87c8488d1c72eb +size 326737 diff --git a/79E2T4oBgHgl3EQflQc0/content/tmp_files/2301.03986v1.pdf.txt b/79E2T4oBgHgl3EQflQc0/content/tmp_files/2301.03986v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a2ba34ed526e92ead5c73bdec35cf4461faffd0 --- /dev/null +++ b/79E2T4oBgHgl3EQflQc0/content/tmp_files/2301.03986v1.pdf.txt @@ -0,0 +1,1239 @@ +arXiv:2301.03986v1 [math.AP] 10 Jan 2023 +THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD +PLASMA +OLGA S. ROZANOVA* +Abstract. A solution of the Riemann problem is constructed for a nonstrictly +hyperbolic inhomogeneous system of equations describing one-dimensional cold +plasma oscillations. Each oscillation period includes one rarefaction wave and +one shock wave containing a delta singularity. The rarefaction wave can be +constructed in a non-unique way, the admissibility principle is proposed. +1. Introduction +In vector form, the system of hydrodynamic of electron liquid, together with +Maxwell’s equations, has the form: +(1) +nt + div (nV) = 0 , +Vt + (V · ∇) V = e +m +� +E + 1 +c [V × B] +� +, +1 +c Et = −4π +c enV + rot B , +1 +cBt = −rot E , +div B = 0 , +where e, m are the charge and mass of the electron (here the electron charge has +a negative sign: e < 0), c is the speed of light; n, V are the density and velocity +of electrons; E, B are the vectors of electric and magnetic fields, x ∈ R3, t ≥ 0, +∇, div, rot are the gradient, divergence and vorticity with respect to the spatial +variables. The system of equations (1) is one of the simplest models of plasma, +which is often called the equations of hydrodynamics of ”cold” plasma, it is well +known and described in sufficient detail in textbooks and monographs (see, for +example, [1], [4]). +This system has an important subclass of solutions, dependent only on one space +variable x, for which V = (V, 0, 0), E = (E, 0, 0), B ≡ 0, e.g. [3]. In dimensionless +form it can be written as +(2) +nt + (n V )x = 0, +Vt + V Vx = −E, +Et = n V. +Assume that the solution is smooth. Then the first and last equations (2) imply +(n + Ex)t = 0. For the background density n ≡ 1 we get +(3) +n = 1 − Ex. +This allows us to obtain a hyperbolic system for the two components of the velocity +V and the electric field E in the form +(4) +Vt + V Vx = −E, +Et + V Ex = V, +2020 Mathematics Subject Classification. Primary 35Q60; Secondary 35L60, 35L67, 34M10. +Key words and phrases. Quasilinear hyperbolic system, Riemann problem, non-uniqueness, +singular shock, plasma oscillations. +1 + +2 +OLGA S. ROZANOVA* +where (V, E) = (V (t, x), E(t, x)), t ∈ R+, x ∈ R. The density n(t, x) > 0 can be +found from (3). +System (4), (3) can be also rewritten as a pressureless repulsive Euler-Poisson +system [5] +nt + (nV )x = 0, +Vt + V Vx = ∇Φ, +∆Φ = n − n0, +n0 = 1, +(5) +where Φ is a repulsive force potential, ∇Φ = −E. +For (4) we consider the Cauchy problem +(6) +(V, E)|t=0 = (V0(x), E0(x)). +If the initial data are C1 - smooth functions, then locally in t there exists a smooth +solution of (4), (6). Nevertheless, it is known that the derivatives of the solution +of such a Cauchy problem can go to infinity for a finite time, which corresponds +to the formation of a shock wave, the criterion for the formation of a singularity +is known [13]. Thus, it makes sense to consider piecewise-smooth functions as the +initial data (6), the simplest example of which is the Riemann initial data +(7) +(V, E)|t=0 = (V 0 +− + [V ]0Θ(x), E0 +− + [E]0Θ(x)), +where Θ(x) is the Heaviside function, constants (V−, E−) are the values to the left +of the jump, ([V ], [E]) the values to the jumps, (V+ = V− + [V ], E+ = E− + [E]) +are the values to the right of the jump, (v0 +±, E0 +±), ([v]0, [E]0) are the corresponding +values at time zero. In this case, the density at the initial moment of time is +(8) +n|t=0 = 1 − [E]0δ(x). +Since the initial data contain a delta function, the Riemann problem for the com- +ponents of the solution (V, E, n) is singular and the Rankine-Hugoniot conditions +cannot be written in the traditional form [15]. In order to ensure that the density +is positive initially, it is necessary to impose the condition [E]0 ≤ 0. +To construct the shock, we write system (4) in the divergent form +(9) +nt + (V n)x = 0, +�nV 2 +2 ++ E2 +2 +� +t ++ +�nV 3 +2 +� +x += 0, +corresponding to the laws of conservation of mass and total energy (for example, +[6]). System (9) (together with (3)) is equivalent to (4), (3) for smooth solutions. +The Riemann problem (9), (3), (7), (8) is completely non-standard and demon- +strates new phenomena in the construction of both a rarefaction wave and a shock +wave. +The difficulty in constructing a solution is associated, in particular, with the +fact that system (4) is inhomogeneous and does not have a constant stationary +state. To the left and right side of the discontinuity, the solution is a 2π - periodic +function of time. This leads to the fact that the rarefaction wave and the shock +wave periodically replace each other. Further, system (4) is hyperbolic, but not +strictly hyperbolic: it has the form +ut + A(u)ux = f(u), +u = (V, E), +f = (−E, V ), +the matrix A has a complete set of eigenvectors with coinciding eigenvalues λ1 = +λ2 = V . Because of this, it has a subclass of solutions in the form of simple waves, +distinguished by the condition +V 2 + E2 = C2 + +THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +3 +with a given constant C. We show that this leads to the non-uniqueness of the +rarefaction wave for the Riemann problem. Therefore, the question arises about +the principles by which one can single out the ”correct” solution. In our work, the +correct one is chosen for which the total energy density is minimal. +When constructing a singular shock wave, we use homogeneous conservative +system of two equations (9), which are linked by the differential relation (3). This +formulation has not been encountered before, although a modification of the method +previously used for the case of equations of the pressureless gas dynamics with +energy [11] can be used to construct a solution to the Riemann problem. The shock +wave satisfies the so-called “supercompression” conditions, which are traditionally +used to distinguish admissible singular shock waves [15]. +The paper is organised as follows. In Sec.2 we discuss the structure of charac- +teristics which is crucial for construction of rarefaction and shock waves. In Sec.3 +we construct the rarefaction wave for Riemann data (7) of a general form and then +show that for the data corresponding to a simple wave the rarefaction can be con- +structed non-uniquely. We also propose two variational conditions of admissibility +of the rarefaction waves for this case. In Sec.4 we give a definition of the strongly +singular solution for an arbitrary piecewise smooth initial data, prove an analog of +the Rankine-Hugoniot conditions (Theorem 1), study the mass and energy transfer +for a singular shock wave (Theorem 2). Then we construct the singular shock for +piecewise smooth initial data (7) and give two examples. The first example corre- +sponds to the case of simple wave, here we compare the result obtained starting +from conservative form (9) and the result obtained from a divergence form, natural +for the Hopf equation. The second example show how it is possible to construct the +shock in the case where the shock position has an extremum on the characteristic +plane. Sec.5 contains a discussion about a physical and mathematical sense of the +results obtained and mention works concerning shock waves in plasma for other +models. +2. Characteristics +The equations for the characteristics corresponding to system (4) have the form +(10) +dV +dt = −E, +dE +dt = V, +dx +dt = V, +whence, first, it follows that along the characteristics +(11) +d(V 2 + E2) +dt += 0, +and also, according to (7), +V±(t) += −E0 +± sin t + V 0 +± cos t, +E±(t) = V 0 +± sin t + E0 +± cos t, +x±(t) += V 0 +± sin t + E0 +±(cos t − 1) + x0, +x0 = 0. +It is easy to see that for [E]0 ̸= 0 the characteristics x−(t) and x+(t), corresponding +to the states to the left and to the right of the discontinuity, intersect once inside +each period 2π. Therefore, on that part of the period where x−(t) < x+(t), it is +necessary to construct a continuous solution, and on the part where x−(t) > x+(t), +that is, there is an intersection of the characteristics, we construct a shock wave. +The moment of time at which x−(t) = x+(t), we denote by T∗, T∗ ∈ (0, 2π). + +4 +OLGA S. ROZANOVA* +Fig.1 gives a schematic representation of the behavior of the characteristics, where +the rarefaction wave comes first. +Note that (11) implies that the value C2 = V 2 + E2 is constant for each specific +characteristic, but in general it is a function of t and x. +Figure 1. +Characteristics and their intersections: rarefaction +waves and shock waves. +3. Construction of a rarefaction wave +Suppose the initial data is such that V 0 +− < V 0 ++, that is, x−(t) < x+(t), and first +the initial data (7) generate a rarefaction wave. Between the characteristics x−(t) +and x+(t), it is necessary to construct a continuous solution (V, E) connecting the +states (V−(t), E−(t)) and (V+(t), E+(t)). Recall that the moment of time at which +x−(t) = x+(t), we denote by T∗, T∗ ∈ (0, 2π). +The rarefaction wave, of course, is not a smooth solution, it satisfies the conser- +vative system (9) with the additional condition (3) in the usual sense of the integral +identity. +3.1. The linear profile solution. It is easy to check that a continuous solution +(V, E) can be constructed by joining the states (V−(t), E−(t)) and (V+(t), E+(t)), +between characteristics with the help of functions linear in x, i.e. +(12) +(V, E) = + + + +(V−(t), E−(t)), +x < x−(t); +(Vr1, Er1) = (a(t)x + b(t), c(t)x + d(t)), +x ∈ [x−(t), x+(t)]; +(V+(t), E+(t)), +x > x+(t), +with +(13) +a(t) = +−[E]0 sin t + [V ]0 cos t +−[V ]0 sin t + [E]0(1 − cos t), +c(t) = +−[V ]0 sin t − [E]0 cos t +−[V ]0 sin t + [E]0(1 − cos t), +(14) +b(t) = (V 0 ++E0 +− − E0 ++V 0 +−)(1 − cos t) +−[V ]0 sin t + [E]0(1 − cos t), +d(t) = +(V 0 ++E0 +− − E0 ++V 0 +−) sin t +−[V ]0 sin t + [E]0(1 − cos t). +Then +n = 1 − c(t)χ(x−(t),x+(t)), + +10: +7.5 +Shock +Rarefaction +2,5 +X +-5 +2,5 +10 +2,5THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +5 +where χ(x−(t),x+(t)) is the characteristic function of the interval (x−(t), x+(t)), for +t ∈ (0, T∗) the density does not contain a delta function, but the singular component +that was present in the initial data is again formed at t = T∗. +3.2. Simple waves. The system (4) has a subclass of solutions distinguished by +the condition +(15) +V 2 + E2 = C2(≡ const) +with a given constant C, the so called simple waves. In this case, (4) reduces on +smooth solutions to one equation +(16) +Vt + V Vx = −σ +� +C2 − V 2, +σ = sign(−Vx) = ±1, +E = σ +� +C2 − V 2, +moreover, Vxx ̸= 0 on no set of positive measure. The last requirement means that +the solution cannot become constant on any interval, but at the points at which +C2 = V 2 the value of σ changes its sign to the opposite. The second conservation +law (9) in this situation turns out to be a consequence of the first. +In the initial conditions (7) the values E0 +− and E0 ++ are expressed as E0 +− = +± +� +C2 − (V 0 +−)2, E0 ++ = ± +� +C2 − (V 0 ++)2, so as to ensure the condition [E]0 ≤ 0. +It is easy to see that a function of the form (12) with an intermediate state +(Vr1, Er1) is not a solution to the equation (16). Let us show that in this case +another continuous solution can be constructed, with another function (Vr2, Er2) +as an intermediate state. +Indeed, the general solution (16), written implicitly, looks like +x − σ +� +C2 − V 2 = F +� +t + arctan +V +σ +√ +C2 − V 2 +� +, +with an arbitrary smooth function F. In order to find the function F corresponding +to the initial data (7), (15), we will construct the function X(t, V ) inverse to Vr2(t, x) +for every fixed t ∈ (0, T∗). For t = 0 such a function is multivalued. +We require that for t = 0 the condition X(0, V ) = 0 holds for V ∈ (V 0 +−, V 0 ++). +Then F = − tan +√ +C2−ξ2 +ξ +, ξ = σ +√ +C2 − V 2. After transformations, we get +X1(t, V ) = C(cos q − cos(q + t)), (V±)t < 0 +(σ = 1), +(17) +X2(t, V ) = C(− cos q + cos(q − t)), (V±)t > 0 +(σ = −1), +(18) +q = arcsin V +C . +Note that in each case the monotonicity of V in x ensures the existence of an inverse +function. +The situation is considered separately when the behavior of the solution between +the right and left characteristics is given by different formulas. Namely, consider +the time T1 at which V ′ +−(t) = 0 and the time T2 at which V ′ ++(t) = 0. Between +T1 and T2 there is a moment T0, at which V+(t) = V−(t), and therefore, the jump +disappears. However, at such a point the characteristics do not intersect, that is, +x+(T0) ̸= x−(T0). To construct a continuous solution in such a situation, we need +auxiliary curves X1(t, q−) and X2(t, q+), where q± = arcsin +V 0 +± +C . + +6 +OLGA S. ROZANOVA* +Then for t ∈ (0, T∗) the continuous solution of problem (7), (16), (15) can be +written as +(19) +Vr2(t, x) = + + + +V−(t), +x < X−(t), +Vi(t, x), +X−(t) < x < X+(t), +V+(t), +x > X+(t), +where +X−(t) = +� +x−(t), +t < T1, t > T0 +X2(t, q+), +t ∈ [T1, T0] +, +X+(t) = +� +x+(t), +t < T0, t > T2 +X1(t, q−), +t ∈ [T0, T2] +, +and Vi(t, x) is the function inverse to Xi(t, V ), i = 1, 2, given by formulas (17), +(18). +Thus, a continuous solution to the problem (4), (7), (15) can be constructed as +(20) +(V, E) = + + + +V−(t), +x < x−(t), +(Vr2, Er2), +x−(t) < x < x+(t), +v+(t), +x > x+(t), +where (Vr2, Er2), where Vr2 is given by (19), and Er2 = ± +� +C2 − V 2 +r2, the sign +matches the one that was selected in the initial data (7). Fig.2 presents the con- +struction of the rarefaction wave on the characteristic plane. +Figure 2. +Characteristics and and values V− and V+ from the +left and right side of the rarefaction wave. +3.3. Nonuniqueness of rarefaction wave. Obviously, (12) and (21) are different +continuous solutions. Moreover, on their basis it is possible to construct an infinite +number of other rarefaction waves. Indeed, one can check that Vr2 is an upward con- +vex function and, for t = t1 ∈ (0, T∗), we can choose any point x1 ∈ (x−(t1), x+(t1)) +and replace on the segment (x−(t1), x1) by a linear function. Next, we find the po- +sition of the right point of the linear segment as a solution to the problem for +t ∈ (0, T∗) as ˙x = Vr2(t, x), x(t1) = x1. Such linear sections can be built in any +number. + +t +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +T1 +To +T2 +V+ +V1 ++ +-2 +V +V +-3 +x+ +V2 +-4 +X-0 +3 +t +-1 +V- +x+ +-2 +-3 +-4 +x- +-5THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +7 +3.4. Admissibility of the rarefaction wave. The question of choosing the “cor- +rect” continuous solution can be solved proceeding from the minimality of the total +energy of the rarefaction wave +E(t) = 1 +2 +x+(t) +� +x−(t) +(nV 2 + E2) dx, +see (9). +For the solution (Vr2, Er2) +E(t) = 1 +2 +x+(t) +� +x−(t) +((1 − Ex)(C2 − E2) + E2) dx = 1 +2(C2∆x − C2[E] + 1 +3[E3]), +where ∆x = x+(t)−x−(t) ≥ 0, [E] = E+ −E− = ∆x+[E]0, [E3] = (E+)3 −(E−)3. +For the solution (Vr1, Er1) +E(t) = 1 +2 +x+(t) +� +x−(t) +((1 − c)(ax + b)2 + (cx + d)2) dx, +where a, b, c, d given as (13), (14). +It can be readily computed that +E(Vr2, Er2) − E(Vr1, Er1) = −1 +6[E]0(C2 − (E0 ++E0 +− + V 0 ++V 0 +−) = +− 1 +12[E]0(([E]0)2 + ([V ]0)2) ≥ 0, +t ∈ (0, T∗). +Here we take into account [E]0 ≤ 0 and (E+)2 + (V+)2 = (E−)2 + (V−)2 = C2. +Thus, if [E]0 < 0, for reasons of less energy E we have to choose (Vr1, Er1). +2. Another way to distinguish an acceptable rarefaction wave is the pointwise +minimality of the local energy +E(t, x) = V 2 + E2. +Indeed, +E(Vr2, Er2) = V 2 +r2 + E2 +r2 = C2 +is constant by the construction of the solution, whereas +E(Vr1, Er1) = (ax + b)2 + (cx + d)2 +has a minimum x∗(t) = − +ab+cd +2(a2+c2) ∈ (x−(t), x+(t)). +Since E(Vr1, Er1) = C2 at +x = x±(t), then +E(Vr1, Er1) < C2 = E(Vr2, Er2), +x(t) ∈ (x−(t), x+(t)), +t ∈ (0, T∗). +Both principles, on the basis of which admissible solutions can be distinguished, +lead to the same conclusion: for the complete system (4), the solution (Vr1, Er1) +must be chosen as a rarefaction wave, while when the condition (15) is applied, +only the possibility (Vr2, Er2) remains. + +8 +OLGA S. ROZANOVA* +4. Construction of a singular shock wave +We need to build a shock wave for the second part of the period 2π, t ∈ (T∗, 2π). +However, in order not to complicate the notation, we, without loss of generality, +shift the time point T∗ to zero. Thus, we are in a situation where the initial data +correspond to a shock wave and t = T∗ is the point of the first intersection of the +characteristics. +Suppose that for t ∈ (0, T∗) we have constructed a solution to the problem as +(21) +(Vs, Es) = +� +V−(t), +x < Φ(t), +V+(t), +x > Φ(t), +that is, we found the position of the shock wave x = Φ(t). Then the density can be +found as n(t, x) = 1 − [E]|x=Φ(t)δ(x − Φ(t)). +Thus, we must take into account the presence of a strongly singular component +of the solution. However, before proceeding to the construction of a solution in this +case, we will give a general definition of a strongly singular solution and obtain its +main properties. +4.1. Definition of a generalized strongly singular solution. Starting from +the divergent form (9), we define a generalized strongly singular solution to the +problem (9), (6) according to [15]. +Let +V (t, x) += +V−(t, x) + [V (t, x)]|x=Φ(t)Θ(x − Φ(t)), +(22) +E(t, x) += +E−(t, x) + [E(t, x)]|x=Φ(t)Θ(x − Φ(t)), +(23) +n(t, x) += +ˆn(t, x) + e(t)δ(x − Φ(t)), +(24) +where [f] = f+ − f−, f± are differentiable functions having one-sided limits, t ≥ 0, +x ∈ R, ˆn(t, x) = 1−{Ex(t, x)}, {Ex} is the derivative of the function E at the points +at which it exists in the usual sense, e(t) := e(t, Φ(t)), e(t) = −[E(t, x)]|x=Φ(t). +Definition 4.1. The triple of distributions (V, E, n), given as (22) - (24) and the +curve γ, given as x = Φ(t), Φ(0) = 0, Φ(t) ∈ C1, is called a generalized singular +solution of the problem (9), +(V, E, n)|t=0 = +(V 0 +−(x) + [V (x)]0Θ(x), E0 +−(x) + [E(x)]0Θ(x), n0(x) = ˆn0(x) + e0δ(x)), +if for all test functions φ(t, x) ∈ D(R × [0, ∞) +∞ +� +0 +� +R +ˆn(φt + V φx)dxdt + +� +γ +e(t)δφ(t, x) +δt +dl +� +1 + ( ˙Φ(t))2 ++ +� +R +ˆn0(x)φ(0, x)dx + e(0)φ(0, x) = 0, +∞ +� +0 +� +R +� +( ˆnV 2 +2 ++ E2)φt + ˆnV 3 +2 +φx +� +dxdt + +� +γ +e(t)( ˙Φ(t))2 +2 +δφ(t, x) +δt +dl +� +1 + ( ˙Φ(t))2 ++ +� +R +� ˆn0(x)(V 0(x))2 +2 ++ (E0(x))2 +� +φ(0, x)dx + e(0)( ˙Φ(0))2 +2 +φ(0, 0) = 0, + +THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +9 +where +� +γ +·dl is the curvilinear integral along the curve γ, the delta-derivative δφ(t,x) +δt +�� +γ +is defined as the tangential derivative on the curve γ, namely +δφ(t, x) +δt +�� +γ = +�∂φ(t, x) +∂t ++ ˙Φ(t)∂φ(t, x) +∂x +�� +γ +� �� +γ = dφ(t, Φ(t)) +dt += +� +1 + ( ˙Φ(t))2 ∂φ(t, x) +dl +, +where l = (−ν2, ν1) = +(1, ˙Φ(t)) +√ +1+( ˙Φ(t))2 is a unit vector tangent to γ. +The action of the delta function δ(γ) concentrated on the curve γ on the test +function is defined according to [8], as +(δ(γ), φ(t, x)) = +� +γ +φ(t, x) +dl +� +1 + ( ˙Φ(t))2 +, +where φ(t, x) ∈ D(R × [0, ∞)). +4.2. Rankine-Hugoniot conditions for delta-shock waves (the Rankine- +Hugoniot deficit). +Theorem 1. Let the domain Ω ∈ R2 be divided by a smooth curve γt = {(t, x) : +x = Φ(t)} into the left and right sides Ω∓. Let the triple of distributions (V, E, n), +given as (22) - (24) and the curve γt be a strongly singular generalized solution for +the system (9). Then this solution satisfies the following analogue of the Rankine- +Hugoniot conditions +d +dte(t) += +� +−[ˆnv] + [ˆn] ˙Φ(t) +� �� +x=Φ(t), +(25) +d +dt +e(t)( ˙Φ(t))2 +2 += +� +− +� ˆnv3 +2 +� ++ +� ˆnv2 + E2 +2 +� +˙Φ(t) +� �� +x=Φ(t). +(26) +The proof of the first statement, (25), is contained in [15], the proof of (26) +repeats the proof of the analogue of the Rankine-Hugoniot conditions for the energy +equation in the ”pressureless” gas dynamics model [11]. Let us briefly recall this. +We denote n = (ν1, ν2) = +( ˙Φ(t)),−1) +√ +1+( ˙Φ(t))2 the unit normal to the curve γt directed +from Ω− to Ω+. +Choose a test function φ(t, x) with support K ⊂ Ω. Then +∞ +� +0 +� +R +� +( ˆnV 2 +2 ++ E2)φt + ˆnV 3 +2 +φx +� +dxdt = +� +Ω−∩K +� +( ˆnV 2 +2 ++ E2)φt + ˆnV 3 +2 +φx +� +dxdt + +� +Ω+∩K +� +( ˆnV 2 +2 ++ E2)φt + ˆnV 3 +2 +φx +� +dxdt. +Integration by parts by the second equation (9) gives +� +Ω±∩K +� +( ˆnV 2 +2 ++ E2)φt + ˆnV 3 +2 +φx +� +dxdt = − +� +Ω±∩K +� +( ˆnV 2 +2 ++ E2)t + ( ˆnV 3 +2 )x +� +φ dxdt ∓ +� +γt +� +ν2( ˆn±(V±)2 +2 ++ (E±)2) + ν1 +ˆn±(V±)3 +2 +� +φ(t, x)dl − +� +Ω±∩K∩R +� ˆn0(x)(V 0(x))2 +2 ++ (E0)2 +� +φ(0, x)dx. + +10 +OLGA S. ROZANOVA* +Thus, +∞ +� +0 +� +R +� +( ˆnV 2 +2 ++ E2)φt + ˆnV 3 +2 +φx +� +dxdt + +� +Ω±∩K∩R +� ˆn0(x)(V 0(x))2 +2 ++ (E0)2 +� +φ(0, x)dx = +− +� +γt +�� ˆnV 2 +2 ++ E2 +� +ν2 + +� ˆnV 3 +2 +� +ν1 +� +φ(t, x)dl. +Further, +� +γ +e(t)( ˙Φ(t))2 +2 +δφ(t, x) +δt +dl +� +1 + ( ˙Φ(t))2 += +(27) +− +� +γ +δ +δt +� +e(t)( ˙Φ(t))2 +2 +� +φ(t, x) +dl +� +1 + ( ˙Φ(t))2 +− e(0)( ˙Φ(0))2 +2 +φ(0, 0). +Adding (27) and (27), taking into account (25), we get +� +γt +�� ˆnV 2 +2 ++ E2 +� +ν2 + +� ˆnV 3 +2 +� +ν1 − δ +δt +� +e(t)( ˙Φ(t))2 +2 +�� +φ(t, x)dl = 0 +for any φ ∈ D(Ω). This implies (26). +We see that the generalized Rankine-Hugoniot conditions are a system of second- +order ordinary differential equations, therefore, to solve the Cauchy problem (4), +(4) (with the divergent form (9)), (22) , (23) should set the initial velocity of the +shock position ˙Φ(0). +Since the system (4) has coinciding eigenvalues λ1(V ) = λ2(V ) = V , the admis- +sibility condition for a singular shock wave coincides with the geometric entropy +condition: +min{V−, V+} ≤ ˙Φ(t) ≤ max{V−, V+}, +(28) +meaning that characteristics from both sides come to the shock. +As we will see below, this condition allows us to obtain a condition on the +derivative at an intermediate point and construct a solution to the Riemann problem +in a unique way. +In addition, in this problem, a final point arises, where the +trajectory of the delta-shaped singularity must come, which also determines the +problem. +4.3. Mass and energy transfer ratios for a singular shock. Suppose that +V, E is a compactly supported classical solution of the (4) system. Then, according +to (9), the total mass +� +R +n(t, x)dx +and the total energy +1 +2 +� +R +(n(t, x)V 2(t, x) + E2(t, x)) dx +are conserved. Note that the total energy consists of kinetic and potential parts. +Let us show that in order to obtain analogs of these conservation laws for a strongly + +THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +11 +singular solution, it is necessary to introduce the mass and energy concentrated on +the shock. Suppose that the line of discontinuity x = Φ(t) is a smooth curve. +We denote +M(t) = +Φ(t) +� +−∞ +n(t, x)dx + ++∞ +� +Φ(t) +n(t, x)dx, +Ek(t) = 1 +2 + + + +Φ(t) +� +−∞ +n(t, x)V 2(t, x) dx + ++∞ +� +Φ(t) +n(t, x)V 2(t, x) dx + + + , +Ep(t) = 1 +2 ++∞ +� +−∞ +E2(t, x) dx, +the mass, kinetic and potential energies concentrated outside the shock. We inter- +pret the amplitude e(t) and the term 1 +2e(t)( ˙Φ(t))2 as the mass m(t) and kinetic +energy w(t) concentrated on the shock. +Theorem 2. Let the solution (22) - (24) be a strongly singular solution to the +system (9). Then the following relations of balance take place: +˙m(t) = − ˙ +M(t), +˙w(t) = − ˙E(t), +(29) +M(t) + m(t) = M(0) + m(0), +E(t) + w(t) = E(0) + w(0). +(30) +Proof. Both equalities (29) can be proved in the same way. Let us prove, for +example, the first of them. Because +˙ +M(t) = [ˆn] +�� +x=Φ(t) ˙Φ(t) + + + + +Φ(t) +� +−∞ ++ ++∞ +� +Φ(t) + + + nt(t, x)dx = +[ˆn] +�� +x=Φ(t) ˙Φ(t) − + + + +Φ(t) +� +−∞ ++ ++∞ +� +Φ(t) + + + (n(t, x)V (t, x))xdx = +[ˆn] +�� +x=Φ(t) ˙Φ(t) − [ˆnV ] +�� +x=Φ(t). +Together with (25) this equality shows that +˙m(t) + +˙ +M(t) = 0, +whence the first equalities in (29) and (30) follow. +4.4. Constructing a strongly singular shock wave for piecewise constant +initial data (7). We proceed to constructing a strongly singular solution in our +particular case. +Since in this situation ˆn = 1 and, accordingly, [ˆn] = 0, the equations according to +which it is possible to determine the amplitude and location of a strongly singular +shock wave are greatly simplified and take the form +˙e(t) += +−[V ] +�� +x=Φ(t), +(31) +d +dte(t)( ˙Φ(t))2 += +� +− +� +V 3� ++ +� +V 2 + E2� ˙Φ(t) +� �� +x=Φ(t). +(32) + +12 +OLGA S. ROZANOVA* +Since the values of V±(t) and E±(t) are known (see (12)), the values of jumps can +be calculated directly: +[V 3] +�� +x=Φ(t) = (([E]0)3 − 3[EV 2]0) cos2 t sin t ++(([V ]0)3 − 3[E2V ]0) cos3 t + 3[EV 2]0 cos t − ([E]0)3 sin t, +and +� +V 2 + E2� �� +x=Φ(t) = +� +V 2 + E2�0 �� +x=x0 = K = const. +Therefore, from (25) we find +e(t) = −[V ]0 sin t − [E]0 cos t. +(33) +Note that for all t for which a shock wave exists, for e(0) > 0 the amplitude e(t) +remains positive. +On the interval (0, T∗) there is a point t∗ at which V−(t∗) = V+(t∗) := U. Then +from the admissibility condition (28) we have V−(t∗) = ˙Φ(t∗) = V+(t∗) = U. It can +be readily found that +T∗ = 2 arctan [V ]0 +[E]0 , +t∗ = T∗ +2 , +U = +E0 +−V 0 ++ − E0 ++V 0 +− +� +([V ]0)2 + ([E]0)2 . +We denote ( ˙Φ(t) = q and ( ˙Φ(t))2 = Q(t). From (32) we get the Cauchy problems +˙q += +− +� +V 3� ++ Kq − ˙eq2 +2eq +, +q(t∗) = U, +(34) +and +˙Q += +− +� +V 3� +e ++ sign U K +e +� +Q − ˙e +eQ, +Q(t∗) = U 2. +(35) +The solutions to problems (34), (35) cannot be found explicitly, however, they +always exist for q ̸= 0. If at a certain point t = t0 ∈ (0, T∗), we have q → 0, then, +as follows from (34),(35), ˙q → ∞, +˙Q → c = const ̸= 0. Indeed, if c = 0, then +� +V 3� +|t=t0 = 0. Nevertheless, it is easy to check that +� +V 3� += 0 if and only if t = t∗. +This implies that q = O( +� +|t − t0|), as t → t0 ̸= t∗. +Thus, if there exists a point t = t0 ∈ (0, T∗) such that Q(t0) = 0, we first find +the unique solution to (35) at the first segment (0, t0) or (t0, T∗) (the segment must +contain t∗), and then find the unique solution to the Cauchy problem +˙Q += +− +� +V 3� +e +− sign U K +e +� +Q − ˙e +eQ, +Q(t∗) = 0, +on the second segment. Then we find q on both segments. +Let us note if Φ(t) has an extremum on (0, T∗), then Φ(t) ∈ C1(0, T∗) and can +be found uniquely, however, ¨Φ(t0) does not exist. +4.5. Examples. 1. +We start with the case (15), when the system (16) can be +reduced to one equation and one of the possible conservative form is +Vt + (V 2 +2 )x = −σ +� +C2 − V 2, +σ = sign(−Vx) = ±1, +which does nor require an introduction of a singular shock. The position of a usual +shock is defined by the Rankine-Hugoniot condition and gives +(36) +˙Φ(t) = V−(t) + V+(t) +2 +. + +THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +13 +Let us choose the initial data as +V 0 +− = 1, V 0 ++ = 0, E0 +− = 0, E0 ++ = −1. +Here T∗ = π +2 , K = U = 0 and e(0) = 1. +Fig.3, left, presents the behavior of the velocity ˙Φ(t) of the singular shock satis- +fying the geometric entropy condition (28) (solid), in comparison with the velocity +of shock based on the Rankine-Hugoniot condition (36) (dash). One can see that +the difference is very small. +Fig.3, center, presents the position of the singular +shock between characteristics (solid), in comparison with the Rankine-Hugoniot +shock (dash), the difference is almost negligible. Fig.3, right, shows the zoom of +this difference near the origin. +Figure 3. Velocity (left) and position (center and right) of the +singular shock (solid) vs. the velocity and position of the Rankine- +Huhoniot shock (dash) for Example 1. +2. The next example is for the following data: +V 0 +− = 1, V 0 ++ = 0.5, E0 +− = 1, E0 ++ = 0.9. +Here T∗ = 2.746801534, U = −.7844645404, K = −.94 and e(0) = 0.1. +This +example is interesting, since ˙Φ(t) changes the sign at a point t0 = 0.69174927 ̸= t∗. +Fig.4, left, presents the behavior of the velocity ˙Φ(t) of the singular shock satisfying +the geometric entropy condition (28), Fig.2, right, presents the position of the +singular shock between characteristics. +Figure 4. +Velocity (left) and position (right) of the singular +shock for Example 2. + +X-(t) +*L +to +(0)+XV-(t) +0.5 +t* +T* +0 +V+(t)0.005 +x(t) +0 +0.01x(t) +X-(t) +x+(t) +t* +T*V_(t) +V+(t) +0 +t$ +T*14 +OLGA S. ROZANOVA* +The solution in the examples are found numerically by means of the Runge- +Kutta-Fehlberg method of fourth-fifth order. +5. Discussion +1. We show that the reduced equations of a cold plasma provide the simplest +example of an inhomogeneous system in which the solution of the Riemann problem +consists of a rarefaction wave and a shock wave periodically replacing each other. +The system is not so interesting from a physical point of view, since it is believed +that the cold plasma equations are valid only for a smooth solution [3]. However, +they are extremely interesting mathematically. Indeed, firstly, due to the non-strict +hyperbolicity of the system, one can construct an example of non-uniqueness of the +rarefaction wave. Second, the natural conservative form of the system can be used +to construct a singular shock wave. +2. The solution of the Cauchy problem (4), (7) can be rewritten in terms of the +solution of the Euler-Poisson equation (5) with discontinuous data (n0, V0). +3. A similar procedure for solving the Riemann problem can also be applied to +other non-strict hyperbolic systems written initially in a non-divergent form. The +method consists in introducing an “artificial density”, which makes it possible to +write the system in a conservative form and define a strong singular solution. +4. The appearance of multiple rarefaction waves was noticed earlier in other +models, for example, [9]. +5. The presence of pressure apparently prevents the existence of a strong singular +solution [7], in other words, the situation is similar to the influence of pressure in +the gas dynamics model without pressure. +6. It should be noted that there are different plasma models and a huge amount +of literature devoted to shock waves there. One of the popular models is the Vlasov- +Maxwell system, which describes a collisionless ionized plasma [10], [14], [12], [2]. +Another assumption about plasma naturally changes the properties of shock waves. +7. The non-strictly hyperbolic system considered here has a simple wave solution +(an invariant manifold), what makes them related to systems of the Temple class +[16]. However, the most interesting feature of the solution of the Riemann problem +for the system of cold plasma equations are rooted in its inhomogeneity, while the +equations of the Temple class are homogeneous and have constant states to the left +and right of the shock wave or rarefaction wave. +Acknowledgements +Supported by the Moscow Center for Fundamental and Applied Mathematics un- +der the agreement 075-15-2019-1621. The author thanks her former student Darya +Kapridova for numerical calculations confirming the construction of a rarefaction +wave in the case of a simple wave solution.. +References +[1] A.F. Alexandrov, L.S. Bogdankevich, and A.A. Rukhadze, “Principles of plasma electrody- +namics,” Springer series in electronics and photonics, Springer, Berlin Heidelberg, 1984. +[2] A. Balogh, R.A. Treumann, “Physics of Collisionless Shocks”, ISSI Scientific Report Series, +12, Springer, 2013. +[3] E.V. Chizhonkov, +“Mathematical Aspects of Modelling Oscillations and Wake Waves in +Plasma” (CRC Press, 2019). +[4] R. C. Davidson, “Methods in nonlinear plasma theory”, Acad. Press, New York, 1972. + +THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA +15 +[5] S. Engelberg, H. Liu, E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana +University Mathematics Journal, Vol.50 (2001), 109–157. +[6] A.A. Frolov, E.V. Chizhonkov, Application of the energy conservation law in the cold plasma +model. Comput. Math. and Math. Phys. 60, 498–513 (2020). +[7] S.S. Ghoshal, B. Haspot, A. Jana, Existence of almost global weak solution for the Euler- +Poisson system in one dimension with large initial data, arXiv:2109.13182 +[8] R.P. Kanwal, “ Generalized Functions: Theory and technique”, Birkha¨user, Boston, Basel, +Berlin, 1998. +[9] A.A. Mailybaev, D. Marchesin, Hyperbolic singularities in rarefaction waves, +Journal of +Dynamics and Differential Equations, 20 (2008), 1–29. +[10] C.S. Morawetz, Magnetohydrodynamic shock structure without collisions, Physics of Fluids, +4 988–1006 (1961). +[11] B. Nilsson, O.S. Rozanova, V.M. Shelkovich, Mass, momentum and energy conservation laws +in zero-pressure gas dynamics and δ-shocks: II, Applicable Analysis, 90(5), 831–842 (2011). +[12] R.V. Polovin, Laminar theory of shock waves in plasma in the absence of collisions, Nuclear +Fusion, 4 (1) (1964). +[13] O.S. Rozanova, E.V. Chizhonkov, +On the conditions for the breaking of oscillations in a cold +plasma, Z. Angew. Math. Phys. 72, 13 (2021). +[14] J. Schaeffer, A restriction on shocks in collisionless plasma, SIAM J. Math. Anal., 46 (4) +2767–2797 (2014). +[15] V.M. Shelkovich, δ and δ′-shock wave types of singular solutions of systems of conservation +laws and transport and concentration processes”, Russian Math. Surveys, 63(3), 473–546 +(2008). +[16] B. Temple, Systems of conservation laws with invariant submanifolds, Transactions of the +American Mathematical Society 280 (2) 781–795 (1983). +Mathematics and Mechanics Department, Lomonosov Moscow State University, Lenin- +skie Gory, Moscow, 119991, Russian Federation, rozanova@mech.math.msu.su + diff --git a/79E2T4oBgHgl3EQflQc0/content/tmp_files/load_file.txt b/79E2T4oBgHgl3EQflQc0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4b92080b380d0fc9bbf7a125ddae51055362e6b --- /dev/null +++ b/79E2T4oBgHgl3EQflQc0/content/tmp_files/load_file.txt @@ -0,0 +1,406 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf,len=405 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='03986v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='AP] 10 Jan 2023 THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' A solution of the Riemann problem is constructed for a nonstrictly hyperbolic inhomogeneous system of equations describing one-dimensional cold plasma oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Each oscillation period includes one rarefaction wave and one shock wave containing a delta singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The rarefaction wave can be constructed in a non-unique way, the admissibility principle is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Introduction In vector form, the system of hydrodynamic of electron liquid, together with Maxwell’s equations, has the form: (1) nt + div (nV) = 0 , Vt + (V · ∇) V = e m � E + 1 c [V × B] � , 1 c Et = −4π c enV + rot B , 1 cBt = −rot E , div B = 0 , where e, m are the charge and mass of the electron (here the electron charge has a negative sign: e < 0), c is the speed of light;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' n, V are the density and velocity of electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' E, B are the vectors of electric and magnetic fields, x ∈ R3, t ≥ 0, ∇, div, rot are the gradient, divergence and vorticity with respect to the spatial variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The system of equations (1) is one of the simplest models of plasma, which is often called the equations of hydrodynamics of ”cold” plasma, it is well known and described in sufficient detail in textbooks and monographs (see, for example, [1], [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This system has an important subclass of solutions, dependent only on one space variable x, for which V = (V, 0, 0), E = (E, 0, 0), B ≡ 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In dimensionless form it can be written as (2) nt + (n V )x = 0, Vt + V Vx = −E, Et = n V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Assume that the solution is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then the first and last equations (2) imply (n + Ex)t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' For the background density n ≡ 1 we get (3) n = 1 − Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This allows us to obtain a hyperbolic system for the two components of the velocity V and the electric field E in the form (4) Vt + V Vx = −E, Et + V Ex = V, 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Primary 35Q60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Secondary 35L60, 35L67, 34M10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Quasilinear hyperbolic system, Riemann problem, non-uniqueness, singular shock, plasma oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 1 2 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* where (V, E) = (V (t, x), E(t, x)), t ∈ R+, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The density n(t, x) > 0 can be found from (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' System (4), (3) can be also rewritten as a pressureless repulsive Euler-Poisson system [5] nt + (nV )x = 0, Vt + V Vx = ∇Φ, ∆Φ = n − n0, n0 = 1, (5) where Φ is a repulsive force potential, ∇Φ = −E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' For (4) we consider the Cauchy problem (6) (V, E)|t=0 = (V0(x), E0(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' If the initial data are C1 - smooth functions, then locally in t there exists a smooth solution of (4), (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Nevertheless, it is known that the derivatives of the solution of such a Cauchy problem can go to infinity for a finite time, which corresponds to the formation of a shock wave, the criterion for the formation of a singularity is known [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Thus, it makes sense to consider piecewise-smooth functions as the initial data (6), the simplest example of which is the Riemann initial data (7) (V, E)|t=0 = (V 0 − + [V ]0Θ(x), E0 − + [E]0Θ(x)), where Θ(x) is the Heaviside function, constants (V−, E−) are the values to the left of the jump, ([V ], [E]) the values to the jumps, (V+ = V− + [V ], E+ = E− + [E]) are the values to the right of the jump, (v0 ±, E0 ±), ([v]0, [E]0) are the corresponding values at time zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In this case, the density at the initial moment of time is (8) n|t=0 = 1 − [E]0δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Since the initial data contain a delta function, the Riemann problem for the com- ponents of the solution (V, E, n) is singular and the Rankine-Hugoniot conditions cannot be written in the traditional form [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In order to ensure that the density is positive initially, it is necessary to impose the condition [E]0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' To construct the shock, we write system (4) in the divergent form (9) nt + (V n)x = 0, �nV 2 2 + E2 2 � t + �nV 3 2 � x = 0, corresponding to the laws of conservation of mass and total energy (for example, [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' System (9) (together with (3)) is equivalent to (4), (3) for smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The Riemann problem (9), (3), (7), (8) is completely non-standard and demon- strates new phenomena in the construction of both a rarefaction wave and a shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The difficulty in constructing a solution is associated, in particular, with the fact that system (4) is inhomogeneous and does not have a constant stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' To the left and right side of the discontinuity, the solution is a 2π - periodic function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This leads to the fact that the rarefaction wave and the shock wave periodically replace each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Further, system (4) is hyperbolic, but not strictly hyperbolic: it has the form ut + A(u)ux = f(u), u = (V, E), f = (−E, V ), the matrix A has a complete set of eigenvectors with coinciding eigenvalues λ1 = λ2 = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Because of this, it has a subclass of solutions in the form of simple waves, distinguished by the condition V 2 + E2 = C2 THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA 3 with a given constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We show that this leads to the non-uniqueness of the rarefaction wave for the Riemann problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Therefore, the question arises about the principles by which one can single out the ”correct” solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In our work, the correct one is chosen for which the total energy density is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' When constructing a singular shock wave, we use homogeneous conservative system of two equations (9), which are linked by the differential relation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This formulation has not been encountered before, although a modification of the method previously used for the case of equations of the pressureless gas dynamics with energy [11] can be used to construct a solution to the Riemann problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The shock wave satisfies the so-called “supercompression” conditions, which are traditionally used to distinguish admissible singular shock waves [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2 we discuss the structure of charac- teristics which is crucial for construction of rarefaction and shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='3 we construct the rarefaction wave for Riemann data (7) of a general form and then show that for the data corresponding to a simple wave the rarefaction can be con- structed non-uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We also propose two variational conditions of admissibility of the rarefaction waves for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='4 we give a definition of the strongly singular solution for an arbitrary piecewise smooth initial data, prove an analog of the Rankine-Hugoniot conditions (Theorem 1), study the mass and energy transfer for a singular shock wave (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then we construct the singular shock for piecewise smooth initial data (7) and give two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The first example corre- sponds to the case of simple wave, here we compare the result obtained starting from conservative form (9) and the result obtained from a divergence form, natural for the Hopf equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The second example show how it is possible to construct the shock in the case where the shock position has an extremum on the characteristic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='5 contains a discussion about a physical and mathematical sense of the results obtained and mention works concerning shock waves in plasma for other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Characteristics The equations for the characteristics corresponding to system (4) have the form (10) dV dt = −E, dE dt = V, dx dt = V, whence, first, it follows that along the characteristics (11) d(V 2 + E2) dt = 0, and also, according to (7), V±(t) = −E0 ± sin t + V 0 ± cos t, E±(t) = V 0 ± sin t + E0 ± cos t, x±(t) = V 0 ± sin t + E0 ±(cos t − 1) + x0, x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' It is easy to see that for [E]0 ̸= 0 the characteristics x−(t) and x+(t), corresponding to the states to the left and to the right of the discontinuity, intersect once inside each period 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Therefore, on that part of the period where x−(t) < x+(t), it is necessary to construct a continuous solution, and on the part where x−(t) > x+(t), that is, there is an intersection of the characteristics, we construct a shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The moment of time at which x−(t) = x+(t), we denote by T∗, T∗ ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='1 gives a schematic representation of the behavior of the characteristics, where the rarefaction wave comes first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Note that (11) implies that the value C2 = V 2 + E2 is constant for each specific characteristic, but in general it is a function of t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Characteristics and their intersections: rarefaction waves and shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Construction of a rarefaction wave Suppose the initial data is such that V 0 − < V 0 +, that is, x−(t) < x+(t), and first the initial data (7) generate a rarefaction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Between the characteristics x−(t) and x+(t), it is necessary to construct a continuous solution (V, E) connecting the states (V−(t), E−(t)) and (V+(t), E+(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Recall that the moment of time at which x−(t) = x+(t), we denote by T∗, T∗ ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The rarefaction wave, of course, is not a smooth solution, it satisfies the conser- vative system (9) with the additional condition (3) in the usual sense of the integral identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The linear profile solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' It is easy to check that a continuous solution (V, E) can be constructed by joining the states (V−(t), E−(t)) and (V+(t), E+(t)), between characteristics with the help of functions linear in x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (12) (V, E) = \uf8f1 \uf8f2 \uf8f3 (V−(t), E−(t)), x < x−(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (Vr1, Er1) = (a(t)x + b(t), c(t)x + d(t)), x ∈ [x−(t), x+(t)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (V+(t), E+(t)), x > x+(t), with (13) a(t) = −[E]0 sin t + [V ]0 cos t −[V ]0 sin t + [E]0(1 − cos t), c(t) = −[V ]0 sin t − [E]0 cos t −[V ]0 sin t + [E]0(1 − cos t), (14) b(t) = (V 0 +E0 − − E0 +V 0 −)(1 − cos t) −[V ]0 sin t + [E]0(1 − cos t), d(t) = (V 0 +E0 − − E0 +V 0 −) sin t −[V ]0 sin t + [E]0(1 − cos t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then n = 1 − c(t)χ(x−(t),x+(t)), 10: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='5 Shock Rarefaction 2,5 X 5 2,5 10 2,5THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA 5 where χ(x−(t),x+(t)) is the characteristic function of the interval (x−(t), x+(t)), for t ∈ (0, T∗) the density does not contain a delta function, but the singular component that was present in the initial data is again formed at t = T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Simple waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The system (4) has a subclass of solutions distinguished by the condition (15) V 2 + E2 = C2(≡ const) with a given constant C, the so called simple waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In this case, (4) reduces on smooth solutions to one equation (16) Vt + V Vx = −σ � C2 − V 2, σ = sign(−Vx) = ±1, E = σ � C2 − V 2, moreover, Vxx ̸= 0 on no set of positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The last requirement means that the solution cannot become constant on any interval, but at the points at which C2 = V 2 the value of σ changes its sign to the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The second conservation law (9) in this situation turns out to be a consequence of the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In the initial conditions (7) the values E0 − and E0 + are expressed as E0 − = ± � C2 − (V 0 −)2, E0 + = ± � C2 − (V 0 +)2, so as to ensure the condition [E]0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' It is easy to see that a function of the form (12) with an intermediate state (Vr1, Er1) is not a solution to the equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let us show that in this case another continuous solution can be constructed, with another function (Vr2, Er2) as an intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Indeed, the general solution (16), written implicitly, looks like x − σ � C2 − V 2 = F � t + arctan V σ √ C2 − V 2 � , with an arbitrary smooth function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In order to find the function F corresponding to the initial data (7), (15), we will construct the function X(t, V ) inverse to Vr2(t, x) for every fixed t ∈ (0, T∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' For t = 0 such a function is multivalued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We require that for t = 0 the condition X(0, V ) = 0 holds for V ∈ (V 0 −, V 0 +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then F = − tan √ C2−ξ2 ξ , ξ = σ √ C2 − V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' After transformations, we get X1(t, V ) = C(cos q − cos(q + t)), (V±)t < 0 (σ = 1), (17) X2(t, V ) = C(− cos q + cos(q − t)), (V±)t > 0 (σ = −1), (18) q = arcsin V C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Note that in each case the monotonicity of V in x ensures the existence of an inverse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The situation is considered separately when the behavior of the solution between the right and left characteristics is given by different formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Namely, consider the time T1 at which V ′ −(t) = 0 and the time T2 at which V ′ +(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Between T1 and T2 there is a moment T0, at which V+(t) = V−(t), and therefore, the jump disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' However, at such a point the characteristics do not intersect, that is, x+(T0) ̸= x−(T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' To construct a continuous solution in such a situation, we need auxiliary curves X1(t, q−) and X2(t, q+), where q± = arcsin V 0 ± C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 6 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* Then for t ∈ (0, T∗) the continuous solution of problem (7), (16), (15) can be written as (19) Vr2(t, x) = \uf8f1 \uf8f2 \uf8f3 V−(t), x < X−(t), Vi(t, x), X−(t) < x < X+(t), V+(t), x > X+(t), where X−(t) = � x−(t), t < T1, t > T0 X2(t, q+), t ∈ [T1, T0] , X+(t) = � x+(t), t < T0, t > T2 X1(t, q−), t ∈ [T0, T2] , and Vi(t, x) is the function inverse to Xi(t, V ), i = 1, 2, given by formulas (17), (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Thus, a continuous solution to the problem (4), (7), (15) can be constructed as (20) (V, E) = \uf8f1 \uf8f2 \uf8f3 V−(t), x < x−(t), (Vr2, Er2), x−(t) < x < x+(t), v+(t), x > x+(t), where (Vr2, Er2), where Vr2 is given by (19), and Er2 = ± � C2 − V 2 r2, the sign matches the one that was selected in the initial data (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2 presents the con- struction of the rarefaction wave on the characteristic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Characteristics and and values V− and V+ from the left and right side of the rarefaction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Nonuniqueness of rarefaction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Obviously, (12) and (21) are different continuous solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Moreover, on their basis it is possible to construct an infinite number of other rarefaction waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Indeed, one can check that Vr2 is an upward con- vex function and, for t = t1 ∈ (0, T∗), we can choose any point x1 ∈ (x−(t1), x+(t1)) and replace on the segment (x−(t1), x1) by a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Next, we find the po- sition of the right point of the linear segment as a solution to the problem for t ∈ (0, T∗) as ˙x = Vr2(t, x), x(t1) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Such linear sections can be built in any number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' t 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='6 T1 To T2 V+ V1 + 2 V V 3 x+ V2 4 X-0 3 t 1 V- x+ 2 3 4 x- 5THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Admissibility of the rarefaction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The question of choosing the “cor- rect” continuous solution can be solved proceeding from the minimality of the total energy of the rarefaction wave E(t) = 1 2 x+(t) � x−(t) (nV 2 + E2) dx, see (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' For the solution (Vr2, Er2) E(t) = 1 2 x+(t) � x−(t) ((1 − Ex)(C2 − E2) + E2) dx = 1 2(C2∆x − C2[E] + 1 3[E3]), where ∆x = x+(t)−x−(t) ≥ 0, [E] = E+ −E− = ∆x+[E]0, [E3] = (E+)3 −(E−)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' For the solution (Vr1, Er1) E(t) = 1 2 x+(t) � x−(t) ((1 − c)(ax + b)2 + (cx + d)2) dx, where a, b, c, d given as (13), (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' It can be readily computed that E(Vr2, Er2) − E(Vr1, Er1) = −1 6[E]0(C2 − (E0 +E0 − + V 0 +V 0 −) = − 1 12[E]0(([E]0)2 + ([V ]0)2) ≥ 0, t ∈ (0, T∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Here we take into account [E]0 ≤ 0 and (E+)2 + (V+)2 = (E−)2 + (V−)2 = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Thus, if [E]0 < 0, for reasons of less energy E we have to choose (Vr1, Er1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Another way to distinguish an acceptable rarefaction wave is the pointwise minimality of the local energy E(t, x) = V 2 + E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Indeed, E(Vr2, Er2) = V 2 r2 + E2 r2 = C2 is constant by the construction of the solution, whereas E(Vr1, Er1) = (ax + b)2 + (cx + d)2 has a minimum x∗(t) = − ab+cd 2(a2+c2) ∈ (x−(t), x+(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Since E(Vr1, Er1) = C2 at x = x±(t), then E(Vr1, Er1) < C2 = E(Vr2, Er2), x(t) ∈ (x−(t), x+(t)), t ∈ (0, T∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Both principles, on the basis of which admissible solutions can be distinguished, lead to the same conclusion: for the complete system (4), the solution (Vr1, Er1) must be chosen as a rarefaction wave, while when the condition (15) is applied, only the possibility (Vr2, Er2) remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 8 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Construction of a singular shock wave We need to build a shock wave for the second part of the period 2π, t ∈ (T∗, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' However, in order not to complicate the notation, we, without loss of generality, shift the time point T∗ to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Thus, we are in a situation where the initial data correspond to a shock wave and t = T∗ is the point of the first intersection of the characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Suppose that for t ∈ (0, T∗) we have constructed a solution to the problem as (21) (Vs, Es) = � V−(t), x < Φ(t), V+(t), x > Φ(t), that is, we found the position of the shock wave x = Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then the density can be found as n(t, x) = 1 − [E]|x=Φ(t)δ(x − Φ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Thus, we must take into account the presence of a strongly singular component of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' However, before proceeding to the construction of a solution in this case, we will give a general definition of a strongly singular solution and obtain its main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Definition of a generalized strongly singular solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Starting from the divergent form (9), we define a generalized strongly singular solution to the problem (9), (6) according to [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let V (t, x) = V−(t, x) + [V (t, x)]|x=Φ(t)Θ(x − Φ(t)), (22) E(t, x) = E−(t, x) + [E(t, x)]|x=Φ(t)Θ(x − Φ(t)), (23) n(t, x) = ˆn(t, x) + e(t)δ(x − Φ(t)), (24) where [f] = f+ − f−, f± are differentiable functions having one-sided limits, t ≥ 0, x ∈ R, ˆn(t, x) = 1−{Ex(t, x)}, {Ex} is the derivative of the function E at the points at which it exists in the usual sense, e(t) := e(t, Φ(t)), e(t) = −[E(t, x)]|x=Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The triple of distributions (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' given as (22) - (24) and the curve γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' given as x = Φ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Φ(0) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Φ(t) ∈ C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' is called a generalized singular solution of the problem (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' n)|t=0 = (V 0 −(x) + [V (x)]0Θ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' E0 −(x) + [E(x)]0Θ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' n0(x) = ˆn0(x) + e0δ(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' if for all test functions φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) ∈ D(R × [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ∞) ∞ � 0 � R ˆn(φt + V φx)dxdt + � γ e(t)δφ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) δt dl � 1 + ( ˙Φ(t))2 + � R ˆn0(x)φ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x)dx + e(0)φ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ∞ � 0 � R � ( ˆnV 2 2 + E2)φt + ˆnV 3 2 φx � dxdt + � γ e(t)( ˙Φ(t))2 2 δφ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) δt dl � 1 + ( ˙Φ(t))2 + � R � ˆn0(x)(V 0(x))2 2 + (E0(x))2 � φ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x)dx + e(0)( ˙Φ(0))2 2 φ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 0) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA 9 where � γ dl is the curvilinear integral along the curve γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' the delta-derivative δφ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='x) δt �� γ is defined as the tangential derivative on the curve γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' namely δφ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) δt �� γ = �∂φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) ∂t + ˙Φ(t)∂φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) ∂x �� γ � �� γ = dφ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Φ(t)) dt = � 1 + ( ˙Φ(t))2 ∂φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' x) dl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' where l = (−ν2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ν1) = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ˙Φ(t)) √ 1+( ˙Φ(t))2 is a unit vector tangent to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The action of the delta function δ(γ) concentrated on the curve γ on the test function is defined according to [8], as (δ(γ), φ(t, x)) = � γ φ(t, x) dl � 1 + ( ˙Φ(t))2 , where φ(t, x) ∈ D(R × [0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Rankine-Hugoniot conditions for delta-shock waves (the Rankine- Hugoniot deficit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let the domain Ω ∈ R2 be divided by a smooth curve γt = {(t, x) : x = Φ(t)} into the left and right sides Ω∓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let the triple of distributions (V, E, n), given as (22) - (24) and the curve γt be a strongly singular generalized solution for the system (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then this solution satisfies the following analogue of the Rankine- Hugoniot conditions d dte(t) = � −[ˆnv] + [ˆn] ˙Φ(t) � �� x=Φ(t), (25) d dt e(t)( ˙Φ(t))2 2 = � − � ˆnv3 2 � + � ˆnv2 + E2 2 � ˙Φ(t) � �� x=Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (26) The proof of the first statement, (25), is contained in [15], the proof of (26) repeats the proof of the analogue of the Rankine-Hugoniot conditions for the energy equation in the ”pressureless” gas dynamics model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let us briefly recall this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We denote n = (ν1, ν2) = ( ˙Φ(t)),−1) √ 1+( ˙Φ(t))2 the unit normal to the curve γt directed from Ω− to Ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Choose a test function φ(t, x) with support K ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then ∞ � 0 � R � ( ˆnV 2 2 + E2)φt + ˆnV 3 2 φx � dxdt = � Ω−∩K � ( ˆnV 2 2 + E2)φt + ˆnV 3 2 φx � dxdt + � Ω+∩K � ( ˆnV 2 2 + E2)φt + ˆnV 3 2 φx � dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Integration by parts by the second equation (9) gives � Ω±∩K � ( ˆnV 2 2 + E2)φt + ˆnV 3 2 φx � dxdt = − � Ω±∩K � ( ˆnV 2 2 + E2)t + ( ˆnV 3 2 )x � φ dxdt ∓ � γt � ν2( ˆn±(V±)2 2 + (E±)2) + ν1 ˆn±(V±)3 2 � φ(t, x)dl − � Ω±∩K∩R � ˆn0(x)(V 0(x))2 2 + (E0)2 � φ(0, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 10 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* Thus, ∞ � 0 � R � ( ˆnV 2 2 + E2)φt + ˆnV 3 2 φx � dxdt + � Ω±∩K∩R � ˆn0(x)(V 0(x))2 2 + (E0)2 � φ(0, x)dx = − � γt �� ˆnV 2 2 + E2 � ν2 + � ˆnV 3 2 � ν1 � φ(t, x)dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Further, � γ e(t)( ˙Φ(t))2 2 δφ(t, x) δt dl � 1 + ( ˙Φ(t))2 = (27) − � γ δ δt � e(t)( ˙Φ(t))2 2 � φ(t, x) dl � 1 + ( ˙Φ(t))2 − e(0)( ˙Φ(0))2 2 φ(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Adding (27) and (27), taking into account (25), we get � γt �� ˆnV 2 2 + E2 � ν2 + � ˆnV 3 2 � ν1 − δ δt � e(t)( ˙Φ(t))2 2 �� φ(t, x)dl = 0 for any φ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This implies (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We see that the generalized Rankine-Hugoniot conditions are a system of second- order ordinary differential equations, therefore, to solve the Cauchy problem (4), (4) (with the divergent form (9)), (22) , (23) should set the initial velocity of the shock position ˙Φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Since the system (4) has coinciding eigenvalues λ1(V ) = λ2(V ) = V , the admis- sibility condition for a singular shock wave coincides with the geometric entropy condition: min{V−, V+} ≤ ˙Φ(t) ≤ max{V−, V+}, (28) meaning that characteristics from both sides come to the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' As we will see below, this condition allows us to obtain a condition on the derivative at an intermediate point and construct a solution to the Riemann problem in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' In addition, in this problem, a final point arises, where the trajectory of the delta-shaped singularity must come, which also determines the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Mass and energy transfer ratios for a singular shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Suppose that V, E is a compactly supported classical solution of the (4) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then, according to (9), the total mass � R n(t, x)dx and the total energy 1 2 � R (n(t, x)V 2(t, x) + E2(t, x)) dx are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Note that the total energy consists of kinetic and potential parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let us show that in order to obtain analogs of these conservation laws for a strongly THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA 11 singular solution, it is necessary to introduce the mass and energy concentrated on the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Suppose that the line of discontinuity x = Φ(t) is a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We denote M(t) = Φ(t) � −∞ n(t, x)dx + +∞ � Φ(t) n(t, x)dx, Ek(t) = 1 2 \uf8eb \uf8ec \uf8ed Φ(t) � −∞ n(t, x)V 2(t, x) dx + +∞ � Φ(t) n(t, x)V 2(t, x) dx \uf8f6 \uf8f7 \uf8f8 , Ep(t) = 1 2 +∞ � −∞ E2(t, x) dx, the mass, kinetic and potential energies concentrated outside the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We inter- pret the amplitude e(t) and the term 1 2e(t)( ˙Φ(t))2 as the mass m(t) and kinetic energy w(t) concentrated on the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let the solution (22) - (24) be a strongly singular solution to the system (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then the following relations of balance take place: ˙m(t) = − ˙ M(t), ˙w(t) = − ˙E(t), (29) M(t) + m(t) = M(0) + m(0), E(t) + w(t) = E(0) + w(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Both equalities (29) can be proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let us prove, for example, the first of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Because ˙ M(t) = [ˆn] �� x=Φ(t) ˙Φ(t) + \uf8eb \uf8ec \uf8ed Φ(t) � −∞ + +∞ � Φ(t) \uf8f6 \uf8f7 \uf8f8 nt(t, x)dx = [ˆn] �� x=Φ(t) ˙Φ(t) − \uf8eb \uf8ec \uf8ed Φ(t) � −∞ + +∞ � Φ(t) \uf8f6 \uf8f7 \uf8f8 (n(t, x)V (t, x))xdx = [ˆn] �� x=Φ(t) ˙Φ(t) − [ˆnV ] �� x=Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Together with (25) this equality shows that ˙m(t) + ˙ M(t) = 0, whence the first equalities in (29) and (30) follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Constructing a strongly singular shock wave for piecewise constant initial data (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We proceed to constructing a strongly singular solution in our particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Since in this situation ˆn = 1 and, accordingly, [ˆn] = 0, the equations according to which it is possible to determine the amplitude and location of a strongly singular shock wave are greatly simplified and take the form ˙e(t) = −[V ] �� x=Φ(t), (31) d dte(t)( ˙Φ(t))2 = � − � V 3� + � V 2 + E2� ˙Φ(t) � �� x=Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (32) 12 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* Since the values of V±(t) and E±(t) are known (see (12)), the values of jumps can be calculated directly: [V 3] �� x=Φ(t) = (([E]0)3 − 3[EV 2]0) cos2 t sin t +(([V ]0)3 − 3[E2V ]0) cos3 t + 3[EV 2]0 cos t − ([E]0)3 sin t, and � V 2 + E2� �� x=Φ(t) = � V 2 + E2�0 �� x=x0 = K = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Therefore, from (25) we find e(t) = −[V ]0 sin t − [E]0 cos t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (33) Note that for all t for which a shock wave exists, for e(0) > 0 the amplitude e(t) remains positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' On the interval (0, T∗) there is a point t∗ at which V−(t∗) = V+(t∗) := U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then from the admissibility condition (28) we have V−(t∗) = ˙Φ(t∗) = V+(t∗) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' It can be readily found that T∗ = 2 arctan [V ]0 [E]0 , t∗ = T∗ 2 , U = E0 −V 0 + − E0 +V 0 − � ([V ]0)2 + ([E]0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We denote ( ˙Φ(t) = q and ( ˙Φ(t))2 = Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' From (32) we get the Cauchy problems ˙q = − � V 3� + Kq − ˙eq2 2eq , q(t∗) = U, (34) and ˙Q = − � V 3� e + sign U K e � Q − ˙e eQ, Q(t∗) = U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' (35) The solutions to problems (34), (35) cannot be found explicitly, however, they always exist for q ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' If at a certain point t = t0 ∈ (0, T∗), we have q → 0, then, as follows from (34),(35), ˙q → ∞, ˙Q → c = const ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Indeed, if c = 0, then � V 3� |t=t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Nevertheless, it is easy to check that � V 3� = 0 if and only if t = t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This implies that q = O( � |t − t0|), as t → t0 ̸= t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Thus, if there exists a point t = t0 ∈ (0, T∗) such that Q(t0) = 0, we first find the unique solution to (35) at the first segment (0, t0) or (t0, T∗) (the segment must contain t∗), and then find the unique solution to the Cauchy problem ˙Q = − � V 3� e − sign U K e � Q − ˙e eQ, Q(t∗) = 0, on the second segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Then we find q on both segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Let us note if Φ(t) has an extremum on (0, T∗), then Φ(t) ∈ C1(0, T∗) and can be found uniquely, however, ¨Φ(t0) does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We start with the case (15), when the system (16) can be reduced to one equation and one of the possible conservative form is Vt + (V 2 2 )x = −σ � C2 − V 2, σ = sign(−Vx) = ±1, which does nor require an introduction of a singular shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The position of a usual shock is defined by the Rankine-Hugoniot condition and gives (36) ˙Φ(t) = V−(t) + V+(t) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' THE RIEMANN PROBLEM FOR EQUATIONS OF A COLD PLASMA 13 Let us choose the initial data as V 0 − = 1, V 0 + = 0, E0 − = 0, E0 + = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Here T∗ = π 2 , K = U = 0 and e(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='3, left, presents the behavior of the velocity ˙Φ(t) of the singular shock satis- fying the geometric entropy condition (28) (solid), in comparison with the velocity of shock based on the Rankine-Hugoniot condition (36) (dash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' One can see that the difference is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='3, center, presents the position of the singular shock between characteristics (solid), in comparison with the Rankine-Hugoniot shock (dash), the difference is almost negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='3, right, shows the zoom of this difference near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Velocity (left) and position (center and right) of the singular shock (solid) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' the velocity and position of the Rankine- Huhoniot shock (dash) for Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The next example is for the following data: V 0 − = 1, V 0 + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='5, E0 − = 1, E0 + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Here T∗ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='746801534, U = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='7844645404, K = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='94 and e(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' This example is interesting, since ˙Φ(t) changes the sign at a point t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='69174927 ̸= t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='4, left, presents the behavior of the velocity ˙Φ(t) of the singular shock satisfying the geometric entropy condition (28), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='2, right, presents the position of the singular shock between characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Velocity (left) and position (right) of the singular shock for Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' X-(t) L to (0)+XV-(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='5 t* T* 0 V+(t)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='005 x(t) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='01x(t) X-(t) x+(t) t* T*V_(t) V+(t) 0 t$ T*14 OLGA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' ROZANOVA* The solution in the examples are found numerically by means of the Runge- Kutta-Fehlberg method of fourth-fifth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Discussion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' We show that the reduced equations of a cold plasma provide the simplest example of an inhomogeneous system in which the solution of the Riemann problem consists of a rarefaction wave and a shock wave periodically replacing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The system is not so interesting from a physical point of view, since it is believed that the cold plasma equations are valid only for a smooth solution [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' However, they are extremely interesting mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Indeed, firstly, due to the non-strict hyperbolicity of the system, one can construct an example of non-uniqueness of the rarefaction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Second, the natural conservative form of the system can be used to construct a singular shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The solution of the Cauchy problem (4), (7) can be rewritten in terms of the solution of the Euler-Poisson equation (5) with discontinuous data (n0, V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' A similar procedure for solving the Riemann problem can also be applied to other non-strict hyperbolic systems written initially in a non-divergent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The method consists in introducing an “artificial density”, which makes it possible to write the system in a conservative form and define a strong singular solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The appearance of multiple rarefaction waves was noticed earlier in other models, for example, [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The presence of pressure apparently prevents the existence of a strong singular solution [7], in other words, the situation is similar to the influence of pressure in the gas dynamics model without pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' It should be noted that there are different plasma models and a huge amount of literature devoted to shock waves there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' One of the popular models is the Vlasov- Maxwell system, which describes a collisionless ionized plasma [10], [14], [12], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Another assumption about plasma naturally changes the properties of shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The non-strictly hyperbolic system considered here has a simple wave solution (an invariant manifold), what makes them related to systems of the Temple class [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' However, the most interesting feature of the solution of the Riemann problem for the system of cold plasma equations are rooted in its inhomogeneity, while the equations of the Temple class are homogeneous and have constant states to the left and right of the shock wave or rarefaction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Acknowledgements Supported by the Moscow Center for Fundamental and Applied Mathematics un- der the agreement 075-15-2019-1621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' The author thanks her former student Darya Kapridova for numerical calculations confirming the construction of a rarefaction wave in the case of a simple wave solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content=' Mathematics and Mechanics Department, Lomonosov Moscow State University, Lenin- skie Gory, Moscow, 119991, Russian Federation, rozanova@mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} +page_content='su' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E2T4oBgHgl3EQflQc0/content/2301.03986v1.pdf'} diff --git a/7dE4T4oBgHgl3EQfcww5/content/tmp_files/2301.05085v1.pdf.txt b/7dE4T4oBgHgl3EQfcww5/content/tmp_files/2301.05085v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b41b2f6e1c07170acec1bb9a7ecb54677f974092 --- /dev/null +++ b/7dE4T4oBgHgl3EQfcww5/content/tmp_files/2301.05085v1.pdf.txt @@ -0,0 +1,2030 @@ +Bandgaps of bent and buckled carbon nanotubes +Alex Kleiner +orcid.org/0000-0002-4694-2218 +(Dated: January 12, 2023) +Carbon nanotube’s large electro-mechanical coupling and robustness makes them attractive for +applications where bending and buckling is present. But the nature of this coupling is not well +understood. Existing theory treats only weak and homogeneous deformations. We generalize it and +derive close-form expressions for bandgaps under non-homogeneous deformation. The theory is first +compared with a number of published DFT simulations, and then applied to the specific case of +bending and buckling – where a kink is present. In the pre-buckling regime, bandgaps change ∝ ±κ4 +where κ is the bending curvature; inside the kink at post-buckling, while near criticality, the kink +is shallow and the gap is ∝ ±κ1/2, where the sign depends on the chiral integers. For a deeper kink +but still with an open cross-section, both the bandgap and local Fermi energy strongly downshift. +CONTENTS +I. Introduction +1 +II. Homogeneous deformation +2 +A. General formulation +2 +B. Small circumferential curvature +3 +C. Large circumferential curvature +3 +D. Strain +4 +E. Torsion +5 +III. Non-homogeneous deformation +6 +A. General formulation +6 +B. Elastic theory of bending and buckling +7 +C. Bending – pre-buckling +8 +D. Bending – critical curvature +8 +E. Bending – post-buckling +9 +IV. Summary +10 +V. Appendices +10 +A. Brazier’s Theory +10 +B. The fundamental bandgap +12 +C. Fermi energy and Cs +12 +References +14 +I. +INTRODUCTION +Three decades after its discovery, carbon nanotubes +has yet to realize its potential. A slow but continuous +progress in fabrication may, however, make feasible a +growing number of application ideas suggested early-on +but never pursued due to economical and fabrication con- +siderations. One such application class would be electro- +mechanics – as it relies on the coupling between two of +the tube’s unique features: one dimensional conductivity +and elasticity. +This coupling is most extreme in the kink of a buckled +tube. The kink’s size is of the order of the tube diame- +ter, but what is its electronic structure and how will it +affect transport as a function of further bending? Cur- +rently, the answer is far from being clear. The kink is a +point of large curvature and non-homogeneous deforma- +tion, and, as we show below, non of these is theoretically +well understood. +The elementary theory predicting bandgaps in car- +bon nanotubes is based on zone-folding of graphene [5]. +It classifies tubes as metallic or semiconducting – with +bandgaps ∝ 1/R according to their chiral vectors (ap- +pendix B). Corrections ∝ 1/R2 due to circumferential +curvature [6][7], predict that metallicity is lifted in all +but one type: the (achiral) armchair tubes. +An obvi- +ous prediction is the bandgaps would trend higher at +ever smaller radii. +But DFT computations (table II) +clearly show that the opposite is true: the narrowest +tubes are in-fact gapless; it is qualitatively attributed to +a curvature-induced hybridization of the σ−π bands, but +it has not been quantitatively incorporated into existing +theory. +This should also play a critical role in the kink, where +very large curvature is present. Thus, our first aim is +to understand the effects of large curvature quantita- +tively. But the other characteristic of the kink – its large +and non-homogeneous deformation, needs also to be ac- +counted for. +The modulation of bandgaps due to homogeneous de- +formation is well-known [8][9]. But that is not sufficient, +even for weak bending without kinks; the strain profile in +bending is asymmetric about the neutral line, i.e: pure +bending has no net strain; this would imply, according +to existing theory, no change in the bandgap – a clear +contradiction with simulations [10][11]. +Thus, in order to reconcile theory and simulations and +derive closed-form solutions for the bandgap in the kink, +we need, in addition to incorporating effects of large cur- +vature, extend the theory of homogeneous deformation +into the non-homogeneous domain. +For completeness, +however, we start by recapping the existing, homoge- +neous, theory of bandgaps in carbon nanotubes. +arXiv:2301.05085v1 [cond-mat.mes-hall] 12 Jan 2023 + +2 +Symbol +Definition +ϵ +axial strain +ζ +axial torsion +κθ +circumferential curvature +κ +bending curvature +κz +local axial curvature +κcr +bending curvature at buckling point +b = 3.5 eV/˚A dγ/dl, where l is bond length +γ = 2.7eV +π-orbitals overlap energy +n, m +nanotube’s chiral integers; m ≤ n +p, q +integers; n − m = 3q + p (B1) +v, w +in-plain and out-of-plain deformation vectors, respectively (appendix A) +ch +ach is the circumference; ch = +√ +n2 + nm + m2 +R +nanotube’s radius, 2πR = ach +ν = 0.19 +Poisson ratio [1]; ν = 0.34 according to [2]; see also [3], [4] +t = 0.66˚A +effective thickness [1]; 0.75˚A according to [2]; see also [3], [4] +α +chiral angle; cos 3α = (2n3 − 2m3 + 3n2m − 3nm2)/2c3 +h +ξ +ovalization parameter (eq. A3) +ξcr +critical ovalization – 2/9 +Eπ +g +bandgap due to the π−band alone +Epre +g +bandgap at the pre-buckling regime +Epost +g +bandgap at the post-buckling regime +Es +energy of the singlet-band above graphene’s Fermi level +Cs +coefficient for Es (appendix C); Cs = 8 ∼ 12eV ˚A2 +R10,0 +the radius of the (10,0) tube; R10,0 = 4˚A +Rc +upper critical radius; above it, unhybridized π-band structure is valid +Rv +lower critical radius; below it, bandgaps are zero +⃗KF +Fermi points +δky +lateral shift of the Fermi points +⃗l, ly, lz +nearest neighbor bond vectors and their lateral and axial projections +U +elastic energy per unit length +Table I: List of symbols and values. +II. +HOMOGENEOUS DEFORMATION +In this section we start by recasting the tight-binding +π-band theory of homogeneous deformation [7][8][9] in a +formalism that will be used here for non-homogeneous +fields; we then consider deviations from the pure π-band, +which become dominant at very small radii, or large cir- +cumferential curvatures. +A. +General formulation +The fundamental theory of bandgaps in carbon nan- +otubes, reproduced here in appendix B, gives increasingly +erroneous gaps as the nanotubes’ diameter decreases. +The reason lies in a symmetry of graphene that is broken +in nanotubes: isotropy of bonds. The assumption that +the three nearest overlap integrals are equal (B5), does +not fully hold in nanotubes, since neighboring π-orbitals +are not parallel, nor are neighboring atoms equally sepa- +rated. Discarding thus assumption (B5), we find the new +Fermi points by solving eq. (B3) while allowing small +changes to the overlap integrals δγj. The solution yields +[9], +δky = sgn[1 − 2p] +2πR +√ +3γ +× +(1) +� +(m − n)δγ1 + (2n + m)δγ2 − (n + 2m)δγ3 +� +. + +3 +where δky is the shift of the Fermi points along the cir- +cumferential coordinate y, relative to their zone-folding +position (B6). Now, knowing the δγj in (1), translates, +through the linear spectrum (B7) to the following gap +equation, +Eπ +g = aγ +R +���� +p +√ +3 + +√ +3Rδky +���� . +(2) +The superscript π serves to label it as a pure π-band, in +contrast to the hybridized band to be considered later. +B. +Small circumferential curvature +One of the symmetries present in graphene but not +in nanotubes is the parallel nature of the π-orbitals. In +nanotubes, neighboring π-orbitals are misaligned by a +small angle: if ⃗lj is the vector connecting them (eqs. B4), +and ljy is its y-component, the angle is βj = ljy/R. Then +γj → γ cos2 βj – which gives +δγj = −γ l2 +jy +2R2 . +(3) +Now plugging this in eqs. (1) and (2) yields +Eπ +g +γ += |p| +a +R +√ +3 + sgn[1 − 2p] a2 cos 3α +16R2 +, +(4) +where cos 3α is given in table I. Eπ +g includes two effects +over the π-band: the 1/R-rule of the fundamental gap, +and the 1/R2 correction due to curvature. Here, in con- +trast with the fundamental prediction, ¨metallic¨ tubes +(p = 0) do open small gaps, and their 1/R2 trend, as +given by eq. (4), was experimentally verified on a few +large zigzag tubes [17]. +C. +Large circumferential curvature +Predictions from the gap equation (4) can be compared +with MD-simulations: table (II) lists results from simula- +tions run by different groups, and fig. (1) compares them +with eq. 4 for zigzag tubes. Two opposite trends in fig. 1 +are evident: in tubes larger then (10,0), the gaps seen to +agree with eq. (4), but smaller tubes reverse this trend +and quickly reach metallicity. +The reason for this was pointed out early-on by Blase +et al. [18]; it concerns a singly-degenerate band (termed +below: S-band) which is a π∗ − σ∗ hybridization. +In +graphene, the S-band is positioned very high above Fermi +level; but since hybridization is ∝ 1/R2, the S−band +downshifts with decreasing radius, and at a certain radius +it crosses the conduction band. This radius was found +[13] by DFT to be the radius of the (10, 0) tube. i.e: in +this tube, the S and conduction bands overlap. Its radius +is, +R10,0 ≈ 4˚A. +(5) +At smaller radii, the S-band cuts through the gap (see +for example: (8,0) and (7,0) in fig. 1) until, at the small- +est radii, it vanishes entirely (see for example: (6,0), (5,0) +and (4,0) in table II). +Let Es be the energy, relative to Fermi level, of the +S−band at the K-points. Its 1/R2 dependence was found +by DFT computations to scale by a factor of Cs ∼ 8 +(ev·˚A2) [15] (see also appendix C). But the hybridization +near the K-points scales also with chirality ∝ cos 3α [19]. +Then, the functional form of Es is +Es = 1 +2Eπ +g (10, 0) − Cs +� +1 +R2 − +1 +R2 +10,0 +� +cos 3α, +(6) +where Eπ +g (10, 0) ≈ 0.93eV – is the un-hybridized bandgap +of the (10,0) tube. The radii, Rc and Rv, at which the +S−band cross the conduction and valence bands, respec- +tively, can be found by setting +Es − 1 +2Eπ +g = 0, +R = Rc, +(7) +Eπ +g − Cs +� +1 +R2 − +1 +R2 +10,0 +� +cos 3α = 0, +R = Rv. (8) +This yields +Rc = +1 +2ac +� +bc + +� +b2c + 4accc +� +, where +(9) +ac = Cs cos 3α +R2 +10,0 ++ Eπ +g (10, 0) +2 +, +bc = |p| γa +2 +√ +3, +cc = +� +Cs + sgn[1 − 2p]γa2 +32 +� +cos 3α, +and, +Rv = +1 +2av +� +−bv + +� +b2v + 4avcv +� +, where +(10) +av = Cs cos 3α +R2 +10,0 +, +bv = |p| γa +√ +3, +cv = +� +Cs − sgn[1 − 2p]γa2 +16 +� +cos 3α, +Eqs. 9, 10 yield three pairs of Rc, Rv – a pair for each +p. These radii thus delimit the bandgap regimes: large +tubes (R > Rc) have their bandgap expressed by the +usual π-band, Eπ +g (eq. 4); smaller tubes (Rv < R < Rc) +have the S-band between the valence and the conduction +bands, hence the bandgaps in this regime become +Eπ +g − Cs +� 1 +R2 − 1 +R2c +� +cos 3α +(11) +where Eπ +g is the non-hybridized (π-band) gap (4) and +CS ∼ 8 eV·˚A2 is a semi-empirical scaling factor (see ap- +pendix C). The smallest tubes (R ≤ Rv), on the other + +4 +n, m +ref. [12] +ref. [13] +ref. [14] +ref. [15] +ref. [16] +eqs. (12) +4, 0 +0 +0 +0 +0 +0 +0 +5, 0 +0 +0 +0 +0 +0 +0 +5, 1 +- +- +- +0.05 +0.13 +0.06 +5, 3 +- +- +- +1.2 +1.18 +1.12 +5, 4 +- +- +- +1.12 +- +1.14 +6, 1 +- +- +- +0.43 +0.41 +0.71 +6, 2 +- +- +- +0.7 +0.67 +0.74 +6, 4 +- +- +- +1.1 +1.09 +1.07 +7, 0 +0.66 +0.34 +0.19 +0.2 +0.21 +0.474 +8, 0 +1.05 +0.487 +0.73 +0.6 +0.59 +0.853 +10, 0 +0.96 +0.87 +0.88 +0.8 +0.77 +0.913 +11, 0 +1.1 +0.8 +1.13 +- +0.93 +0.945 +13, 0 +0.79 +- +0.73 +- +0.64 +0.714 +14, 0 +0.85 +- +0.9 +- +0.72 +0.733 +16, 0 +- +0.57 +0.61 +- +0.54 +0.586 +17, 0 +- +- +- +- +0.58 +0.599 +19, 0 +- +- +- +- +0.46 +0.497 +20, 0 +- +- +- +- +0.5 +0.506 +Table II: Bandgaps (in eV) of semiconducting tubes. +hand, are metallic since the S-band crossed the valence +band. The bandgaps are then given by, +Eg = +� +� +� +� +� +� +� +� +� +� +� +|p|γa +R +√ +3 + sgn[1 − 2p] γa2 +16R2 cos 3α, +R ≥ Rc, +|p|γa +R +√ +3 + +� +sgn[1 − 2p] γa2 +16 − Cs +� +cos 3α +R2 + ++ Cs cos 3α +R2 +c +, +Rv < R < Rc, +0, +R ≤ Rv. +(12) +This is the complete solution for bandgaps under no de- +formation. It gives comparable results with various pub- +lished simulations (table II); fig. (1) plots the bandgaps +of zigzag tubes according to eqs. (12) and the simula- +tions. +D. +Strain +Having found the general gap equations (12), we wish +to find its response to homogeneous deformation, such as +axial strain, ϵz. Following the procedure in [9] and [8], +the bond vectors (B4) ⃗lj = (ljy, ljz), transform as +⃗lj → D⃗lj, +(13) +where D is a deformation matrix, given, for axial strain, +by +D = +� +1 − νϵz +0 +0 +1 + ϵz +� +(14) +where ν is the tube’s Poisson ratio (table I). This defor- +mation alters the bond distances, |lj| → |lj|+δ|lj|, which +in turn, alters the overlap integrals γj → γj + δγj ≈ +γj + bδ|lj|, where b ≈ 3.5eV/˚A([9]). This yields, using +(13) and (14), +δγj = ϵzb +√ +3 +a +(−νl2 +jy + l2 +jz). +(15) +Now plugging the above three δγjs in eq. (1), we get +δky = 1 +2γ sgn[2p − 1]ϵzb(1 + ν) cos 3α. +(16) +This equation is the explicit form of eq. (1) under axial +strain. On substitution in eq. (2) we get, +Eπ +g = |p|γa +R +√ +3 + +� +γa2 +8R2 − ϵz(1 + ν)ba +√ +3 +� +× 1 +2sgn[1 − 2p] cos 3α +(17) +But as discussed above, at small radii the π-band is +crossed by and hybridized by the singlet band, down- +shifting the gap (11), until it vanishes. This yields the +three regimes, as in eqs. 12, +Eg(ϵz) = +� +� +� +� +� +Eπ +g , +R ≥ Rc, +Eπ +g − Cs +� +1 +R2 − +1 +R2c +� +cos 3α, Rv < R < Rc +0, +R ≤ Rv. +(18) + +5 +Figure 1: Bandgaps of semiconducting zigzag tubes vs. 1/R2. The two type of semiconductors, p = ∓1 (for +definition see eq. B1), are marked with red and blue colors, respectively. The integers above/below the x-axis are +aliases for the zigzags (n, 0); they are positioned under their DFT bandgaps (dots connected by vertical lines); the +dots, labels a, b, c, d and e, refer to the correspond publications (see table II). The full curves are plots of eq. (12). +The broken curve is the zone-folding ”1/R-rule” (eq. B8). It is evident that, for large tubes, this rule holds with +slight modification, but for smaller tubes the gap reverses trend and finally vanishes. The critical radius of trend +reversal is ∼ 4.2˚A, which lies between the tubes (11, 0) and (10, 0). The regions A, B and C correspond to the +regimes R ≥ Rc, Rv < R < Rc and R ≤ Rv, respectively. +An unequivocal experimental verification of eqs. (18) is +not known to us, but the strain and chiral angle depen- +dence of Eπ +g was probed by [20] and qualitatively con- +firmed. Simulations of compressive strain in zigzag tubes +[21] appear to agree with both the linear behaviour and +p-dependence of Eπ +g as given by eq. (17). +E. +Torsion +To find the bandgap under an axial torsion, ζ, we start +by setting its deformation matrix, +D = +� +1 +0 +−ζ 1 +� +. +(19) +following the steps applied to strain (14-18), we get for +the un-hybridized π-bandgap +Eπ +g = +|p|γa +R +√ +3 + 1 +2sgn[1 − 2p] × +� +γa2 cos 3α +8R2 +− ζba +√ +3 sin 3α +� +. +(20) +As with strain, this bandgap is valid only in the regime +where the S-band is above the (π) conduction band (R > +Rc, eq. 9). Otherwise, hybridization with the S-band, +as discussed above, strongly downshifts it. In complete +equivalence to strain (18), the torsion bandgaps yield the +general formulae, +Eg(ζ) = +� +� +� +� +� +Eπ +g , +R ≥ Rc, +Eπ +g − Cs +� +1 +R2 − +1 +R2c +� +cos 3α, Rv < R < Rc, +0, +R ≤ Rv, +(21) +where Eπ +g is the torsion-modified bandgap (20) of the +π-band alone. + +A +B +c +1.5 +p=-1 +Cite +a +1 +worh +e +C +p = +1 +1 +be +C +R +a +e +p=-1 +e +Thil +0.5 +woI +b +p= +1 +e +b +e +1714 +11 +8 +5 +1916 +13 +10 +7 +0 +0.1 +0.2 +0.3 +R6 +Figure 2: Rc and Rv (eqs. 9, 10) for p = +1 (red) and p = −1 (blue) tubes. The bandgap of tubes whose radius lies +above Rc is not affected by hybridization. While the gap of tubes that lie between Rc and Rv (colored background) +is affected – and reduced. The couple of tubes lying below Rv have zero gap. The regions A, B and C , as in fig. +(2), correspond to the regimes R ≥ Rc, Rv < R < Rc and R ≤ Rv, respectively. +III. +NON-HOMOGENEOUS DEFORMATION +So far, the bandgaps found here concerned tubes with +circular cross-section and axial uniformity. But that is +not the case in many situations. The cross-section of a +bent tube, for example, is compressed in the inner side +of bending-curvature and stretched in the other. Its de- +formation profile is thus varying throughout the unit cell +of the tube. To treat this and similar cases, we need to +generalize the formalism of section (II A) to include these +non-homogeneous deformation. +A. +General formulation +In the theoretical treatment of section II A, it was tac- +itly assumed that the deformation is everywhere identi- +cal. This allowed us to deform a single graphene unit +cell and extract the shift of the Fermi points from there. +But that can not be done when the deformation varies +throughout the tube’s unit cell. Hence, if N is the num- +ber of graphene unit cells in the nanotube unit cell, the +position of the Fermi points, normally found by solving +eq. (B3), is now found by the sum over the N A-atoms +N +� +A=1 +3 +� +j=1 +γAjei⃗k·⃗lAj = 0. +(22) +As before, we seek the lateral shift ∆Ky which is a +simple sum of δky(A) (1) of the constituent A-atoms, +∆Ky = 1 +N +N +� +A=1 +δky(A) +(23) +where δky(A) is now a function of the local strain and +curvature at the position of the corresponding A’th atom. +The overlap integrals γAj in eq. 22 are functions of the +in-plain and out-of-plain deformations of the bond vec- +tors ⃗lAj. In-plain deformation, such as strain and torsion, +changes the γ’s by changing the distance between neigh- +boring π-orbitals, while out-of-plain deformation, such as +the tube’s curvature, lowers the γ’s by having the orbitals +misaligned. +In this section we include both deformations without +assuming their homogeneity throughout the tube. First +consider curvature. On a plain cutting through the cross +section, let ±φy be the angles between neighboring π + +R(A) +(11,0) +Re(p +4 +(7,5) + (8,3) +O (6,5) +(10.0) Rc(p +(7,3) +O (6,4) +(8,1) +. +● (7,2) +A +(8,0) +3 +0 (5,3) +(6,2) +(7.0) 0 +(4.3) +(6,1) 0 +R,(p +(5,1) +0 (4,2) +R(p +2 +(3,2) +(5.0) +B +(4,0) 0 +c +0 +9'0 +cos307 +orbitals and the normal passing through the middle point +between them; and let ±φz be the corresponding angles +projected on a plain along the axis. +If κθ, κz are the +coordinate curvatures defined as (radius of curvature)−1 +within the respective plains, and writing bond vectors +(B4) in the form ⃗l = lyˆy + lzˆz, one gets |φy| = |ly|κθ/2 +and |φz| = |lz|κz/2. The overlap integral becomes γ → +γ cos φy cos φz, so that for small change δγ is given by +δγcurv = −γ +8 +� +l2 +yκ2 +θ + l2 +zκ2 +z +� +. +(24) +Let us now include local axial strain. Its deformation +matrix, given by eq. (14), induces a change δγstrain ac- +cording to eq. (15). The total change in the local overlap +integrals is then, +δγ = δγcurv + δγstrain ≡ l2 +yDy + l2 +zDz, +(25) +where +Dy = νb +√ +3 +a +ϵz − γ +8 κ2 +θ, +(26) +Dz = −b +√ +3 +a ϵz − γ +8 κ2 +z. +(27) +Substituting ly, lz from eqs. B4 in eq. (25) we find, +δγA1 = a2 +4c2 +h +� +(n + m)2Dy + (n − m)2 +3 +Dz +� +δγA2 = a2 +4c2 +h +� +m2Dy + (2n + m)2 +3 +Dz +� +(28) +δγA3 = a2 +4c2 +h +� +n2Dy + (2m + n)2 +3 +Dz +� +Inserting now eqs. 28 in 1 +δky = sgn[1 − 2p]a(Dz − Dy) +2 +√ +3γ +cos 3α. +(29) +This, by eq. (23), we sum over all A-atoms in the tube’s +unit cell, +∆Ky = sgn[1 − 2p]a cos 3α +2 +√ +3γN +N +� +A=1 +(Dz − Dy). +(30) +Converting now the sum to an integral, +∆Ky = sgn[1 − 2p]a cos 3α +2 +√ +3πγ +� 2π +0 +(Dz − Dy)dθ. +(31) +Once ∆ky is known, the energy gap follows immediately +from the dispersion relation (B7) and the zone-folding +gap (B8), +Eπ +g = aγ +R +���� +p +√ +3 + +√ +3R∆ky +���� . +(32) +Equation (32) is the non-homogeneous version of eq. +(2), and likewise, it includes only the π-band. But now +the non-homogeneity of the deformation must be in- +cluded through the integration in (31), we thus replace +1/R2 in eq. (6) accordingly, +1 +R2 → 1 +2π +� 2π +0 +κ2 +θ dθ. +(33) +Now the full set of gap equations for non-homogeneous +deformation can be obtained by using equations (12) with +the substitution of eq. (32) for Eπ +g , and a replacement as +in (33). +B. +Elastic theory of bending and buckling +The structural properties of bending and buckling of +SWCNT’s had been simulated [1][4][22][23][24] [25] and +experimented [22][2] in the years following nanotubes’ +discovery. They established that SWCNT’s abide by con- +tinuous elasticity theory given an effective wall ¨thick- +ness¨ of 0.66˚A (values by other groups in table I). Which +is always much smaller then the tube’s diameter. Hence +SWCNTs are also ¨shells¨; precisely: slender cylindrical +shells. +The elastic theory of their bending, up to the onset of +buckling, was developed a century ago by Brazier [26]. It +is summarized in appendix A for convenience. +Brazier’s fundamental insight was that, in order to +reduce shear under bending, the tube’s circular cross- +section is ovalized (fig. 5). At the critical bending, how- +ever, the elastic cost of increased ovalization exceeds the +reduction in shear energy – and the tube buckles. +In the pre-buckling regime, Brazier’s theory gives the +exact shape of the ovalized cross-section, parametrized +by ξ in fig. +5. +It also predicts, as reproduced in the +appendix, that at the onset of buckling the ovalization +is ξ = 2/9 (eq. A19), independent of Poisson’s ratio or +other material properties. +At the post-buckling regime, the elastic energy associ- +ated with ovalization (beyond the buckling point)is +∆U = G(ξ − ξcr)2, +ξ ≥ ξcr +(34) +where G is a constant. +But MD simulations [1][22] +showed that in this regime, the post-buckling energy den- +sity is linear with bending curvature, that is +∆U = Q(κ − κcr), +κ ≥ κcr, +(35) +where Q is a constant and κcr is the critical bending +curvature (A17). Comparing eqs. (34) and (35), +ξ − ξcr = Q +G(κ − κcr)1/2, +κ ≥ κcr. +(36) +This gives the proportionality relation in the kink, be- +tween bending curvature, κ, and ovalization of the cross- +section , ξ. Outside of the kink, however, This implies +that outside the kink, strain is independent of further +bending – its energy is absorbed in the kink, which acts + +8 +as a hinge between the two fixed sections of the tube +[27][28]. +A realistic closed-form model of such kinks is not +known to us. But simulations revealed [29] that for bend- +ing not too far from the onset of buckling, where the kink +is shallow, the cross-section is shaped as an oval, just +as the oval parametrized by Brazier in the pre-buckling +regime (appendix A). Since this regime is an intermidiate +stage between the onset of buckling and a fully developed +kink, it is also called the transient-regime [29]. +This regime ends when the opposite walls of the kink +approach contact; at this point the distance between +the walls is comparable with the inter-planar distance +in graphite dg ∼ 3.35˚A. In Brazier’s parametrization this +distance is 2R(1 − ξ) (fig. 5). We thus have +ξclose = 1 − dg +2R, +(37) +where the superscript close signifies the closing of the +cross-section in the kink and bringing the opposite walls +to near contact. ξclose marks the upper limit of flattening +within which our analysis is expected to hold. For a tube +of 1nm in diameter, ξclose ∼ 2/3. +The cross-section of a kink in the transient regime is +thus bounded by the onset of buckling, at ξ = 2/9, and +ξclose, having the curvature dependence according to eq. +(36), namely, +ξkink = 2 +9 + +� +κ − κcr +κclose − κcr +�1/2 � +ξclose − 2 +9 +� +, +(38) +where κcr ≤ κ ≤ κclose. +Equation 38 demonstrates the evolution of ξkink, being +continuous at the onset of buckling ξkink = 2/9, through +further bending where ξkink ∝ κ1/2, and finally reaching +the closing point at ξclose. +Having found ξkink, the circumferential curvature in +the center of the kink is given, as in the pre-buckling +regime, through the out-of-plane deformation w (eq. A3) +and eq. (A4) by, +κkink +θ += 1 +R(1 + 3ξkink cos 2θ). +(39) +The axial strain under bending, ϵz, is anti-symmetric +with respect to the neutral plane; it thus has no contri- +bution to the integration in eq. (31); also the bending +curvature, κz ≪ 1/R, can be neglected in the integra- +tion (31), although, as will be shown next, it affects κθ. +Hence we set in eqs. (26-27) Dy = − γ +8 κ2 +θ and Dz = 0. +According to the elastic theory of bending (appendix +A), the circular cross-section of a bent tube becomes in- +creasingly oval with bending. Parametrizing the ovaliza- +tion by ξ (fig. 5), let the circumferential integral of the +square of the curvature (eq. A4) be I, then +I ≡ 1 +2π +� 2π +0 +κ2 +θdθ = 1 + 9 +2ξ2 +R2 +, +(40) +where the pre-buckling regime corresponds to 0 ≤ ξ ≤ +2/9. The point ξ = 2/9 corresponds to the critical cur- +vature at buckling (eq. A19). +C. +Bending – pre-buckling +Since the ovalization in this regime is a quadratic func- +tion of the bending curvature (eq. A14), the replacement +rule (33) becomes, +I = 1 +R2 → 1 + 9 +2ξ2 +R2 += 1 +R2 +� +1 + (Lκ)4� +. +(41) +where +L ≡ (1 − ν2)1/2 +�9 +2 +�1/4 R2 +t +where ν and t are given in table I. With this replacement, +eqs. (12) yield the explicit gap equations under bending, +Epre +g +(A) = |p|aγ +R +√ +3 + sgn[1 − 2p] γa2 +16R2 +� +1 + (Lκ)4� +cos 3α, +(42) +where the superscript ”pre” refers to pre-buckling and +”A” corresponds to R ≥ Rc. Region B then corresponds +to radii in the range Rv < R < Rc; the bandgaps there +are, +Epre +g +(B) = |p|aγ +R +√ +3 + (sgn[1 − 2p] γa2 − 16Cs) +16R2 +× +� +1 + (Lκ)4� +cos 3α, +(43) +and finally , at the smallest radii range, C, where R ≤ Rv, +Epre +g +(C) = 0. +(44) +Eqs. +(42–44) give the bandgaps for all radii in the +pre-buckling regime; the three ranges of radii, A, B and +C are also shown in figs. (1) and (2). These equations +reveal that, depending on the sign of p, bending may +increase, decrease or even close the gap – a prediction +which also coincides with simulations of ovalized cross- +sections [30][31]. +D. +Bending – critical curvature +At the critical point of buckling the ovalization param- +eter ξcr = 2/9 (appendix A). This gives for the rhs of eq. +(40) I = +11 +9R2 . It is interesting to note that for metallic +tubes (p = 0), replacing the rhs of eq. 41 for this value +of I and substituting in eq. 42, gives, +∆Eg(κ = κcr) +Eg(κ = 0) += 2 +9, +p = 0, R ≥ Rc. +(45) +This universal ratio for metallic tubes (armchair tubes +excluded as their Eg = 0), relates their bandgaps at the +critical point of buckling with their straight value. Figure +(3) depicts this in terms of the change in the bandgap as +a function of the bending curvature. + +9 +Figure 3: Bandgap vs. bending curvature for primary +metallic tubes, p = 0, (armchairs excluded). Eg is the +gap of the straight tube (eq. 4); ∆Eg is the additional +gap due to pure bending (42-44) and (48-50); κ is the +bending curvature and κcr is the critical curvature (eq. +A17). In the pre-buckling regime the gap ∝ κ4 up to +buckling point where it is higher by a factor of 2/9 +compared with the straight state (eq. 45). At +post-buckling, the bulk relaxes by a small amount δ (46) +and remains there, while in the kink it is ∝ (κ − κcr)1/2. +E. +Bending – post-buckling +Since the elastic bending moment reaches maximum +at buckling point [32], the buckling transition is accom- +panied by a small relaxation of the strain energy [1] [22] +outside of the kink. This relaxation also lowers Brazier’s +ovalization and, as will be shown next, also the associated +bandgap. +The pre-buckling elastic energy per unit length (eq. +A16) can be approximated near critical point by U ∼ +1 +2R3tEπκ2 . Denoting the post-buckling elastic energy +relaxation per unit length by ∆U, the relaxed state corre- +sponds to a lower curvature given by δκ = ∆U/(∂U/∂κ). +The associated shift in the band-gap ratio ∆Eg/Eg +for metallic tubes (eq. +45) can be found by expand- +ing this ratio near on the pre-buckling side of critical- +ity, where the function is well behaved δ(∆Eg/Eg) = +∂(∆Eg/Eg)/∂κ)δκ where the derivative is taken at the +buckling curvature κ = κcr (eq. A17); this yields, +δ +�∆Eg +Eg +� += R∆U +3πD , +(46) +where D is the elastic rigidity (eq. +A11) and ∆U is +the elastic energy relaxation per unit length at buck- +ling. Equation 46 links, for metallic tubes, the mechani- +cal post-buckling relaxation step with the post-buckling +electronic band-gap change in the bulk. In contrast with +the relaxation in the bulk, the bandgap in the kink is +now a stronger function of bending curvature κ; this can +be seen by applying the replacement rule (33) on the gap +eqs. (12) using κkink +θ +(39). +Staying, as we do throughout this work, within the +transient regime where the kink remains open (ξkink < +ξclose, eqs. 37, 38), bandgaps in the kink fall in three +regimes; these are related to the three regimes in straight +tubes (fig. 1), corresponding to whether the gap is de- +termined by the π-band alone (regime A), the modified +regime where the singlet (S) band hybridizes with the π +band (regime B), or zero – where the S-band downshifted +enough to completely close the gap (regime C). All tubes +fall into one of these regimes depending on their diame- +ter (largest to smallest – regimes A to C, respectively). +Here, however, the regimes are determined by the cur- +vature within the kink which is bounded by I < 1/R2 +c +(regime A), 1/R2 +v > I > 1/R2 +c (regime B), and I > 1/R2 +v +(regime C), where I is given by eq. (40) and Rc, Rv are +given by eqs. (9 – 10), respectively. +Explicitly, regime B is bounded by +2 +9 +�R2 +R2c +− 1 +� +< (ξkink)2 < 2 +9 +�R2 +R2v +− 1 +� +, +(47) +while regime A applies at the lower bound and regime C +in the upper. +The bandgaps of the kink in the various regimes of +bending can now be found by replacing 1/R2 in eqs. (12) +with (1 + 9 +2(ξkink)2)/R2 (eq. 41), which gives +Ekink +g +(A) = |p|aγ +R +√ +3 + γa2 +16R2 sgn[1 − 2p] +× +� +1 + 9 +2(ξkink)2 +� +cos 3α, +(48) +where ξkink is given by eq. (38). In region B, eqs. (12) +then give, +Ekink +g +(B) = |p|aγ +R +√ +3 + (sgn[1 − 2p] γa2 − 16Cs) +16R2 +× +� +1 + 9 +2(ξkink)2 +� +cos 3α, +(49) +and as before , in regime C, +Ekink +g +(C) = 0. +(50) +Eqs. (48–50) give the bandgaps in the center of the +kink for the respective radii ranges. +Here, as in pre- +buckling regime, the bandgap as a function of bending +can increase, decrease, or vanish, depending on the sign of +p. The difference with the pre-buckling regime, however, +is that here it is ∝ (κ−κcr)1/2; this behaviour is depicted +for zigzag tubes where p = +1 (fig. 4a) and p = −1 (fig. +4b) and for metallic tubes of all chiral angles in fig. (3). +It may be useful to compare the bandgaps of the kink +with the bulk (far from the kink – where |z| ≫ R). As- +suming that in the bulk ξ remains near criticality, the + +AEE +Kink +94 +Bulk10 +(a) p = +1 +(b) p = −1 +Figure 4: Bandgaps and Fermi energies vs. bending for semiconducting zigzag tubes of the two types (p = ±1). +Following eq. (42), the initial gaps first decrease (left) / increase (right) ∝ κ4; at post-buckling it evolves faster: +∝ (κ − κcr)1/2; at increased curvature – where the singlet band crosses the conduction band – the gap strongly +downshifts in all cases (49); at this point, also the Fermi energy (C2) downshifts. The point of EF = 0 in the plots +corresponds to the value of EF in graphene. +bandgap in regime B (eq. 49) compared with the bulk +gives, +Ekink +g +−Ebulk +g += − 9Cs +2R2 +� +(ξkink)2 − +�2 +9 +�2� +cos 3α, (51) +where ξkink is given by eq. (38). +It is worth noting that for armchair tubes (p = 0, α = +30o), these equations (as well as the pre-buckling ones) +predict no bandgaps. That holds, however, as long as the +underlying assumption through this work – that the kink +remains open [29] – holds. +IV. +SUMMARY +We presented here a comprehensive theory of bandgaps +in carbon nanotubes, including strong curvature and +large non-homogeneous deformation. This theory recon- +ciles the fundamental theory of bandgaps in nanotubes +(appendix B) and its well-known corrections (at small +deformation and small curvature), with contrasting re- +sults from a host of DFT computations of very small or +highly deformed tubes; the present theory shows them to +be special cases of the same general equations (12). +A formalism was derived to calculate the gaps due to +a general non-homogeneous circumferential deformation +(by starting with eq. +12 and making the replacement +33). We then applied this formalism to study bending, +including buckling and a kink (with a caveat of staying +within the transient regime, i.e: where the opposite walls +of the kink do not touch). +The results detail the gap evolution under both weak +and strong bending. +In the pre-buckling regime, the +bandgap shifts ∝ ±κ4. A notable result is that, by the +onset of buckling, the gaps of primary-metallic tubes, in- +dependent of chiral vector or radius, increase by a ratio +of 2/9 compared with their un-bent value, (eq. 45 and +fig. 3). +In the post-buckling regime (κ > κcr, eq. A17), at first, +the bandgap in the kink shifts ∝ ±(κ−κcr)1/2, up-to the +point where the singlet band crosses the conduction π- +band initiating a steep downshift to zero (eqs. 48-50, and +fig. 4). The downshift in the bandgap is accompanied, +in this regime, by a substantial downshift of the Fermi +energy (appendix C). +V. +APPENDICES +Appendix A: Brazier’s Theory +This appendix has a number of relevant derivations +from the elastic theory of thin cylindrical shells under +pure bending. The theory was derived by Brazier [26]. +For the exposition we follow reference [32]. +Consider a slender cylindrical shell, initially straight, +under bending. +The strain profile is anti-symmetric +about the neutral plane, compressive in the inner side +and tensile in the outer; the energy per unit length is +given by +Uz = 1 +2Iκ2 +(A1) +where κ is the bending curvature and I is the second +moment of area; for a perfectly circular cross-section +I ≡ I0 = πR3t. +(A2) + +Eg +EF +0- +0.5 +(13,0) +(16,0) +(22,0) +(31,0) +(13,0) +(16,0) +(22,0) +0 +0.5 +1.5 +2 +2.5 +0.5 +-1(11,0) +Eg +0.5+ +(14,0) +(17,0) +(20,0) +K/A +(11,0) +(17.0) +(20.0) +0 +0.5 +(14.0) +1.5 +2 +2.5 +3 +-0.5 +-1,11 +(a) Cross-section +(b) Profile +Figure 5: Bending causes the circular cross-section to become oval, turning R → R(1 + ξ cos 2θ) where ξ is the +ovalization parameter; at ξ = 2/9 (blue shape) the tube reaches criticality and buckles. +where t is the thickness of the tube. +It will be shown below that by flattening the cross +section the tube reduces its energy. It is reasonable to +assume that small flattening can be expressed by an out- +of-plain displacement of the form +w = Rξ cos 2θ. +(A3) +where ξ (see fig. 5) is a dimensionless measure of the flat- +tening, or ovalization. The curvature due to ovalization +is +δκθ = −∂2w +∂y2 − w +R2 = 3ξ +R cos 2θ, +(A4) +where y = Rθ is the circumferential coordinate of the +original circle, the second derivative is the usual curva- +ture due to local changes of displacement while the sec- +ond component is due to the change of radius. The in- +plain deformation v, together with w (A3), determines +the circumferential strain ϵy = ∂v/∂y + w/R. As a first +approximation Brazier assumed the surface to be inex- +tentional, ie: ϵy = 0, yielding +v = −R +2 ξ sin 2θ. +(A5) +where we assumed no net rotation by taking the constant +of integration to be zero. +It may be commented that +for the electronic structure calculation we take a much +looser requirement, instead of assuming the surface to be +inextentional it is assumed only that the circumference +is unchanged; i.e: +� +ϵydy = 0. +The second moment of area in eq. A1 is actually de- +fined as +I = +� 2π +0 +Rt(R sin θ + η)2dθ, +(A6) +where t is the surface thickness and η is given by +η = w sin θ + v cos θ = −Rξ sin3 θ +(A7) +where the last step used eqs. A3 and A5. Inserting this +in eq. A6 we finally get +I = I0 +� +1 − 3 +2ξ + 5 +8ξ2 +� +(A8) +where I0 is given by A2. +The strain energy per unit +length (eq. A1) is then given by +Uz = 1 +2κ2R3tEπ +� +1 − 3 +2ξ + +�5 +8ξ2 +�� +(A9) +where E is the Young’s modulus; the ξ2 component in +the square brackets is truncated in Brazier’s analysis. +The energy per unit length due to Brazier deformation +is +UB = 1 +2D +� 2π +0 +δκ2 +θRdθ, +(A10) +where δκθ is given by eq. A4 and D is the elastic rigidity +given by +D = +Et3 +12(1 − ν2). +(A11) +Integration of eq. A10 gives +UB = +3πEt3 +8(1 − ν2) +ξ2 +R . +(A12) +The total elastic energy per unit length is +U = Uz + UB. +(A13) +Now ξ can be found by requiring ∂U/∂ξ = 0. The result +is +ξ = (1 − ν2)R4 +t2 +κ2. +(A14) + +n +R +0 +s)2R +neutral pla +2RC +cell12 +The values of ν and t were found by a number of groups +(see table I), so we can write +ξ = BR4κ2, +(A15) +where B = 2.2(˚A−2) by the values of [1] compared with +B = 1.57(˚A−2) by [2]. +The total energy (eq. A13) is finally given by +U = 1 +2R3tEπκ2 − +�3R7(1 − ν2)Eπ +8t +� +κ4 +(A16) +The tube begins to buckle when the bending moment +M = dU/dκ, reaches maximum; hence the curvature at +buckling point can be found by putting d2U/dκ2 = 0, +which gives +κcr = +t +3R2 +� +2 +1 − ν2 . +(A17) +κcr is the critical curvature at the onset of buckling. Us- +ing the same values as for eq. A15, we find +κcr = A +R2 , +(A18) +where A = 0.316˚A using the values of [1], compared +with A = 0.376˚A by [2]. MD simulations confirmed the +general shape eq. (A18) and found A to range between +0.185 [3] and 0.387˚A [1] Regardless of the actual numer- +ical value of A, substituting A17 in A14 gives the exact +ovalization parameter at buckling point +ξcr = 2 +9. +(A19) +This is a remarkable result of Brazier’s theory; it states +that at the critical point of buckling, all tubes, indepen- +dent of thickness, radius or Young’s modulus, become +ovalized by the same ratio of 2/9. +Appendix B: The fundamental bandgap +The band structure of carbon nanotubes is based on +graphene’s band structure [33] sliced-up with lateral +quantization lines. The exact wrapping of graphene into +a nanotube is determined by the tube’s chiral integers +(n, m) [34]. +Now, any two integers can be related by +other two integers (q, p), such that, +n − m = 3q + p +(B1) +where p takes one of the values (0, +1, −1). +The lateral k-vectors, ky, must lie on quantization lines +given by +ky = ν +R, +where +ν = 0, ±1, ±2, · · · . +(B2) +Graphene’s Fermi surface is found by the k-vectors +that solve, +3 +� +j=1 +γjei⃗k·⃗lj = 0 +(B3) +where γj are the three overlap integrals of the nearest +π-orbitals, and lj are their bond vectors, given by +⃗l1 = +a2 +4πR +� +(n + m)ˆy − 1 +√ +3(n − m)ˆz +� +, +⃗l2 = +a2 +4πR +� +−mˆy + 1 +√ +3(2n + m)ˆz +� +, +(B4) +⃗l3 = +a2 +4πR +� +−nˆy − 1 +√ +3(n + 2m)ˆz +� +, +where (n, m) are the chiral integers, and (ˆy, ˆz) are the +circumferential and axial coordinates, respectively. +Now under the assumption that graphene is isotropic, +we can take all +γj ≡ γ. +(B5) +The solution of eq. (B3) is then given by the two points, +⃗KF 1 = 1 +3R +� +(m + 2n)ˆy + m +√ +3ˆz +� +⃗KF 2 = 1 +3R +� +(n − m)ˆy + (m + n) +√ +3ˆz +� +(B6) +These are the Fermi points of graphene, in the nanotube’s +natural coordinates. +The spectrum near a Fermi point is linear, +E = ± +√ +3 +2 aγ δk, +(B7) +where E is the energy above the Fermi level and δk is +the distance from the nearest Fermi point. +Substituting (B1) in (B6) reveals that when p = 0 +they do lie on a quantization line (B2) – these tubes are +thus, according to zone-folding, metallic. For p = ±1, on +the other hand, the distance to the nearest quantization +line is 1/3R; these tube are thus semiconducting with +an energy gap given by substituting 1/3R in the linear +spectrum (B7), +Eg = |p| aγ +√ +3R. +(B8) +Appendix C: Fermi energy and Cs +The lowering of the singlet (S) band below the con- +duction (π) band causes, as DFT simulations demon- +strate (table IV), a downshift not only of the bandgap +but also the Fermi energy. While it is generally under- +stood to be a result of a large circumferential curvature + +13 +of small tubes, we wish to quantify it by extrapolating +the published data for tubes in this radii range (regime +B: section II C). +In this regime, the bandgap depends on the position +of the singlet band above the Fermi points (eq. +6), +which depends, in turn, on the proportionality factor Cs. +The DFT bandgaps of semiconducting, straight and un- +deformed tubes (table II) in this regime (R ≲ 4˚A) give +Cs ≈ 8(eV · ˚A2). +ξ +11, 0 10, 0 +8, 0 +0 +0.82 +0.87 0.487 +0.0625 +0.8 +0.85 +0.41 +0.125 +0.75 0.825 +0.3 +0.1825 +0.65 +0.7 +0.075 +0.25 +0.525 0.55 +0 +0.3125 +0.375 0.32 +0 +0.375 +0.225 0.075 +0 +0.4375 +0 +0 +0 +Table III: Energy gaps (eV) in the respective tubes as a +function of the deformation parameter ξ; data extracted +from figure 1a in ref. [31] +Figure 6: Bandgap vs. total circumferential curvature +square for the zigzags (11,0), (10,0) and (8,0). Data +points are given in table (III); lines are plotted by eq. +(12) (regime B) where 1/R2 was replaced by I (see eq. +40) with Cs ∼ 8 − 10 (eV·˚A2). The crosses label the +critical points of buckling (given by eq. A18). +Moving to Fermi energy, DFT simulations show (ta- +ble IV) that the Fermi-energy of tubes in this regime are +downshifted too in ∝ 1/R2 (fig. 7 ), while larger tubes +have it identical to graphene. The S-band in this regime +is the effective conduction band, and thus, the Fermi en- +ergy lies in the middle between it and the valence π-band +(which is not shifted), +EF = 1 +2(Es + Ev) = 1 +2 +� +Es − Eπ +g +2 +� +, +(C1) +where the singlet band energy Es is given by eq. (6) and +the pure π bandgap Eπ +g is given by eq. (4). This gives +explicitly, +EF = c1 − |p| +γa +4 +√ +3R +− cos 3α +R2 +� +c2 + sgn[1 − 2p]γa2 +64 +� +. +(C2) +where the constants are +c1 = Eg(10, 0) +4 ++ +Cs +2R2 +10,0 +≈ 0.53 eV, +c2 = Cs +2 ≈ 4 eV · ˚A +2, +where we used Eg(10, 0) = 0.87 eV (table II), R10,0 = 4˚A, +and Cs = 8 eV ·˚A2. +n, m +ref. [12] ref. [13] +4, 0 +−1.23 +−1.29 +5, 0 +−0.78 +−0.64 +7, 0 +−0.21 +−0.38 +8, 0 +−0.02 +−0.14 +10, 0 +0.06 +−0.04 +Table IV: Fermi energies (in eV) of semiconducting +zigzag tubes. What was actually computed are +work-functions, (WF)tube; Fermi energies shown here +were then found relative to graphene by +Etube +F += (WF)graphene − (WF)tube, where +(WF)graphene = 4.55eV in ref. [12] and 4.66eV in ref. +[13]. +Now this can be further simplified if we neglect the +second term in the parenthesis in eq. +(C2) and, since +EF (R ≥ Rc) = 0, following the downshift of the singlet +band Es (eq. 6), +EF = Cs +2 +� 1 +R2c +− 1 +R2 +� +cos 3α, +R ≤ Rc, +(C3) +where Rc is given, by eq. (9); for zigzag tubes (α = 0), +Rc = R(10, 0) = 4˚A. Eq. (C3) is depicted in fig. (7) for +semiconducting zigzag tubes in this range, together with +DFT data. +Considering circumferentially ovalized tubes, follow- +ing our procedure we first replace 1/R2 with (1 + +9ξ2/2)/R2 ≡ I (eq. 40), where ξ is the ovalization pa- +rameter. Explicitly, +EF (ξ) = Cs +2R2 +�� R +Rc +�2 +− 1 + 9 +2ξ2 +� +cos 3α, +R ≤ Rc. +(C4) + +1 (10,0) +(11,0) +0.75 +0.6eV +0.595eV +0.5 +(8,0) +0.25 +I(A +0.114 +Figure 7: Fermi energy of a number of small zigzag +tubes vs. 1/R2. The line is a plot of eq. (C3) with +Cs = 8 (eV·˚A2). Data is in table IV where red dots +taken from ref. [12], green dots from [13]. +Comparing this with the DFT bandgaps (table III), plot- +ted in fig. 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S., Physical Properties of Carbon +Nanotubes (Imperial college Press, 1999). + diff --git a/7dE4T4oBgHgl3EQfcww5/content/tmp_files/load_file.txt b/7dE4T4oBgHgl3EQfcww5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85e0ba23c957d8eb0b6f208f4ea2caf557be3044 --- /dev/null +++ b/7dE4T4oBgHgl3EQfcww5/content/tmp_files/load_file.txt @@ -0,0 +1,853 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf,len=852 +page_content='Bandgaps of bent and buckled carbon nanotubes Alex Kleiner orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='org/0000-0002-4694-2218 (Dated: January 12, 2023) Carbon nanotube’s large electro-mechanical coupling and robustness makes them attractive for applications where bending and buckling is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But the nature of this coupling is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Existing theory treats only weak and homogeneous deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' We generalize it and derive close-form expressions for bandgaps under non-homogeneous deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The theory is first compared with a number of published DFT simulations, and then applied to the specific case of bending and buckling – where a kink is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In the pre-buckling regime, bandgaps change ∝ ±κ4 where κ is the bending curvature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' inside the kink at post-buckling, while near criticality, the kink is shallow and the gap is ∝ ±κ1/2, where the sign depends on the chiral integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' For a deeper kink but still with an open cross-section, both the bandgap and local Fermi energy strongly downshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Introduction 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Homogeneous deformation 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' General formulation 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Small circumferential curvature 3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Large circumferential curvature 3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Strain 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Torsion 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Non-homogeneous deformation 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' General formulation 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Elastic theory of bending and buckling 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Bending – pre-buckling 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Bending – critical curvature 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Bending – post-buckling 9 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Summary 10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Appendices 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Brazier’s Theory 10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The fundamental bandgap 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Fermi energy and Cs 12 References 14 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' INTRODUCTION Three decades after its discovery, carbon nanotubes has yet to realize its potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A slow but continuous progress in fabrication may, however, make feasible a growing number of application ideas suggested early-on but never pursued due to economical and fabrication con- siderations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' One such application class would be electro- mechanics – as it relies on the coupling between two of the tube’s unique features: one dimensional conductivity and elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This coupling is most extreme in the kink of a buckled tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The kink’s size is of the order of the tube diame- ter, but what is its electronic structure and how will it affect transport as a function of further bending?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Cur- rently, the answer is far from being clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The kink is a point of large curvature and non-homogeneous deforma- tion, and, as we show below, non of these is theoretically well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The elementary theory predicting bandgaps in car- bon nanotubes is based on zone-folding of graphene [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It classifies tubes as metallic or semiconducting – with bandgaps ∝ 1/R according to their chiral vectors (ap- pendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Corrections ∝ 1/R2 due to circumferential curvature [6][7], predict that metallicity is lifted in all but one type: the (achiral) armchair tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' An obvi- ous prediction is the bandgaps would trend higher at ever smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But DFT computations (table II) clearly show that the opposite is true: the narrowest tubes are in-fact gapless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' it is qualitatively attributed to a curvature-induced hybridization of the σ−π bands, but it has not been quantitatively incorporated into existing theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This should also play a critical role in the kink, where very large curvature is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Thus, our first aim is to understand the effects of large curvature quantita- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But the other characteristic of the kink – its large and non-homogeneous deformation, needs also to be ac- counted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The modulation of bandgaps due to homogeneous de- formation is well-known [8][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But that is not sufficient, even for weak bending without kinks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the strain profile in bending is asymmetric about the neutral line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='e: pure bending has no net strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' this would imply, according to existing theory, no change in the bandgap – a clear contradiction with simulations [10][11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Thus, in order to reconcile theory and simulations and derive closed-form solutions for the bandgap in the kink, we need, in addition to incorporating effects of large cur- vature, extend the theory of homogeneous deformation into the non-homogeneous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' For completeness, however, we start by recapping the existing, homoge- neous, theory of bandgaps in carbon nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='05085v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='mes-hall] 12 Jan 2023 2 Symbol Definition ϵ axial strain ζ axial torsion κθ circumferential curvature κ bending curvature κz local axial curvature κcr bending curvature at buckling point b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 eV/˚A dγ/dl, where l is bond length γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='7eV π-orbitals overlap energy n, m nanotube’s chiral integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' m ≤ n p, q integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' n − m = 3q + p (B1) v, w in-plain and out-of-plain deformation vectors, respectively (appendix A) ch ach is the circumference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' ch = √ n2 + nm + m2 R nanotube’s radius, 2πR = ach ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='19 Poisson ratio [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='34 according to [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' see also [3], [4] t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='66˚A effective thickness [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='75˚A according to [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' see also [3], [4] α chiral angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' cos 3α = (2n3 − 2m3 + 3n2m − 3nm2)/2c3 h ξ ovalization parameter (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A3) ξcr critical ovalization – 2/9 Eπ g bandgap due to the π−band alone Epre g bandgap at the pre-buckling regime Epost g bandgap at the post-buckling regime Es energy of the singlet-band above graphene’s Fermi level Cs coefficient for Es (appendix C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Cs = 8 ∼ 12eV ˚A2 R10,0 the radius of the (10,0) tube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' R10,0 = 4˚A Rc upper critical radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' above it, unhybridized π-band structure is valid Rv lower critical radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' below it, bandgaps are zero ⃗KF Fermi points δky lateral shift of the Fermi points ⃗l, ly, lz nearest neighbor bond vectors and their lateral and axial projections U elastic energy per unit length Table I: List of symbols and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' HOMOGENEOUS DEFORMATION In this section we start by recasting the tight-binding π-band theory of homogeneous deformation [7][8][9] in a formalism that will be used here for non-homogeneous fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' we then consider deviations from the pure π-band, which become dominant at very small radii, or large cir- cumferential curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' General formulation The fundamental theory of bandgaps in carbon nan- otubes, reproduced here in appendix B, gives increasingly erroneous gaps as the nanotubes’ diameter decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The reason lies in a symmetry of graphene that is broken in nanotubes: isotropy of bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The assumption that the three nearest overlap integrals are equal (B5), does not fully hold in nanotubes, since neighboring π-orbitals are not parallel, nor are neighboring atoms equally sepa- rated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Discarding thus assumption (B5), we find the new Fermi points by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (B3) while allowing small changes to the overlap integrals δγj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The solution yields [9], δky = sgn[1 − 2p] 2πR √ 3γ × (1) � (m − n)δγ1 + (2n + m)δγ2 − (n + 2m)δγ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 3 where δky is the shift of the Fermi points along the cir- cumferential coordinate y, relative to their zone-folding position (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Now, knowing the δγj in (1), translates, through the linear spectrum (B7) to the following gap equation, Eπ g = aγ R ���� p √ 3 + √ 3Rδky ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (2) The superscript π serves to label it as a pure π-band, in contrast to the hybridized band to be considered later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Small circumferential curvature One of the symmetries present in graphene but not in nanotubes is the parallel nature of the π-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In nanotubes, neighboring π-orbitals are misaligned by a small angle: if ⃗lj is the vector connecting them (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B4), and ljy is its y-component, the angle is βj = ljy/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Then γj → γ cos2 βj – which gives δγj = −γ l2 jy 2R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (3) Now plugging this in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (1) and (2) yields Eπ g γ = |p| a R √ 3 + sgn[1 − 2p] a2 cos 3α 16R2 , (4) where cos 3α is given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Eπ g includes two effects over the π-band: the 1/R-rule of the fundamental gap, and the 1/R2 correction due to curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Here, in con- trast with the fundamental prediction, ¨metallic¨ tubes (p = 0) do open small gaps, and their 1/R2 trend, as given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (4), was experimentally verified on a few large zigzag tubes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Large circumferential curvature Predictions from the gap equation (4) can be compared with MD-simulations: table (II) lists results from simula- tions run by different groups, and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (1) compares them with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 4 for zigzag tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Two opposite trends in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 1 are evident: in tubes larger then (10,0), the gaps seen to agree with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (4), but smaller tubes reverse this trend and quickly reach metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The reason for this was pointed out early-on by Blase et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' it concerns a singly-degenerate band (termed below: S-band) which is a π∗ − σ∗ hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In graphene, the S-band is positioned very high above Fermi level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' but since hybridization is ∝ 1/R2, the S−band downshifts with decreasing radius, and at a certain radius it crosses the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This radius was found [13] by DFT to be the radius of the (10, 0) tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='e: in this tube, the S and conduction bands overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Its radius is, R10,0 ≈ 4˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (5) At smaller radii, the S-band cuts through the gap (see for example: (8,0) and (7,0) in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 1) until, at the small- est radii, it vanishes entirely (see for example: (6,0), (5,0) and (4,0) in table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Let Es be the energy, relative to Fermi level, of the S−band at the K-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Its 1/R2 dependence was found by DFT computations to scale by a factor of Cs ∼ 8 (ev·˚A2) [15] (see also appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But the hybridization near the K-points scales also with chirality ∝ cos 3α [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Then, the functional form of Es is Es = 1 2Eπ g (10, 0) − Cs � 1 R2 − 1 R2 10,0 � cos 3α, (6) where Eπ g (10, 0) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='93eV – is the un-hybridized bandgap of the (10,0) tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The radii, Rc and Rv, at which the S−band cross the conduction and valence bands, respec- tively, can be found by setting Es − 1 2Eπ g = 0, R = Rc, (7) Eπ g − Cs � 1 R2 − 1 R2 10,0 � cos 3α = 0, R = Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (8) This yields Rc = 1 2ac � bc + � b2c + 4accc � , where (9) ac = Cs cos 3α R2 10,0 + Eπ g (10, 0) 2 , bc = |p| γa 2 √ 3, cc = � Cs + sgn[1 − 2p]γa2 32 � cos 3α, and, Rv = 1 2av � −bv + � b2v + 4avcv � , where (10) av = Cs cos 3α R2 10,0 , bv = |p| γa √ 3, cv = � Cs − sgn[1 − 2p]γa2 16 � cos 3α, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 9, 10 yield three pairs of Rc, Rv – a pair for each p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' These radii thus delimit the bandgap regimes: large tubes (R > Rc) have their bandgap expressed by the usual π-band, Eπ g (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' smaller tubes (Rv < R < Rc) have the S-band between the valence and the conduction bands, hence the bandgaps in this regime become Eπ g − Cs � 1 R2 − 1 R2c � cos 3α (11) where Eπ g is the non-hybridized (π-band) gap (4) and CS ∼ 8 eV·˚A2 is a semi-empirical scaling factor (see ap- pendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The smallest tubes (R ≤ Rv), on the other 4 n, m ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [12] ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [13] ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [14] ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [15] ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [16] eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) 4, 0 0 0 0 0 0 0 5, 0 0 0 0 0 0 0 5, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='06 5, 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='12 5, 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='14 6, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='71 6, 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='7 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='497 20, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='506 Table II: Bandgaps (in eV) of semiconducting tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' hand, are metallic since the S-band crossed the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The bandgaps are then given by, Eg = � � � � � � � � � � � |p|γa R √ 3 + sgn[1 − 2p] γa2 16R2 cos 3α, R ≥ Rc, |p|γa R √ 3 + � sgn[1 − 2p] γa2 16 − Cs � cos 3α R2 + + Cs cos 3α R2 c , Rv < R < Rc, 0, R ≤ Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) This is the complete solution for bandgaps under no de- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It gives comparable results with various pub- lished simulations (table II);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (1) plots the bandgaps of zigzag tubes according to eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) and the simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Strain Having found the general gap equations (12), we wish to find its response to homogeneous deformation, such as axial strain, ϵz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Following the procedure in [9] and [8], the bond vectors (B4) ⃗lj = (ljy, ljz), transform as ⃗lj → D⃗lj, (13) where D is a deformation matrix, given, for axial strain, by D = � 1 − νϵz 0 0 1 + ϵz � (14) where ν is the tube’s Poisson ratio (table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This defor- mation alters the bond distances, |lj| → |lj|+δ|lj|, which in turn, alters the overlap integrals γj → γj + δγj ≈ γj + bδ|lj|, where b ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5eV/˚A([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This yields, using (13) and (14), δγj = ϵzb √ 3 a (−νl2 jy + l2 jz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (15) Now plugging the above three δγjs in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (1), we get δky = 1 2γ sgn[2p − 1]ϵzb(1 + ν) cos 3α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (16) This equation is the explicit form of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (1) under axial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' On substitution in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (2) we get, Eπ g = |p|γa R √ 3 + � γa2 8R2 − ϵz(1 + ν)ba √ 3 � × 1 2sgn[1 − 2p] cos 3α (17) But as discussed above, at small radii the π-band is crossed by and hybridized by the singlet band, down- shifting the gap (11), until it vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This yields the three regimes, as in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 12, Eg(ϵz) = � � � � � Eπ g , R ≥ Rc, Eπ g − Cs � 1 R2 − 1 R2c � cos 3α, Rv < R < Rc 0, R ≤ Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (18) 5 Figure 1: Bandgaps of semiconducting zigzag tubes vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 1/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The two type of semiconductors, p = ∓1 (for definition see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B1), are marked with red and blue colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The integers above/below the x-axis are aliases for the zigzags (n, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' they are positioned under their DFT bandgaps (dots connected by vertical lines);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the dots, labels a, b, c, d and e, refer to the correspond publications (see table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The full curves are plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The broken curve is the zone-folding ”1/R-rule” (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It is evident that, for large tubes, this rule holds with slight modification, but for smaller tubes the gap reverses trend and finally vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The critical radius of trend reversal is ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='2˚A, which lies between the tubes (11, 0) and (10, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The regions A, B and C correspond to the regimes R ≥ Rc, Rv < R < Rc and R ≤ Rv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' An unequivocal experimental verification of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (18) is not known to us, but the strain and chiral angle depen- dence of Eπ g was probed by [20] and qualitatively con- firmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Simulations of compressive strain in zigzag tubes [21] appear to agree with both the linear behaviour and p-dependence of Eπ g as given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Torsion To find the bandgap under an axial torsion, ζ, we start by setting its deformation matrix, D = � 1 0 −ζ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (19) following the steps applied to strain (14-18), we get for the un-hybridized π-bandgap Eπ g = |p|γa R √ 3 + 1 2sgn[1 − 2p] × � γa2 cos 3α 8R2 − ζba √ 3 sin 3α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (20) As with strain, this bandgap is valid only in the regime where the S-band is above the (π) conduction band (R > Rc, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Otherwise, hybridization with the S-band, as discussed above, strongly downshifts it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In complete equivalence to strain (18), the torsion bandgaps yield the general formulae, Eg(ζ) = � � � � � Eπ g , R ≥ Rc, Eπ g − Cs � 1 R2 − 1 R2c � cos 3α, Rv < R < Rc, 0, R ≤ Rv, (21) where Eπ g is the torsion-modified bandgap (20) of the π-band alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A B c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 p=-1 Cite a 1 worh e C p = +1 1 be C R a e p=-1 e Thil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 woI b p= +1 e b e 1714 11 8 5 1916 13 10 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='3 R6 Figure 2: Rc and Rv (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 9, 10) for p = +1 (red) and p = −1 (blue) tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The bandgap of tubes whose radius lies above Rc is not affected by hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' While the gap of tubes that lie between Rc and Rv (colored background) is affected – and reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The couple of tubes lying below Rv have zero gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The regions A, B and C , as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (2), correspond to the regimes R ≥ Rc, Rv < R < Rc and R ≤ Rv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' NON-HOMOGENEOUS DEFORMATION So far, the bandgaps found here concerned tubes with circular cross-section and axial uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But that is not the case in many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The cross-section of a bent tube, for example, is compressed in the inner side of bending-curvature and stretched in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Its de- formation profile is thus varying throughout the unit cell of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' To treat this and similar cases, we need to generalize the formalism of section (II A) to include these non-homogeneous deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' General formulation In the theoretical treatment of section II A, it was tac- itly assumed that the deformation is everywhere identi- cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This allowed us to deform a single graphene unit cell and extract the shift of the Fermi points from there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But that can not be done when the deformation varies throughout the tube’s unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Hence, if N is the num- ber of graphene unit cells in the nanotube unit cell, the position of the Fermi points, normally found by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (B3), is now found by the sum over the N A-atoms N � A=1 3 � j=1 γAjei⃗k·⃗lAj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (22) As before, we seek the lateral shift ∆Ky which is a simple sum of δky(A) (1) of the constituent A-atoms, ∆Ky = 1 N N � A=1 δky(A) (23) where δky(A) is now a function of the local strain and curvature at the position of the corresponding A’th atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The overlap integrals γAj in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 22 are functions of the in-plain and out-of-plain deformations of the bond vec- tors ⃗lAj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In-plain deformation, such as strain and torsion, changes the γ’s by changing the distance between neigh- boring π-orbitals, while out-of-plain deformation, such as the tube’s curvature, lowers the γ’s by having the orbitals misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In this section we include both deformations without assuming their homogeneity throughout the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' First consider curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' On a plain cutting through the cross section, let ±φy be the angles between neighboring π R(A) (11,0) Re(p 4 (7,5) (8,3) O (6,5) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='0) Rc(p (7,3) O (6,4) (8,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (7,2) A (8,0) 3 0 (5,3) (6,2) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='0) 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='3) (6,1) 0 R,(p (5,1) 0 (4,2) R(p 2 (3,2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content="0) B (4,0) 0 c 0 9'0 cos307 orbitals and the normal passing through the middle point between them;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' and let ±φz be the corresponding angles projected on a plain along the axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' If κθ, κz are the coordinate curvatures defined as (radius of curvature)−1 within the respective plains, and writing bond vectors (B4) in the form ⃗l = lyˆy + lzˆz, one gets |φy| = |ly|κθ/2 and |φz| = |lz|κz/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The overlap integral becomes γ → γ cos φy cos φz, so that for small change δγ is given by δγcurv = −γ 8 � l2 yκ2 θ + l2 zκ2 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (24) Let us now include local axial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Its deformation matrix, given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (14), induces a change δγstrain ac- cording to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The total change in the local overlap integrals is then, δγ = δγcurv + δγstrain ≡ l2 yDy + l2 zDz, (25) where Dy = νb √ 3 a ϵz − γ 8 κ2 θ, (26) Dz = −b √ 3 a ϵz − γ 8 κ2 z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (27) Substituting ly, lz from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B4 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (25) we find, δγA1 = a2 4c2 h � (n + m)2Dy + (n − m)2 3 Dz � δγA2 = a2 4c2 h � m2Dy + (2n + m)2 3 Dz � (28) δγA3 = a2 4c2 h � n2Dy + (2m + n)2 3 Dz � Inserting now eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 28 in 1 δky = sgn[1 − 2p]a(Dz − Dy) 2 √ 3γ cos 3α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (29) This, by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (23), we sum over all A-atoms in the tube’s unit cell, ∆Ky = sgn[1 − 2p]a cos 3α 2 √ 3γN N � A=1 (Dz − Dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (30) Converting now the sum to an integral, ∆Ky = sgn[1 − 2p]a cos 3α 2 √ 3πγ � 2π 0 (Dz − Dy)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (31) Once ∆ky is known, the energy gap follows immediately from the dispersion relation (B7) and the zone-folding gap (B8), Eπ g = aγ R ���� p √ 3 + √ 3R∆ky ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (32) Equation (32) is the non-homogeneous version of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (2), and likewise, it includes only the π-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But now the non-homogeneity of the deformation must be in- cluded through the integration in (31), we thus replace 1/R2 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (6) accordingly, 1 R2 → 1 2π � 2π 0 κ2 θ dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (33) Now the full set of gap equations for non-homogeneous deformation can be obtained by using equations (12) with the substitution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (32) for Eπ g , and a replacement as in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Elastic theory of bending and buckling The structural properties of bending and buckling of SWCNT’s had been simulated [1][4][22][23][24] [25] and experimented [22][2] in the years following nanotubes’ discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' They established that SWCNT’s abide by con- tinuous elasticity theory given an effective wall ¨thick- ness¨ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='66˚A (values by other groups in table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Which is always much smaller then the tube’s diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Hence SWCNTs are also ¨shells¨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' precisely: slender cylindrical shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The elastic theory of their bending, up to the onset of buckling, was developed a century ago by Brazier [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It is summarized in appendix A for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Brazier’s fundamental insight was that, in order to reduce shear under bending, the tube’s circular cross- section is ovalized (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' At the critical bending, how- ever, the elastic cost of increased ovalization exceeds the reduction in shear energy – and the tube buckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In the pre-buckling regime, Brazier’s theory gives the exact shape of the ovalized cross-section, parametrized by ξ in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It also predicts, as reproduced in the appendix, that at the onset of buckling the ovalization is ξ = 2/9 (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A19), independent of Poisson’s ratio or other material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' At the post-buckling regime, the elastic energy associ- ated with ovalization (beyond the buckling point)is ∆U = G(ξ − ξcr)2, ξ ≥ ξcr (34) where G is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But MD simulations [1][22] showed that in this regime, the post-buckling energy den- sity is linear with bending curvature, that is ∆U = Q(κ − κcr), κ ≥ κcr, (35) where Q is a constant and κcr is the critical bending curvature (A17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Comparing eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (34) and (35), ξ − ξcr = Q G(κ − κcr)1/2, κ ≥ κcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (36) This gives the proportionality relation in the kink, be- tween bending curvature, κ, and ovalization of the cross- section , ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Outside of the kink, however, This implies that outside the kink, strain is independent of further bending – its energy is absorbed in the kink, which acts 8 as a hinge between the two fixed sections of the tube [27][28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A realistic closed-form model of such kinks is not known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' But simulations revealed [29] that for bend- ing not too far from the onset of buckling, where the kink is shallow, the cross-section is shaped as an oval, just as the oval parametrized by Brazier in the pre-buckling regime (appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Since this regime is an intermidiate stage between the onset of buckling and a fully developed kink, it is also called the transient-regime [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This regime ends when the opposite walls of the kink approach contact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' at this point the distance between the walls is comparable with the inter-planar distance in graphite dg ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='35˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In Brazier’s parametrization this distance is 2R(1 − ξ) (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' We thus have ξclose = 1 − dg 2R, (37) where the superscript close signifies the closing of the cross-section in the kink and bringing the opposite walls to near contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' ξclose marks the upper limit of flattening within which our analysis is expected to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' For a tube of 1nm in diameter, ξclose ∼ 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The cross-section of a kink in the transient regime is thus bounded by the onset of buckling, at ξ = 2/9, and ξclose, having the curvature dependence according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (36), namely, ξkink = 2 9 + � κ − κcr κclose − κcr �1/2 � ξclose − 2 9 � , (38) where κcr ≤ κ ≤ κclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Equation 38 demonstrates the evolution of ξkink, being continuous at the onset of buckling ξkink = 2/9, through further bending where ξkink ∝ κ1/2, and finally reaching the closing point at ξclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Having found ξkink, the circumferential curvature in the center of the kink is given, as in the pre-buckling regime, through the out-of-plane deformation w (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A3) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A4) by, κkink θ = 1 R(1 + 3ξkink cos 2θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (39) The axial strain under bending, ϵz, is anti-symmetric with respect to the neutral plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' it thus has no contri- bution to the integration in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (31);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' also the bending curvature, κz ≪ 1/R, can be neglected in the integra- tion (31), although, as will be shown next, it affects κθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Hence we set in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (26-27) Dy = − γ 8 κ2 θ and Dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' According to the elastic theory of bending (appendix A), the circular cross-section of a bent tube becomes in- creasingly oval with bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Parametrizing the ovaliza- tion by ξ (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 5), let the circumferential integral of the square of the curvature (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A4) be I, then I ≡ 1 2π � 2π 0 κ2 θdθ = 1 + 9 2ξ2 R2 , (40) where the pre-buckling regime corresponds to 0 ≤ ξ ≤ 2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The point ξ = 2/9 corresponds to the critical cur- vature at buckling (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Bending – pre-buckling Since the ovalization in this regime is a quadratic func- tion of the bending curvature (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A14), the replacement rule (33) becomes, I = 1 R2 → 1 + 9 2ξ2 R2 = 1 R2 � 1 + (Lκ)4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (41) where L ≡ (1 − ν2)1/2 �9 2 �1/4 R2 t where ν and t are given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' With this replacement, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) yield the explicit gap equations under bending, Epre g (A) = |p|aγ R √ 3 + sgn[1 − 2p] γa2 16R2 � 1 + (Lκ)4� cos 3α, (42) where the superscript ”pre” refers to pre-buckling and ”A” corresponds to R ≥ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Region B then corresponds to radii in the range Rv < R < Rc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the bandgaps there are, Epre g (B) = |p|aγ R √ 3 + (sgn[1 − 2p] γa2 − 16Cs) 16R2 × � 1 + (Lκ)4� cos 3α, (43) and finally , at the smallest radii range, C, where R ≤ Rv, Epre g (C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (44) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (42–44) give the bandgaps for all radii in the pre-buckling regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the three ranges of radii, A, B and C are also shown in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' These equations reveal that, depending on the sign of p, bending may increase, decrease or even close the gap – a prediction which also coincides with simulations of ovalized cross- sections [30][31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Bending – critical curvature At the critical point of buckling the ovalization param- eter ξcr = 2/9 (appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This gives for the rhs of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (40) I = 11 9R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It is interesting to note that for metallic tubes (p = 0), replacing the rhs of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 41 for this value of I and substituting in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 42, gives, ∆Eg(κ = κcr) Eg(κ = 0) = 2 9, p = 0, R ≥ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (45) This universal ratio for metallic tubes (armchair tubes excluded as their Eg = 0), relates their bandgaps at the critical point of buckling with their straight value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Figure (3) depicts this in terms of the change in the bandgap as a function of the bending curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 9 Figure 3: Bandgap vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' bending curvature for primary metallic tubes, p = 0, (armchairs excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Eg is the gap of the straight tube (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' ∆Eg is the additional gap due to pure bending (42-44) and (48-50);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' κ is the bending curvature and κcr is the critical curvature (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In the pre-buckling regime the gap ∝ κ4 up to buckling point where it is higher by a factor of 2/9 compared with the straight state (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' At post-buckling, the bulk relaxes by a small amount δ (46) and remains there, while in the kink it is ∝ (κ − κcr)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Bending – post-buckling Since the elastic bending moment reaches maximum at buckling point [32], the buckling transition is accom- panied by a small relaxation of the strain energy [1] [22] outside of the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This relaxation also lowers Brazier’s ovalization and, as will be shown next, also the associated bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The pre-buckling elastic energy per unit length (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A16) can be approximated near critical point by U ∼ 1 2R3tEπκ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Denoting the post-buckling elastic energy relaxation per unit length by ∆U, the relaxed state corre- sponds to a lower curvature given by δκ = ∆U/(∂U/∂κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The associated shift in the band-gap ratio ∆Eg/Eg for metallic tubes (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 45) can be found by expand- ing this ratio near on the pre-buckling side of critical- ity, where the function is well behaved δ(∆Eg/Eg) = ∂(∆Eg/Eg)/∂κ)δκ where the derivative is taken at the buckling curvature κ = κcr (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' this yields, δ �∆Eg Eg � = R∆U 3πD , (46) where D is the elastic rigidity (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A11) and ∆U is the elastic energy relaxation per unit length at buck- ling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Equation 46 links, for metallic tubes, the mechani- cal post-buckling relaxation step with the post-buckling electronic band-gap change in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In contrast with the relaxation in the bulk, the bandgap in the kink is now a stronger function of bending curvature κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' this can be seen by applying the replacement rule (33) on the gap eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) using κkink θ (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Staying, as we do throughout this work, within the transient regime where the kink remains open (ξkink < ξclose, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 37, 38), bandgaps in the kink fall in three regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' these are related to the three regimes in straight tubes (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 1), corresponding to whether the gap is de- termined by the π-band alone (regime A), the modified regime where the singlet (S) band hybridizes with the π band (regime B), or zero – where the S-band downshifted enough to completely close the gap (regime C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' All tubes fall into one of these regimes depending on their diame- ter (largest to smallest – regimes A to C, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Here, however, the regimes are determined by the cur- vature within the kink which is bounded by I < 1/R2 c (regime A), 1/R2 v > I > 1/R2 c (regime B), and I > 1/R2 v (regime C), where I is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (40) and Rc, Rv are given by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (9 – 10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Explicitly, regime B is bounded by 2 9 �R2 R2c − 1 � < (ξkink)2 < 2 9 �R2 R2v − 1 � , (47) while regime A applies at the lower bound and regime C in the upper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The bandgaps of the kink in the various regimes of bending can now be found by replacing 1/R2 in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) with (1 + 9 2(ξkink)2)/R2 (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 41), which gives Ekink g (A) = |p|aγ R √ 3 + γa2 16R2 sgn[1 − 2p] × � 1 + 9 2(ξkink)2 � cos 3α, (48) where ξkink is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In region B, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) then give, Ekink g (B) = |p|aγ R √ 3 + (sgn[1 − 2p] γa2 − 16Cs) 16R2 × � 1 + 9 2(ξkink)2 � cos 3α, (49) and as before , in regime C, Ekink g (C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (50) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (48–50) give the bandgaps in the center of the kink for the respective radii ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Here, as in pre- buckling regime, the bandgap as a function of bending can increase, decrease, or vanish, depending on the sign of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The difference with the pre-buckling regime, however, is that here it is ∝ (κ−κcr)1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' this behaviour is depicted for zigzag tubes where p = +1 (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 4a) and p = −1 (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 4b) and for metallic tubes of all chiral angles in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It may be useful to compare the bandgaps of the kink with the bulk (far from the kink – where |z| ≫ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' As- suming that in the bulk ξ remains near criticality, the AEE Kink 94 Bulk10 (a) p = +1 (b) p = −1 Figure 4: Bandgaps and Fermi energies vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' bending for semiconducting zigzag tubes of the two types (p = ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Following eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (42), the initial gaps first decrease (left) / increase (right) ∝ κ4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' at post-buckling it evolves faster: ∝ (κ − κcr)1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' at increased curvature – where the singlet band crosses the conduction band – the gap strongly downshifts in all cases (49);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' at this point, also the Fermi energy (C2) downshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The point of EF = 0 in the plots corresponds to the value of EF in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' bandgap in regime B (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 49) compared with the bulk gives, Ekink g −Ebulk g = − 9Cs 2R2 � (ξkink)2 − �2 9 �2� cos 3α, (51) where ξkink is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It is worth noting that for armchair tubes (p = 0, α = 30o), these equations (as well as the pre-buckling ones) predict no bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' That holds, however, as long as the underlying assumption through this work – that the kink remains open [29] – holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' SUMMARY We presented here a comprehensive theory of bandgaps in carbon nanotubes, including strong curvature and large non-homogeneous deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This theory recon- ciles the fundamental theory of bandgaps in nanotubes (appendix B) and its well-known corrections (at small deformation and small curvature), with contrasting re- sults from a host of DFT computations of very small or highly deformed tubes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the present theory shows them to be special cases of the same general equations (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A formalism was derived to calculate the gaps due to a general non-homogeneous circumferential deformation (by starting with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 12 and making the replacement 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' We then applied this formalism to study bending, including buckling and a kink (with a caveat of staying within the transient regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='e: where the opposite walls of the kink do not touch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The results detail the gap evolution under both weak and strong bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In the pre-buckling regime, the bandgap shifts ∝ ±κ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A notable result is that, by the onset of buckling, the gaps of primary-metallic tubes, in- dependent of chiral vector or radius, increase by a ratio of 2/9 compared with their un-bent value, (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 45 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In the post-buckling regime (κ > κcr, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A17), at first, the bandgap in the kink shifts ∝ ±(κ−κcr)1/2, up-to the point where the singlet band crosses the conduction π- band initiating a steep downshift to zero (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 48-50, and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The downshift in the bandgap is accompanied, in this regime, by a substantial downshift of the Fermi energy (appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' APPENDICES Appendix A: Brazier’s Theory This appendix has a number of relevant derivations from the elastic theory of thin cylindrical shells under pure bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The theory was derived by Brazier [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' For the exposition we follow reference [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Consider a slender cylindrical shell, initially straight, under bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The strain profile is anti-symmetric about the neutral plane, compressive in the inner side and tensile in the outer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the energy per unit length is given by Uz = 1 2Iκ2 (A1) where κ is the bending curvature and I is the second moment of area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' for a perfectly circular cross-section I ≡ I0 = πR3t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A2) Eg EF 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 (13,0) (16,0) (22,0) (31,0) (13,0) (16,0) (22,0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 1(11,0) Eg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5+ (14,0) (17,0) (20,0) K/A (11,0) (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='0) (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 1,11 (a) Cross-section (b) Profile Figure 5: Bending causes the circular cross-section to become oval, turning R → R(1 + ξ cos 2θ) where ξ is the ovalization parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' at ξ = 2/9 (blue shape) the tube reaches criticality and buckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' where t is the thickness of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It will be shown below that by flattening the cross section the tube reduces its energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It is reasonable to assume that small flattening can be expressed by an out- of-plain displacement of the form w = Rξ cos 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A3) where ξ (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 5) is a dimensionless measure of the flat- tening, or ovalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The curvature due to ovalization is δκθ = −∂2w ∂y2 − w R2 = 3ξ R cos 2θ, (A4) where y = Rθ is the circumferential coordinate of the original circle, the second derivative is the usual curva- ture due to local changes of displacement while the sec- ond component is due to the change of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The in- plain deformation v, together with w (A3), determines the circumferential strain ϵy = ∂v/∂y + w/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' As a first approximation Brazier assumed the surface to be inex- tentional, ie: ϵy = 0, yielding v = −R 2 ξ sin 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A5) where we assumed no net rotation by taking the constant of integration to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' It may be commented that for the electronic structure calculation we take a much looser requirement, instead of assuming the surface to be inextentional it is assumed only that the circumference is unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='e: � ϵydy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The second moment of area in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A1 is actually de- fined as I = � 2π 0 Rt(R sin θ + η)2dθ, (A6) where t is the surface thickness and η is given by η = w sin θ + v cos θ = −Rξ sin3 θ (A7) where the last step used eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A3 and A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Inserting this in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A6 we finally get I = I0 � 1 − 3 2ξ + 5 8ξ2 � (A8) where I0 is given by A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The strain energy per unit length (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A1) is then given by Uz = 1 2κ2R3tEπ � 1 − 3 2ξ + �5 8ξ2 �� (A9) where E is the Young’s modulus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' the ξ2 component in the square brackets is truncated in Brazier’s analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The energy per unit length due to Brazier deformation is UB = 1 2D � 2π 0 δκ2 θRdθ, (A10) where δκθ is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A4 and D is the elastic rigidity given by D = Et3 12(1 − ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A11) Integration of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A10 gives UB = 3πEt3 8(1 − ν2) ξ2 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A12) The total elastic energy per unit length is U = Uz + UB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A13) Now ξ can be found by requiring ∂U/∂ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The result is ξ = (1 − ν2)R4 t2 κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A14) n R 0 s)2R neutral pla 2RC cell12 The values of ν and t were found by a number of groups (see table I), so we can write ξ = BR4κ2, (A15) where B = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='2(˚A−2) by the values of [1] compared with B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='57(˚A−2) by [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The total energy (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A13) is finally given by U = 1 2R3tEπκ2 − �3R7(1 − ν2)Eπ 8t � κ4 (A16) The tube begins to buckle when the bending moment M = dU/dκ, reaches maximum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' hence the curvature at buckling point can be found by putting d2U/dκ2 = 0, which gives κcr = t 3R2 � 2 1 − ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A17) κcr is the critical curvature at the onset of buckling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Us- ing the same values as for eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A15, we find κcr = A R2 , (A18) where A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='316˚A using the values of [1], compared with A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='376˚A by [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' MD simulations confirmed the general shape eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A18) and found A to range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='185 [3] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='387˚A [1] Regardless of the actual numer- ical value of A, substituting A17 in A14 gives the exact ovalization parameter at buckling point ξcr = 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (A19) This is a remarkable result of Brazier’s theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' it states that at the critical point of buckling, all tubes, indepen- dent of thickness, radius or Young’s modulus, become ovalized by the same ratio of 2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Appendix B: The fundamental bandgap The band structure of carbon nanotubes is based on graphene’s band structure [33] sliced-up with lateral quantization lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The exact wrapping of graphene into a nanotube is determined by the tube’s chiral integers (n, m) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Now, any two integers can be related by other two integers (q, p), such that, n − m = 3q + p (B1) where p takes one of the values (0, +1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The lateral k-vectors, ky, must lie on quantization lines given by ky = ν R, where ν = 0, ±1, ±2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (B2) Graphene’s Fermi surface is found by the k-vectors that solve, 3 � j=1 γjei⃗k·⃗lj = 0 (B3) where γj are the three overlap integrals of the nearest π-orbitals, and lj are their bond vectors, given by ⃗l1 = a2 4πR � (n + m)ˆy − 1 √ 3(n − m)ˆz � , ⃗l2 = a2 4πR � −mˆy + 1 √ 3(2n + m)ˆz � , (B4) ⃗l3 = a2 4πR � −nˆy − 1 √ 3(n + 2m)ˆz � , where (n, m) are the chiral integers, and (ˆy, ˆz) are the circumferential and axial coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Now under the assumption that graphene is isotropic, we can take all γj ≡ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (B5) The solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (B3) is then given by the two points, ⃗KF 1 = 1 3R � (m + 2n)ˆy + m √ 3ˆz � ⃗KF 2 = 1 3R � (n − m)ˆy + (m + n) √ 3ˆz � (B6) These are the Fermi points of graphene, in the nanotube’s natural coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The spectrum near a Fermi point is linear, E = ± √ 3 2 aγ δk, (B7) where E is the energy above the Fermi level and δk is the distance from the nearest Fermi point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Substituting (B1) in (B6) reveals that when p = 0 they do lie on a quantization line (B2) – these tubes are thus, according to zone-folding, metallic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' For p = ±1, on the other hand, the distance to the nearest quantization line is 1/3R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' these tube are thus semiconducting with an energy gap given by substituting 1/3R in the linear spectrum (B7), Eg = |p| aγ √ 3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (B8) Appendix C: Fermi energy and Cs The lowering of the singlet (S) band below the con- duction (π) band causes, as DFT simulations demon- strate (table IV), a downshift not only of the bandgap but also the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' While it is generally under- stood to be a result of a large circumferential curvature 13 of small tubes, we wish to quantify it by extrapolating the published data for tubes in this radii range (regime B: section II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' In this regime, the bandgap depends on the position of the singlet band above the Fermi points (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 6), which depends, in turn, on the proportionality factor Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The DFT bandgaps of semiconducting, straight and un- deformed tubes (table II) in this regime (R ≲ 4˚A) give Cs ≈ 8(eV · ˚A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' ξ 11, 0 10, 0 8, 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='0625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='1825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='55 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='3125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='075 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='4375 0 0 0 Table III: Energy gaps (eV) in the respective tubes as a function of the deformation parameter ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' data extracted from figure 1a in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [31] Figure 6: Bandgap vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' total circumferential curvature square for the zigzags (11,0), (10,0) and (8,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Data points are given in table (III);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' lines are plotted by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (12) (regime B) where 1/R2 was replaced by I (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 40) with Cs ∼ 8 − 10 (eV·˚A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The crosses label the critical points of buckling (given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' A18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Moving to Fermi energy, DFT simulations show (ta- ble IV) that the Fermi-energy of tubes in this regime are downshifted too in ∝ 1/R2 (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 7 ), while larger tubes have it identical to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The S-band in this regime is the effective conduction band, and thus, the Fermi en- ergy lies in the middle between it and the valence π-band (which is not shifted), EF = 1 2(Es + Ev) = 1 2 � Es − Eπ g 2 � , (C1) where the singlet band energy Es is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (6) and the pure π bandgap Eπ g is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' This gives explicitly, EF = c1 − |p| γa 4 √ 3R − cos 3α R2 � c2 + sgn[1 − 2p]γa2 64 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (C2) where the constants are c1 = Eg(10, 0) 4 + Cs 2R2 10,0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='53 eV, c2 = Cs 2 ≈ 4 eV · ˚A 2, where we used Eg(10, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='87 eV (table II), R10,0 = 4˚A, and Cs = 8 eV ·˚A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' n, m ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [12] ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [13] 4, 0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='23 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='29 5, 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='64 7, 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='38 8, 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='14 10, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='04 Table IV: Fermi energies (in eV) of semiconducting zigzag tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' What was actually computed are work-functions, (WF)tube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Fermi energies shown here were then found relative to graphene by Etube F = (WF)graphene − (WF)tube, where (WF)graphene = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='55eV in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [12] and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='66eV in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Now this can be further simplified if we neglect the second term in the parenthesis in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (C2) and, since EF (R ≥ Rc) = 0, following the downshift of the singlet band Es (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 6), EF = Cs 2 � 1 R2c − 1 R2 � cos 3α, R ≤ Rc, (C3) where Rc is given, by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' for zigzag tubes (α = 0), Rc = R(10, 0) = 4˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (C3) is depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (7) for semiconducting zigzag tubes in this range, together with DFT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Considering circumferentially ovalized tubes, follow- ing our procedure we first replace 1/R2 with (1 + 9ξ2/2)/R2 ≡ I (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 40), where ξ is the ovalization pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Explicitly, EF (ξ) = Cs 2R2 �� R Rc �2 − 1 + 9 2ξ2 � cos 3α, R ≤ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (C4) 1 (10,0) (11,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='6eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='595eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='5 (8,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='25 I(A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content='114 Figure 7: Fermi energy of a number of small zigzag tubes vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' 1/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' The line is a plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (C3) with Cs = 8 (eV·˚A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Data is in table IV where red dots taken from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [12], green dots from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Comparing this with the DFT bandgaps (table III), plot- ted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' (6), yields Cs ∼ 8 (eV·˚A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' Finally, the fact that the analytic treatment in this work coincides with DFT on three different quantities: bandgaps of straight tubes, ovalized tubes, and Fermi energy of straight tubes, by having a single adjustable parameter, Cs ∼ 8 (eV·˚A2), is, in our opinion, a strong indication of the correctness of this treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} +page_content=' I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf'} diff --git a/AtAyT4oBgHgl3EQfd_hU/content/tmp_files/2301.00312v1.pdf.txt b/AtAyT4oBgHgl3EQfd_hU/content/tmp_files/2301.00312v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd18ba355f360946ab229d2ec1e94cd3bbd2702a --- /dev/null +++ b/AtAyT4oBgHgl3EQfd_hU/content/tmp_files/2301.00312v1.pdf.txt @@ -0,0 +1,951 @@ +Collision of Environmental Injustice and Sea Level Rise: +Assessment of Risk Inequality in Flood-induced Pollutant +Dispersion from Toxic Sites in Texas + + +Zhewei Liu1, Ali Mostafavi1* + + +1 UrbanResilience.AI Lab, Zachry Department of Civil and Environmental Engineering, +Texas A&M University, College Station, TX, 77843 + +*mostafavi@tamu.edu + + + + +Abstract: +Global sea-level rise causes increasing threats of coastal flood and subsequent pollutant dispersion. +However, there are still few studies on the disparity arising from such threats and the extent to which +different communities could be exposed to flood-induced pollution dispersion from toxic sites under +future sea level rise. To address this gap, this study selects Texas (a U.S. state with a large number of +toxic sites and significant flood hazards) as the study area and investigates impacts of flood-induced +pollutant dispersion on different communities under current (2018) and future (2050) flood hazard +scenarios. The results show, currently, north coastline in Texas bears higher threats and vulnerable +communities (i.e., low income, minorities and unemployed) are disproportionally exposed to these +threats. In addition, the future sea-level rise and the exacerbated flood hazards will put additional +threats on more (about 10%) Texas residents, among which vulnerable communities will still be +disproportionately exposed to the increased threats. Our study reveals the facts that potential coastal +pollutant dispersion will further aggravate the environmental injustice issues at the intersection of +toxic sites and flood hazards for vulnerable populations and exacerbate risk inequalities. Given the +dire impacts of flood-induced pollution dispersion on communities’ health, the findings have +important implications for specific actions from the policy makers to mitigate the inequitable risks. + +Key words: environmental justice; sea level rise; flood-induced pollutant dispersion; Texas; + + + + +1. Introduction +About 60% of non-federal National Priorities List (NPL) sites (i.e., the priority list of hazardous waste +sites) in the United States, are in the areas that may be affected by impacted by the climate change effects +(Gómez 2021). The projected sea level rise and the accompanying coastal flood can cause pollutants +dispersion from the sits and threaten public health. Moreover, the overburdened communities of color and +low-income may disproportionally dwell in the regions around these sites and bear particular health +impacts (Crawford 1994, Carter and Kalman 2020), which worsens the living conditions of socially +vulnerable communities and gives rise to exacerbated environmental injustice issues in the context of +global climate change. +Efforts have been made by previous studies to investigate flood risk and its environmental justice +implications. Studies have found that socially vulnerable communities from different backgrounds +experience drastic inequality in exposure to the flood risk, and such inequity show varied extent across +different regions (Collins, Grineski et al. 2019). For example, in the context of UK, some studies found +that no displayed socioeconomic disparity regarding to the risks of riverine/inland pre-flood (e.g., +residence in 100-year flood zones), but regions with lower socioeconomic status are exposed to higher +coastal risks (Fielding 2007, Walker and Burningham 2011).While studies in the US has documented +mixed findings: socially advantageous communities bear more risk towards pre-land risks in some regions +such as Miami, while disadvantageous communities bear more risk in other regions, e.g., Houston +(Chakraborty, Collins et al. 2014). The built environment and amenities can be possible factors +contributing to the difference across different regions (Chakraborty, Collins et al. 2014, Grineski, Collins +et al. 2017, Collins, Grineski et al. 2018). +However, one important limitation of the existing literature is the insufficient attention to evaluating the +intersection of flood exposure and environmental justice issues surrounding toxic sites. In particular, the +existing studies have paid limited attention to exacerbated flood hazard exposure of toxic sites under +future sea-level rise scenarios. Recognizing this important gap, this study examines the coastal area of +Texas, USA, which has a large number of toxic sites and significant flood hazard exposure as a case study +region to answer the following research questions: +• +RQ1: To what extent coastal regions are threatened by pollutant dispersion from flood under +current flood hazard scenario? +• +RQ2: What is the extent of hazard exposure disparity among vulnerable populations? +• +RQ3: To what extent the future flood hazards exacerbated by sea level rise increase the threats of +pollutant dispersion exposure? +• +RQ4: How the increase in threats is disproportionately affecting vulnerable populations? +To address these questions, we utilize the socioeconomic datasets, and coastal inundation maps in the +current and future scenarios for the analysis. The results reveal that, in the current scenario (2018), the +areas along the north coastline in Texas (such as Houston and Beaumont), are subject to higher threats to +pollution dispersion, and that the communities of low-income, minority, unemployed suffer from greater +threats. Moreover, the future sea level rise by 2050 will increase the flood threats to more people, among +whom vulnerable populations will be disproportionately exposed to flood-induced pollutant dispersion +from toxic sites. +The remaining sections of this paper is structured as follows: Section 2 summarizes the previous relevant +studies; Section 3 elaborates the datasets and methodology; Section 4 details the experimental results; + +Section 5 discusses the study implications and proposes practical suggestions for policy makers; Finally, +Section 6 discusses the study limitations and future research directions. +2. Background +Floods threaten human life in various ways. One notable way is the impacts on human health, including +injuries, diseases and psychological trauma (Du, FitzGerald et al. 2010, Stanke, Murray et al. 2012, +Zhong, Yang et al. 2018, Graham, White et al. 2019, Palinkas and Wong 2020) . The landscape of the +flooded areas can be permanently changed and contaminated, and expose the residents to long-term +chronic diseases (Euripidou and Murray 2004, Karaye, Stone et al. 2019), and also damage the properties +and built environments (Brody, Zahran et al. 2008, Merz, Kreibich et al. 2010, Alipour, Ahmadalipour et +al. 2020, Kousky, Palim et al. 2020). +Environmental justice is another concerned issue associated with flood hazards. Due to the socioeconomic +status and cultural background, different demographic communities suffer from unequal flood hazards +exposure and impacts. The studies from different regions of the world have shown different findings. +Some UK-based studies conclude that people of lower socioeconomic status bear higher risk from coastal +flood (Fielding 2007, Walker and Burningham 2011). Studies from US reveal that in Miami, the non- +Hispanic Black and Hispanic are disproportionately exposed to inland flood risk, and underrepresented +with coastal flood risk (Chakraborty, Collins et al. 2014). And the study from Canada shows that senior +people (over 65 years old) are the groups that are more vulnerable to coastal climate change in Atlantic +Canada (Manuel, Rapaport et al. 2015). These findings demonstrate the significance of examining the +intersection of environmental justice issues and flood hazards, especially under future climate change +impacts such as sea level rise. +A peculiar type of flood impacts is pollutant dispersion from industrial and toxic sites that can be +inundated due to flooding. Studies from the US show that vulnerable communities are exposed to +disproportional risks from superfund sites (Carter and Kalman 2020). Communities with greater +proportion of minority and low-income residents experience higher health risk, because of their closer +residence to superfund sites (Crawford 1994). Some reports argue that race was the most significant factor +in predicting the hazardous facilities (Christ 1987). Floods exacerbate the exposure of residents to +pollutants released from these industrial and toxic sites and such exposures could lead to long-term +chronic health impacts. Increased flood hazards due to sea level rise can make these hazards and impacts +even worse. However, limited studies exist to evaluate the extent to which current and future flood +hazards disproportionately affect populations living in proximity of industrial and toxic sites. +There are also some other determinants for the threats of flood, such as water-based amenities, self- +protection procedures. Usually, the regions with water-based amenities are associated with higher risk of +flood, due to the abundance of water body in the amenities (Collins 2010, Collins, Grineski et al. 2018). +The structures of the building, such as elevation and flood-proofing materials, will also reduce the flood +impacts (Botzen, Aerts et al. 2013, de Moel, van Vliet et al. 2014). The urban built environment factors, +like ground imperviousness, land use type, distance to the streamline, etc., will also affect the locations’ +vulnerability to flood (Mobley, Sebastian et al. 2019, Dong, Yu et al. 2020). Besides these environmental +factors, another factor is the subjective human perceptions on flood, and the related information collected +from social sensing (Yuan, Yang et al. 2021, Yuan, Fan et al. 2022). People’s awareness of flood risk will +influence selection of home locations, and flood-proof materials, which eventually affect the loss caused +by the flood (Lindell and Hwang 2008, Heitz, Spaeter et al. 2009, Kellens, Zaalberg et al. 2011, Harlan, +Sarango et al. 2019, Ridha, Ross et al. 2022). + +3. Datasets and Methodology +3.1 Datasets +The study focuses on coastal areas in Texas as our study area. Coastal areas of Texas have a large number +of industrial and toxic sites and also highly exposed to flood hazards. Four kinds of datasets are used in +the study: (1) Industrial facilities, which is provided by the United States Environmental Protection +Agency (USEPA) and include locations of facilities that emit air pollutants during industrial process +(EPA 2022); (2) Toxic facilities, which is also provided by USEPA, and include locations of toxic +facilities such as National Priority List Sites (NPL) and Toxic Release Inventory Sites (TRI) (EPA 2022); +(3) flood map caused by sea level rise, which is provided by Deltares global flood map (Microsoft 2022) +and provides the inundation maps of flood along coastal and the flood depth, in current and predicted +future scenarios ( in the year of 2018 and 2050 separately); this open dataset flood map is one of the most +reliable publicly available inundation maps available, (4) socioeconomic census data at the census tract +level, provided by the United States Census Bureau (USCB). A description of the datasets is displayed in +Table 1. +Table 1. Datasets in the study and key attributes +Datasets +Source +Description +Industries & Toxic +facilities +USEPA +• +The industrial facilities include the +facilities that emit air pollutants +during industrial process. +• +The toxic facilities include the type +and locations of NPL and TRI sites + + +Flood map by sea level +rise in 2018 and 2050 +Deltares global +flood map +The location of inundation along the +coastline and the flood depth +Socioeconomic census +data at the census tract +level +USCB +Statistics data for each census tract, +including total population, income, minority +population, etc. +3.2 Methodology +This study aims to compare the extent to which certain groups of population are threatened by flood- +induced pollutant dispersion from industries and toxic facilities, under current and future flood scenarios. +Consequently, the first step is to identify the flooded facilities. We overlay the spatial distribution of +industries and toxic facilities with the distribution of flood map along the coastline (as illustrated in +Section 3.1). As the flood map produced by Deltares uses discrete points to represent the flood, we create +a buffer of 0.1 mile based on the points in the flood map and the facilities that fall within the buffer are +identified as the flooded facilities. +Then, the second step is to quantify the population that can be potentially threatened by the pollutant +dispersion from the flooded facilities. To this end, we further create 1 mile, 3 miles and 5 miles buffers +(Carter and Kalman 2020), based on the flooded facilities and the threatened population for each census +tract is the tract’s total population multiplied by the proportion of tract’s area that falls within the flooded +buffer zone. An explanation in formula is given in following equation: +𝐴_𝑃𝑜𝑝𝑢𝑖 = 𝑇_𝑃𝑜𝑝𝑢𝑖 ∗ +𝐴𝑟𝑒𝑎_𝑖𝑛_𝑏𝑢𝑓𝑓𝑒𝑟𝑖,𝑛_𝑚 +𝑇_𝐴𝑟𝑒𝑎𝑖 + + +Where 𝐴_𝑃𝑜𝑝𝑢𝑖 is the threatened population in the 𝑖𝑡ℎ census tract, 𝑇_𝑃𝑜𝑝𝑢𝑖 is the tract’s total +population, 𝐴𝑟𝑒𝑎_𝑖𝑛_𝑏𝑢𝑓𝑓𝑒𝑟𝑖,𝑛_𝑚 is the tract’s area that falls within n-mile (n= 1, 3, and 5) buffer zone of +the flooded facilities, and 𝑇_𝐴𝑟𝑒𝑎𝑖 is the tract total area. We examine vulnerable populations, that are (1) +below poverty, (2) of ethnical minority, (3) unemployed and (4) without high school diploma, to provide +a comprehensive overview about how different communities, especially disadvantageous communities +who live in proximity of industrial and toxic sites and suffer from the flooded caused by sea level rise, +which put new threats on the environmental justice. The flood maps in 2018 and 2050 are used separately +for comparison under current and future scenarios. +4. Results +4.1 Census tracts and population threatened by the pollutant dispersion under +current scenario (2018) +Three different distances (i.e., 1 mile, 3 miles and 5 miles) are used to create buffers for the flooded +facilities based on the flood map of 2018, and then identify the affected tracts (shown in Figure 1). It +shows that as the distance of buffer expands, the number of increased tracts steadily increases. The north +Texas along the coastline are the regions severely affected, including cities such as Houston, Beaumont, +Nederland and Port Arthur. + + + +(a) (b) (c) +Figure 1. The census tracts threatened by the pollutant dispersion within (a) 1-mile buffer, (b) 3-mile +buffer and (c) 5-mile buffer, in 2018. +The affected tracts along the North coastline are also the regions with higher population density and +hence, smaller census tracts, compared with other regions in the central and south Texas, leading to more +population being exposed to the threat of potential pollutant dispersion due to flood risk. The number of +the threatened population in each census are shown in Figure 2. It can be seen that Houston and +Beaumont are the regions with large amount of population threatened by the pollutant dispersion (Details +in Appendix). +A similar pattern can be drawn from Figure 3, which compares the ratio of threatened population +(threatened population/total population) in each census tract. The results indicate that, due to the high +population density and high concentration of industrial and toxic facilities, majority of the populations +along the coastline in Beaumont, Nederland and Port Arthur, are exposed to flood-induced pollutant +dispersion threat. The residents in Houston and Corpus are also potentially threatened, especially along +the coastal areas, where industrial and toxic facilities locate. + +N +Censustract +within1-mile +bufferof +flooded +industrial& +toxic facilities +0 +3672 +144MilesN +Censustract +within3-mile +bufferof +flooded +industrial& +toxicfacilities +0 +36 +72 +144MilesN +Censustract +within5-mile +bufferof +flooded +industrial& +toxicfacilities +0 +36 +72 +144Miles + + + + + (a) (b) (c) +Figure 2. The population threatened by the pollutant dispersion in each census tract, within (a) 1-mile +buffer, (b) 3-mile buffer and (c) 5-mile buffer, in 2018. + + + + (d) (e) (f) +Figure 3. The ratio of the population threatened by the pollutant dispersion in each census tract, +within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer, in 2018. +The vulnerable communities are especially subject to the threat of potential risk. As stated in Section 3.2, +this study focus on four types of socially vulnerable groups and examines the extent to which they are +particularly threatened by the pollutant dispersion. Table 2 shows the mean income, proportions of +minority population, below-poverty population, unemployed population and population without high- +school diploma, in all the census tracts in Texas, and the tracts within certain distances of buffers of the +flooded sites. A radar chart is further shown in Figure 4, with all the proportion normalized for visual +interpretation. It can be clearly seen that, in the regions that are in the proximity of the flooded sites, there +is a high concentration of socially vulnerable populations, whose living situation and health condition +may be further exacerbated due to the potential pollutant dispersion from the flooded facilities. + + + + + + + +N +Beaumont +Houston +Affected +Population +0.0 +Corpus +7,200 +0 +48.5 +97 +194MilesN +Beaumon +Houston +Affected +Population +0.0 +Corpus +9,300 +0 +48.5 +97 +194MilesN +Beaumont +V +Houston +Affected +Population +0.0 +Corpus +14,000 +0 +48.5 +97 +194MilesN +Beaumont +Houston +Ratioof +Affected +Population +0.00076 +Corpus +1.0 +0 +48.5 +97 +194MilesBeaumont +N +Houston +Ratioof +Affected +Population +0.000014 +Corpus +1.0 +0 +48.5 +97 +194MilesBeaumontN +Houston +Ratioof +Affected +Population +0.00020 +Corpus +1.0 +0 +48.5 +97 +194MilesTable 2. The proportions of socially vulnerable communities in different regions + +Mean of All +Census Tracts +Mean of Tracts +within 1 mile of +flooded sites +Mean of Tracts +within 3 mile of +flooded sites +Mean of Tracts +within 5 mile of +flooded sites +Per capita income +29,684 +24,736 +24,296 +24,102 +Proportion of +minority +57.6% +60.5% +65.6% +69.5% +Proportion of +below-poverty +population +15.1% +19.7% +20.6% +19.3% +Proportion of +unemployed +population +2.7% +3.1% +3.2% +3.3% +Proportion of +population +without +high-school +diploma +10.7% +14.0% +16.0% +16.0% + + + +Figure 4. A radar chart of the Table 2, with the proportions normalized for visual comparison. + +Normalized proportion +of below-poverty population +Normalized proportion +of unemployed +Normalized proportion +0 +0.2 0.4 0.6 0.8//1 +population +of minority +All Census Tracts + Tracts within 1 mile of flooded sites +Normalized proportion of +- Tracts within 3 mile of flooded sites +population without +Tracts within 5 mile of flooded sites +high-school diploma4.2 Comparison of threats by flood-induced pollutant dispersion under current +(2018) vs. future (2050) scenarios +To analyze the extent to which the exposure of socially vulnerable communities may be aggravated due to +the future seal level rise, the flood map in 2050 is used to identify the census tracts and population +threated by the pollutant dispersion in the future, and a comparison is made between the current and +future scenarios (shown in Figure 5). The results show that, compared to the current flood scenario in +2018, there will be about 10% more population threatened by flood-induced pollutant dispersion in 2050. + +Figure 5. Population threatened by the flooded industrial & toxic facilities, separately in 2018 and +2050. +The increased exposure of populations that are threated in each census tract, from 2018 to 2050, is further +illustrated in Figure 6. It can be seen that the sea level rise in the future will inevitably cause more +population exposed to the threat of pollutant dispersion from flooded industrial and toxic facilities. The +threats are especially obvious for certain regions in Houston and Beaumont. We further calculate the ratio +of increased population (increased_population_under_threat/total_population) in each census tract (Figure +7). The results show that, among the cities in Texas, Houston will be the one that especially suffer from +the increased flood-induced pollutant dispersion threats caused by the seal level rise. Compared with +situation in 2018, certain tracts in Houston will have 50% more population affected by the pollutant +dispersion in 2050 (Details in Appendix). + +1000000 +925614 +900000 +812553 +800000 +POPULATION +700000 +600000 +501777 +500000 +461401 +400000 +300000 +200000 +136871 +125219 +100000 +0 +1-mile +3-mile +5-mile +12018-year +125219 +461401 +812553 +2050-year +136871 +501777 +925614 +BUFFER SIZE + + +(a) (b) (c) +Figure 6. From 2018 to 2050, the increased population threatened by the pollutant dispersion in each +census tract, within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer. + + + +(a) (b) (c) +Figure 7. From 2018 to 2050, the ratio of increased population threatened by the pollutant dispersion +to total population, in each census tract within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile +buffer. Compared with situation in 2018, certain regions in Houston get 50% more population under +flood-induced pollutant dispersion threat in 2050. +4.3 Who will suffer more, from the projected seal level rise? +We investigate the demographics of the additional 10% more population (Section 4.2) that will be +threatened by the increased pollutant dispersion threats from the flooded facilities in 2050, and compare +the results with the demographics of all the census tracts in Texas (Figure 8). The result shows that for the +increased population that are under threat, the means of proportions of unemployed population (3.2% for +1-mile buffer zone, 3.3% for 3-mile buffer zone and 3.5% for 5-mile buffer zone) and population without +high-school diploma (15.0% for 1-mile buffer zone, 13.2% for 3-mile buffer zone, and 13.0% for 5-mile +buffer zone) are consistently higher than those of the overall population in Texas (2.7% and 10.7% +respectively). On the other hand, for the census tracts that are within certain distances of the flooded +facilities in 2050, the proportion of below-poverty (19.5 %for 3-mile buffer zone and 18.9 % for 5-mile +zone) and minority (60.0% for 3-mile buffer zone and 63.6 % for 5-mile buffer zone) are also consistently +higher than those of overall population (15.1% and 57.7%). +These results reveal that, socially vulnerable communities not only have more exposure from the potential +pollutant dispersion from the flooded facilities under current flooding hazards, but also will bear a greater +threat due to the increasing seal level rise in the future flood scenarios. + +N +Increased +Population +0.0 +2,200 +041.2582.5 +165MilesN +Increased +Population +0.0 +2,000 +41.2582.5 +165MilesN +Increased +Population +0.0 +2,900 +041.2582.5 +165MilesN +Increased +PopulationRatio +0.0 +0.70 +0 +41.2582.5 +165MilesN +Increased +PopulationRatio +0.0 +0.34 +41.2582.5 +165MilesN +Increased +PopulationRatio +0.0 +0.60 +41.2582.5 +165Miles + + (a) + +(b) + +25 +Meanpercentageofbelow-povertyintheincreasedpopulation +20.0 +Meanpercentageof minorityintheincreased population +Meanpercentageofbelow-povertyofallthetracts +Mean percentageof pminority of allthetracts +17.5 +20 +15.0 +15 +12.5 +Count +10 +7.5 +5.0 +5 +2.5 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +Percentageofbelow-poverty +0.2 +0.4 +0.6 +0.8 +1.0 +Percentageof minority +25 +25 +Meanpercentageof unemployed inthe increasedpopulation +Mean percentage of no-high-school-diploma in the increased population +Mean percentage of unemployed of all the tracts +Mean percentage of no-high-school-diploma ofall thetracts +20 +20 +15 +15 +Count +Count +10 +10 +5+ +5 +0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Percentageofunemployed +Percentage of no-high-school-diplomaMeanpercentage of minorityinthe increasedpopulation +Meanpercentageofbelow-povertyintheincreasedpopulation +Meanpercentageof pminorityofall thetracts +Meanpercentage ofbelow-poverty ofall thetracts +40 +25 +20 +30 +Count +20 +10 +10 +5 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Percentageofbelow-poverty +0.2 +0.4 +0.6 +0.8 +1.0 +Percentage of minority +40 +Meanpercentageofunemployedinthe increasedpopulation +Mean percentage of no-high-school-diploma in the increased population +Mean percentageof unemployed ofall the tracts +-3oMean percentage of no-high-school-diploma of all the tracts +30 +25 +25 +15 +10 +10 +5 +5 +0 +0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Percentageofunemployed +Percentage of no-high-school-diploma +(c) +Figure 8. The histogram of the proportion of the (1) minority population, (2) below-poverty +population, (3) unemployed population and (4) population without high school diploma among the +increased population that are under threat, among the census tracts threatened by the pollutant +dispersion from the flooded facilities. + +5. Discussion +5.1 Regions along the north coastline in Texas are subject to higher threats +The threatened population investigated in this study are under the compound influence of the spatial +distribution of population, flood hazards and industrial and toxic site facilities. For example, this study +reveals that certain regions along the North coastline in Texas, such as Houston, Beaumont, Nederland +and Port Arthur, are especially threatened by the pollutant dispersion from the flooded facilities. +However, the underlying factors behind respective regions’ threat extent are different. As shown in Figure +9, for regions in Beaumont, Nederland and Port Arthur, their threats mainly derive from the high +concentration of industrial and toxic facilities along the coastline, which has put nearly all the residents in +these regions under threat (Figure 3). While for Houston, the risk mainly derives from the high population +density in the risky areas. Although the flooded facilities in Houston are much less compared to +Beaumont, Nederland and Port Arthur, the high population density around the flooded facilities, still put +the coastal areas in Houston under great threat. Such difference among different cities reveals the +heterogeneous cause for the threats, underneath the seemingly similar phenomenon. + +Mean percentage of below-poverty in the increased population +Mean percentage of minority in the increased population +-40Meanpercentageofbelow-povertyofallthetracts +Mean percentage of pminority of all the tracts +50 +30 +40 +unt +20 +10- +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +Percentageofbelow-poverty +0.2 +0.4 +0.6 +0.8 +1.0 +Percentage of minority +- Mean percentage of unemployed in the increased population +Mean percentage of no-high-school-diploma in the increased population +Meanpercentageofunemployedofallthetracts +10 +Mean percentage of no-high-school-diploma of all the tracts +35 +30 +30, +Count +20 +15 +10 +10 +5 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Percentageof unemployed +Percentage of no-high-school-diploma +Figure 9. The flooded facilities in 2018. There are 230 flooded facilities in total: 113 flooded TRI +facilities, 2 flooded NPL facilities, 115 flooded industrial facilities. The detailed information of the +flooded sites can be found in appendix. + +5.2 Socially vulnerable populations disproportionately suffer from the threats and +will suffer more in the future +Our study draws a different conclusion from the previous studies. The previous findings conclude that +non-vulnerable population are associated with higher coastal flood threats (Chakraborty, Collins et al. +2014). On the contrary, our study concludes that the socially vulnerable communities have a greater threat +exposure from the coastal flood threats and the associated pollutant dispersion from flooded industrial and +toxic facilities. Such difference originates from that our study defines the threat of coastal flood from a +new perspective: previous works mainly identify the threat based on the ‘pre-flood’ risk (100-year flood +zones); while our study derives the coastal flood with reference to the inundation due to the sea level rise, +and identifies the threat from potential pollutant dispersion from the flooded facilities. Such perspective +for viewing additional dimensions of threats of coastal flood has long been understudied by the previous +works. +Contrary to the conclusions of previous works, our finding points out that, due to their social and +economic status, vulnerable populations (e.g., minority, low-income/below-poverty, unemployed, without +high-school diploma) are more likely to live closer to the industrial and toxic facilities, and actually suffer +more from the threat of the pollutant dispersion, than other communities. Such threat is twofold, which +means not only that the threatened population under current scenario have a higher proportion of socially +vulnerable people, but also that, the socially vulnerable populations will still constitute a large proportion +of the increased population that will be exposed to such threats in the future. +The threat of pollutant dispersion from flooded facilities may potentially put risk on the health of the +disadvantageous communities, whose living and health condition will be further aggravated given their +unfavorable economic status and insufficient coverage of medical insurance. These findings provide + +· Flooded TRI facilities +• Flooded NPL facilities +: Flooded Industrial facilitiesimportant and novel complementary insights to the existing knowledge of social inequality caused by the +flooding and environmental hazards. +5.3 Practical implications +The threat of flood-induced pollutant dispersion, is a compound outcome affected by the spatial +distribution of flood hazards, population and industrial/toxic site facilities. Hence, policies needed to +alleviate the threats of flood-induced pollutant dispersion should account for the interactions among these +three determinants. Based on the results from this study, we identified the following strategies that call for +the actions from public officials, regulators, and decision makers to mitigate the threat from the pollutant +dispersion and improve environmental justice in flood context. +The first strategy is relocating industrial and toxic facilities away the regions that are susceptible to the +flood inundation along the coastline as well as populous regions. Relocation of communities is a strategy +commonly discussed in view of the flood threats (Sipe and Vella 2014). The future sea level rise +inevitably erodes the coastline and posits greater threat on the facilities along the coastal areas. We argue +that a straightforward solution to this issue is to relocate the facilities away from the coastal areas, or +compromisingly in coastal areas with less population concentrations, which is particularly important for +the future practice of urban planning. +Second, it is important to emphasize the necessity for adopting measures of coastal flood protection, +especially for the facilities that are infeasible to relocate within short term due to historical or economic +reasons. Coastal protection measures are effective in reducing the flood threats (Hallegatte, Green et al. +2013). This may be more economically friendly than relocation of facilities, in the short term. Yet +considering the long-term trend of sea level rise, the coastal flood protection may need to be strengthened +or rebuilt to deal with the increasing threat, which may eventually result in larger costs. Besides, the local +environmental factors such as elevation, imperviousness, distance to the streamline/coastline should be +considered for customized protection that fits the local environment. +Finally, we call for greater attention to equity in policies and regulations related to environmental impacts +of industrial and toxic sites, urban development plans, and flood risk reduction. Our analysis has shown +that socially vulnerable populations (i.e., low income, ethnically minority, unemployed, without high- +school diploma) are disproportionately exposed to a greater threat of flood-induced pollutant dispersion +and the subsequent health impacts. The rising sea levels will further worsen this threat exposure, and +cause more profound negative effects on the health and wellbeing of these sub-populations. Urban +development plans, environmental regulations, and flood risk reduction investments should be designed +and implemented in ways to equitably reduce the exposure to these threats for socially vulnerable +populations (Maantay and Maroko 2009, Montgomery and Chakraborty 2015). +5.4 Limitation and future directions +This study and its findings contribute to a better understanding of the intersection of environmental justice +issues and flood hazards under sea level rise in coastal areas. Nevertheless, the analysis in this study has +some limitations and further efforts are needed for more robust results. +First, the socioeconomic census dataset used in this study can be updated in the future. Due to the data +availability issue, we use the census data in current scenario to quantify the future population threatened +by the pollutant dispersion from the flooded facilities. This may introduce imprecision into our results as +the fluctuation of demography in the long term are not included. Future studies are needed to examine +forecasted census information in examining future threats and risks to populations. + +Second, the types of facilities used in this study can be expanded. Again, due to the data availability, this +study includes the NPL and TRI at the toxic facilities. Some other types of facilities such as Risk +Management Plan Sites (RMP) and Treatment Storage and Disposal Sites (TSD) may be further +integrated into the facilities to provide a more comprehensive picture for the toxic facilities that are +potentially threatened by the seal level rise. +Third, the accuracy of the flood map in this study can be refined. Deltares global flood map (Section 3.1) +utilizes a series of models to produce the inundation maps of flood depth, yet lacking in consideration of +certain factors such as the implementation of future coastal flood protection measures. As updated flood +inundation predictions become available, similar analyses can be repeated to refresh the insights obtained +in this study. +6. Concluding remarks +Pollutant dispersion caused by the coastal floods is an increasing risk to public health and wellbeing, +especially with the growing impacts of climate change. However, most of the previous work on flood risk +primarily focused on risks from floodplains, with insufficient attention to flood-induced pollutant +dispersion exposure. In addition, limited number of studies have examined the effects of future sea level +rise on flood-induced pollutant dispersion in coastal areas and the extent to which this risk affects the +overall societal environmental justice by threating different socio-demographic subpopulations. +To address this gap, our study examined the coastal areas of Texas as a case study and utilized the flood +inundation under current and future scenarios, to investigate the threats of coastal pollutant dispersion and +its environmental justice implications. Our analysis shows that socially vulnerable communities are +disproportionately exposed to the threats currently, and will continue to be inequitably exposed to flood- +induced pollutant dispersion in the future. Specifically, socially vulnerable communities with populations +of minority, low-income, unemployed, and without high-school diploma, have disparate threat exposure. +To mitigate such threat exposures at the intersection of environmental injustice and sea level rise, we +suggest that policies and plans should consider mitigation strategies, such as facility relocation, coastal +protection, and home buyout programs. The analysis from this study provides new insights regarding +ways coastal flooding endangers public health as well as our overall societal inequality. This research +bridges the gap at the intersection of environmental justice and climate change impacts by investigating +the extent to which pollutant dispersion due to the sea level rise threatens different communities. + +Acknowledgements +The authors would like to acknowledge funding support from the Texas A&M X-Grant Presidential +Excellence Fund. + +Appendix. + +Attached as a separate file. + + +Reference: +Alipour, A., et al. (2020). "Leveraging machine learning for predicting flash flood damage in the +Southeast US." Environmental Research Letters 15(2): 024011. +Botzen, W., et al. (2013). "Individual preferences for reducing flood risk to near zero through elevation." +Mitigation and Adaptation Strategies for Global Change 18(2): 229-244. +Brody, S. D., et al. (2008). "Identifying the impact of the built environment on flood damage in Texas." +Disasters 32(1): 1-18. +Carter, J. and C. Kalman (2020). 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"Coastal climate change and aging communities in Atlantic Canada: A +methodological overview of community asset and social vulnerability mapping." The Canadian +Geographer/Le Géographe canadien 59(4): 433-446. +Merz, B., et al. (2010). "Review article" Assessment of economic flood damage"." Natural Hazards and +Earth System Sciences 10(8): 1697-1724. +Microsoft (2022). "Deltares Global Flood Maps." Retrieved September 10, 2022, from +https://planetarycomputer.microsoft.com/dataset/deltares-floods. +Mobley, W., et al. (2019). "Estimating flood extent during Hurricane Harvey using maximum entropy to +build a hazard distribution model." Journal of Flood Risk Management 12: e12549. +Montgomery, M. C. and J. Chakraborty (2015). "Assessing the environmental justice consequences of +flood risk: a case study in Miami, Florida." Environmental Research Letters 10(9): 095010. +Palinkas, L. A. and M. Wong (2020). "Global climate change and mental health." Current opinion in +psychology 32: 12-16. + +Ridha, T., et al. (2022). "Climate change impacts on infrastructure: Flood risk perceptions and evaluations +of water systems in coastal urban areas." International Journal of Disaster Risk Reduction 73: +102883. +Sipe, N. and K. Vella (2014). "Relocating a flood-affected community: good planning or good politics?" +Journal of the American Planning Association 80(4): 400-412. +Stanke, C., et al. (2012). "The effects of flooding on mental health: Outcomes and recommendations from +a review of the literature." PLoS currents 4. +Walker, G. and K. Burningham (2011). "Flood risk, vulnerability and environmental justice: Evidence +and evaluation of inequality in a UK context." Critical social policy 31(2): 216-240. +Yuan, F., et al. (2022). "Smart flood resilience: Harnessing community-scale big data for predictive flood +risk monitoring, rapid impact assessment, and situational awareness." Environmental Research: +Infrastructure and Sustainability 2(2): 025006. +Yuan, F., et al. (2021). "Unraveling the temporal importance of community-scale human activity features +for rapid assessment of flood impacts." IEEE Access 10: 1138-1150. +Zhong, S., et al. (2018). "The long-term physical and psychological health impacts of flooding: a +systematic mapping." Science of the total environment 626: 165-194. + + diff --git a/AtAyT4oBgHgl3EQfd_hU/content/tmp_files/load_file.txt b/AtAyT4oBgHgl3EQfd_hU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0ced53a54cc01f9a894c47a982749f00102ad469 --- /dev/null +++ b/AtAyT4oBgHgl3EQfd_hU/content/tmp_files/load_file.txt @@ -0,0 +1,630 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf,len=629 +page_content='Collision of Environmental Injustice and Sea Level Rise: Assessment of Risk Inequality in Flood-induced Pollutant Dispersion from Toxic Sites in Texas Zhewei Liu1, Ali Mostafavi1* 1 UrbanResilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='AI Lab, Zachry Department of Civil and Environmental Engineering, Texas A&M University, College Station, TX, 77843 mostafavi@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='edu Abstract: Global sea-level rise causes increasing threats of coastal flood and subsequent pollutant dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' However, there are still few studies on the disparity arising from such threats and the extent to which different communities could be exposed to flood-induced pollution dispersion from toxic sites under future sea level rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' To address this gap, this study selects Texas (a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' state with a large number of toxic sites and significant flood hazards) as the study area and investigates impacts of flood-induced pollutant dispersion on different communities under current (2018) and future (2050) flood hazard scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The results show, currently, north coastline in Texas bears higher threats and vulnerable communities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', low income, minorities and unemployed) are disproportionally exposed to these threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' In addition, the future sea-level rise and the exacerbated flood hazards will put additional threats on more (about 10%) Texas residents, among which vulnerable communities will still be disproportionately exposed to the increased threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Our study reveals the facts that potential coastal pollutant dispersion will further aggravate the environmental injustice issues at the intersection of toxic sites and flood hazards for vulnerable populations and exacerbate risk inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Given the dire impacts of flood-induced pollution dispersion on communities’ health, the findings have important implications for specific actions from the policy makers to mitigate the inequitable risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Key words: environmental justice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' sea level rise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' flood-induced pollutant dispersion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Texas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Introduction About 60% of non-federal National Priorities List (NPL) sites (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', the priority list of hazardous waste sites) in the United States, are in the areas that may be affected by impacted by the climate change effects (Gómez 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The projected sea level rise and the accompanying coastal flood can cause pollutants dispersion from the sits and threaten public health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Moreover, the overburdened communities of color and low-income may disproportionally dwell in the regions around these sites and bear particular health impacts (Crawford 1994, Carter and Kalman 2020), which worsens the living conditions of socially vulnerable communities and gives rise to exacerbated environmental injustice issues in the context of global climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Efforts have been made by previous studies to investigate flood risk and its environmental justice implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Studies have found that socially vulnerable communities from different backgrounds experience drastic inequality in exposure to the flood risk, and such inequity show varied extent across different regions (Collins, Grineski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' For example, in the context of UK, some studies found that no displayed socioeconomic disparity regarding to the risks of riverine/inland pre-flood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', residence in 100-year flood zones), but regions with lower socioeconomic status are exposed to higher coastal risks (Fielding 2007, Walker and Burningham 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='While studies in the US has documented mixed findings: socially advantageous communities bear more risk towards pre-land risks in some regions such as Miami, while disadvantageous communities bear more risk in other regions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', Houston (Chakraborty, Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The built environment and amenities can be possible factors contributing to the difference across different regions (Chakraborty, Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2014, Grineski, Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2017, Collins, Grineski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' However, one important limitation of the existing literature is the insufficient attention to evaluating the intersection of flood exposure and environmental justice issues surrounding toxic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' In particular, the existing studies have paid limited attention to exacerbated flood hazard exposure of toxic sites under future sea-level rise scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Recognizing this important gap, this study examines the coastal area of Texas, USA, which has a large number of toxic sites and significant flood hazard exposure as a case study region to answer the following research questions: RQ1: To what extent coastal regions are threatened by pollutant dispersion from flood under current flood hazard scenario?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' RQ2: What is the extent of hazard exposure disparity among vulnerable populations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' RQ3: To what extent the future flood hazards exacerbated by sea level rise increase the threats of pollutant dispersion exposure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' RQ4: How the increase in threats is disproportionately affecting vulnerable populations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' To address these questions, we utilize the socioeconomic datasets, and coastal inundation maps in the current and future scenarios for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The results reveal that, in the current scenario (2018), the areas along the north coastline in Texas (such as Houston and Beaumont), are subject to higher threats to pollution dispersion, and that the communities of low-income, minority, unemployed suffer from greater threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Moreover, the future sea level rise by 2050 will increase the flood threats to more people, among whom vulnerable populations will be disproportionately exposed to flood-induced pollutant dispersion from toxic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The remaining sections of this paper is structured as follows: Section 2 summarizes the previous relevant studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Section 3 elaborates the datasets and methodology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Section 4 details the experimental results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Section 5 discusses the study implications and proposes practical suggestions for policy makers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Finally, Section 6 discusses the study limitations and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Background Floods threaten human life in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' One notable way is the impacts on human health, including injuries, diseases and psychological trauma (Du, FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2010, Stanke, Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2012, Zhong, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2018, Graham, White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2019, Palinkas and Wong 2020) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The landscape of the flooded areas can be permanently changed and contaminated, and expose the residents to long-term chronic diseases (Euripidou and Murray 2004, Karaye, Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2019), and also damage the properties and built environments (Brody, Zahran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2008, Merz, Kreibich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2010, Alipour, Ahmadalipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2020, Kousky, Palim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Environmental justice is another concerned issue associated with flood hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Due to the socioeconomic status and cultural background, different demographic communities suffer from unequal flood hazards exposure and impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The studies from different regions of the world have shown different findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Some UK-based studies conclude that people of lower socioeconomic status bear higher risk from coastal flood (Fielding 2007, Walker and Burningham 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Studies from US reveal that in Miami, the non- Hispanic Black and Hispanic are disproportionately exposed to inland flood risk, and underrepresented with coastal flood risk (Chakraborty, Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' And the study from Canada shows that senior people (over 65 years old) are the groups that are more vulnerable to coastal climate change in Atlantic Canada (Manuel, Rapaport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' These findings demonstrate the significance of examining the intersection of environmental justice issues and flood hazards, especially under future climate change impacts such as sea level rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' A peculiar type of flood impacts is pollutant dispersion from industrial and toxic sites that can be inundated due to flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Studies from the US show that vulnerable communities are exposed to disproportional risks from superfund sites (Carter and Kalman 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Communities with greater proportion of minority and low-income residents experience higher health risk, because of their closer residence to superfund sites (Crawford 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Some reports argue that race was the most significant factor in predicting the hazardous facilities (Christ 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Floods exacerbate the exposure of residents to pollutants released from these industrial and toxic sites and such exposures could lead to long-term chronic health impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Increased flood hazards due to sea level rise can make these hazards and impacts even worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' However, limited studies exist to evaluate the extent to which current and future flood hazards disproportionately affect populations living in proximity of industrial and toxic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' There are also some other determinants for the threats of flood, such as water-based amenities, self- protection procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Usually, the regions with water-based amenities are associated with higher risk of flood, due to the abundance of water body in the amenities (Collins 2010, Collins, Grineski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The structures of the building, such as elevation and flood-proofing materials, will also reduce the flood impacts (Botzen, Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2013, de Moel, van Vliet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The urban built environment factors, like ground imperviousness, land use type, distance to the streamline, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', will also affect the locations’ vulnerability to flood (Mobley, Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2019, Dong, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Besides these environmental factors, another factor is the subjective human perceptions on flood, and the related information collected from social sensing (Yuan, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2021, Yuan, Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' People’s awareness of flood risk will influence selection of home locations, and flood-proof materials, which eventually affect the loss caused by the flood (Lindell and Hwang 2008, Heitz, Spaeter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2009, Kellens, Zaalberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2011, Harlan, Sarango et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2019, Ridha, Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Datasets and Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 Datasets The study focuses on coastal areas in Texas as our study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Coastal areas of Texas have a large number of industrial and toxic sites and also highly exposed to flood hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Four kinds of datasets are used in the study: (1) Industrial facilities, which is provided by the United States Environmental Protection Agency (USEPA) and include locations of facilities that emit air pollutants during industrial process (EPA 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (2) Toxic facilities, which is also provided by USEPA, and include locations of toxic facilities such as National Priority List Sites (NPL) and Toxic Release Inventory Sites (TRI) (EPA 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (3) flood map caused by sea level rise, which is provided by Deltares global flood map (Microsoft 2022) and provides the inundation maps of flood along coastal and the flood depth, in current and predicted future scenarios ( in the year of 2018 and 2050 separately);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' this open dataset flood map is one of the most reliable publicly available inundation maps available, (4) socioeconomic census data at the census tract level, provided by the United States Census Bureau (USCB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' A description of the datasets is displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Datasets in the study and key attributes Datasets Source Description Industries & Toxic facilities USEPA The industrial facilities include the facilities that emit air pollutants during industrial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The toxic facilities include the type and locations of NPL and TRI sites Flood map by sea level rise in 2018 and 2050 Deltares global flood map The location of inundation along the coastline and the flood depth Socioeconomic census data at the census tract level USCB Statistics data for each census tract, including total population, income, minority population, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 Methodology This study aims to compare the extent to which certain groups of population are threatened by flood- induced pollutant dispersion from industries and toxic facilities, under current and future flood scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Consequently, the first step is to identify the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' We overlay the spatial distribution of industries and toxic facilities with the distribution of flood map along the coastline (as illustrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' As the flood map produced by Deltares uses discrete points to represent the flood, we create a buffer of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 mile based on the points in the flood map and the facilities that fall within the buffer are identified as the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Then, the second step is to quantify the population that can be potentially threatened by the pollutant dispersion from the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' To this end, we further create 1 mile, 3 miles and 5 miles buffers (Carter and Kalman 2020), based on the flooded facilities and the threatened population for each census tract is the tract’s total population multiplied by the proportion of tract’s area that falls within the flooded buffer zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' An explanation in formula is given in following equation: 𝐴_𝑃𝑜𝑝𝑢𝑖 = 𝑇_𝑃𝑜𝑝𝑢𝑖 ∗ 𝐴𝑟𝑒𝑎_𝑖𝑛_𝑏𝑢𝑓𝑓𝑒𝑟𝑖,𝑛_𝑚 𝑇_𝐴𝑟𝑒𝑎𝑖 Where 𝐴_𝑃𝑜𝑝𝑢𝑖 is the threatened population in the 𝑖𝑡ℎ census tract, 𝑇_𝑃𝑜𝑝𝑢𝑖 is the tract’s total population, 𝐴𝑟𝑒𝑎_𝑖𝑛_𝑏𝑢𝑓𝑓𝑒𝑟𝑖,𝑛_𝑚 is the tract’s area that falls within n-mile (n= 1, 3, and 5) buffer zone of the flooded facilities, and 𝑇_𝐴𝑟𝑒𝑎𝑖 is the tract total area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' We examine vulnerable populations, that are (1) below poverty, (2) of ethnical minority, (3) unemployed and (4) without high school diploma, to provide a comprehensive overview about how different communities, especially disadvantageous communities who live in proximity of industrial and toxic sites and suffer from the flooded caused by sea level rise, which put new threats on the environmental justice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The flood maps in 2018 and 2050 are used separately for comparison under current and future scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 Census tracts and population threatened by the pollutant dispersion under current scenario (2018) Three different distances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', 1 mile, 3 miles and 5 miles) are used to create buffers for the flooded facilities based on the flood map of 2018, and then identify the affected tracts (shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' It shows that as the distance of buffer expands, the number of increased tracts steadily increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The north Texas along the coastline are the regions severely affected, including cities such as Houston, Beaumont, Nederland and Port Arthur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (a) (b) (c) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The census tracts threatened by the pollutant dispersion within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer, in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The affected tracts along the North coastline are also the regions with higher population density and hence, smaller census tracts, compared with other regions in the central and south Texas, leading to more population being exposed to the threat of potential pollutant dispersion due to flood risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The number of the threatened population in each census are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' It can be seen that Houston and Beaumont are the regions with large amount of population threatened by the pollutant dispersion (Details in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' A similar pattern can be drawn from Figure 3, which compares the ratio of threatened population (threatened population/total population) in each census tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The results indicate that, due to the high population density and high concentration of industrial and toxic facilities, majority of the populations along the coastline in Beaumont, Nederland and Port Arthur, are exposed to flood-induced pollutant dispersion threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The residents in Houston and Corpus are also potentially threatened, especially along the coastal areas, where industrial and toxic facilities locate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' N Censustract within1-mile bufferof flooded industrial& toxic facilities 0 3672 144MilesN Censustract within3-mile bufferof flooded industrial& toxicfacilities 0 36 72 144MilesN Censustract within5-mile bufferof flooded industrial& toxicfacilities 0 36 72 144Miles (a) (b) (c) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The population threatened by the pollutant dispersion in each census tract, within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer, in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (d) (e) (f) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The ratio of the population threatened by the pollutant dispersion in each census tract, within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer, in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The vulnerable communities are especially subject to the threat of potential risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' As stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2, this study focus on four types of socially vulnerable groups and examines the extent to which they are particularly threatened by the pollutant dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Table 2 shows the mean income, proportions of minority population, below-poverty population, unemployed population and population without high- school diploma, in all the census tracts in Texas, and the tracts within certain distances of buffers of the flooded sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' A radar chart is further shown in Figure 4, with all the proportion normalized for visual interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' It can be clearly seen that, in the regions that are in the proximity of the flooded sites, there is a high concentration of socially vulnerable populations, whose living situation and health condition may be further exacerbated due to the potential pollutant dispersion from the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' N Beaumont Houston Affected Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Corpus 7,200 0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 97 194MilesN Beaumon Houston Affected Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Corpus 9,300 0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 97 194MilesN Beaumont V Houston Affected Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Corpus 14,000 0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 97 194MilesN Beaumont Houston Ratioof Affected Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00076 Corpus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 97 194MilesBeaumont N Houston Ratioof Affected Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='000014 Corpus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 97 194MilesBeaumontN Houston Ratioof Affected Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00020 Corpus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 97 194MilesTable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The proportions of socially vulnerable communities in different regions Mean of All Census Tracts Mean of Tracts within 1 mile of flooded sites Mean of Tracts within 3 mile of flooded sites Mean of Tracts within 5 mile of flooded sites Per capita income 29,684 24,736 24,296 24,102 Proportion of minority 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5% Proportion of below-poverty population 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='7% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3% Proportion of unemployed population 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='7% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3% Proportion of population without high-school diploma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='7% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0% Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' A radar chart of the Table 2, with the proportions normalized for visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Normalized proportion of below-poverty population Normalized proportion of unemployed Normalized proportion 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='8//1 population of minority All Census Tracts Tracts within 1 mile of flooded sites Normalized proportion of Tracts within 3 mile of flooded sites population without Tracts within 5 mile of flooded sites high-school diploma4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 Comparison of threats by flood-induced pollutant dispersion under current (2018) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' future (2050) scenarios To analyze the extent to which the exposure of socially vulnerable communities may be aggravated due to the future seal level rise, the flood map in 2050 is used to identify the census tracts and population threated by the pollutant dispersion in the future, and a comparison is made between the current and future scenarios (shown in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The results show that, compared to the current flood scenario in 2018, there will be about 10% more population threatened by flood-induced pollutant dispersion in 2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Population threatened by the flooded industrial & toxic facilities, separately in 2018 and 2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The increased exposure of populations that are threated in each census tract, from 2018 to 2050, is further illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' It can be seen that the sea level rise in the future will inevitably cause more population exposed to the threat of pollutant dispersion from flooded industrial and toxic facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The threats are especially obvious for certain regions in Houston and Beaumont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' We further calculate the ratio of increased population (increased_population_under_threat/total_population) in each census tract (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The results show that, among the cities in Texas, Houston will be the one that especially suffer from the increased flood-induced pollutant dispersion threats caused by the seal level rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Compared with situation in 2018, certain tracts in Houston will have 50% more population affected by the pollutant dispersion in 2050 (Details in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 1000000 925614 900000 812553 800000 POPULATION 700000 600000 501777 500000 461401 400000 300000 200000 136871 125219 100000 0 1-mile 3-mile 5-mile 12018-year 125219 461401 812553 2050-year 136871 501777 925614 BUFFER SIZE (a) (b) (c) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' From 2018 to 2050, the increased population threatened by the pollutant dispersion in each census tract, within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (a) (b) (c) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' From 2018 to 2050, the ratio of increased population threatened by the pollutant dispersion to total population, in each census tract within (a) 1-mile buffer, (b) 3-mile buffer and (c) 5-mile buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Compared with situation in 2018, certain regions in Houston get 50% more population under flood-induced pollutant dispersion threat in 2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3 Who will suffer more, from the projected seal level rise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' We investigate the demographics of the additional 10% more population (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2) that will be threatened by the increased pollutant dispersion threats from the flooded facilities in 2050, and compare the results with the demographics of all the census tracts in Texas (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The result shows that for the increased population that are under threat, the means of proportions of unemployed population (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2% for 1-mile buffer zone, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3% for 3-mile buffer zone and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5% for 5-mile buffer zone) and population without high-school diploma (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0% for 1-mile buffer zone, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2% for 3-mile buffer zone, and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0% for 5-mile buffer zone) are consistently higher than those of the overall population in Texas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='7% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='7% respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' On the other hand, for the census tracts that are within certain distances of the flooded facilities in 2050, the proportion of below-poverty (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 %for 3-mile buffer zone and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='9 % for 5-mile zone) and minority (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0% for 3-mile buffer zone and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 % for 5-mile buffer zone) are also consistently higher than those of overall population (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1% and 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' These results reveal that, socially vulnerable communities not only have more exposure from the potential pollutant dispersion from the flooded facilities under current flooding hazards, but also will bear a greater threat due to the increasing seal level rise in the future flood scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' N Increased Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 2,200 041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 165MilesN Increased Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 2,000 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 165MilesN Increased Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 2,900 041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 165MilesN Increased PopulationRatio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='70 0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 165MilesN Increased PopulationRatio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='34 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 165MilesN Increased PopulationRatio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='60 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 165Miles (a) (b) 25 Meanpercentageofbelow-povertyintheincreasedpopulation 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Meanpercentageof minorityintheincreased population Meanpercentageofbelow-povertyofallthetracts Mean percentageof pminority of allthetracts 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 20 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 15 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 Count 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Percentageofbelow-poverty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Percentageof minority 25 25 Meanpercentageof unemployed inthe increasedpopulation Mean percentage of no-high-school-diploma in the increased population Mean percentage of unemployed of all the tracts Mean percentage of no-high-school-diploma ofall thetracts 20 20 15 15 Count Count 10 10 5+ 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='35 Percentageofunemployed Percentage of no-high-school-diplomaMeanpercentage of minorityinthe increasedpopulation Meanpercentageofbelow-povertyintheincreasedpopulation Meanpercentageof pminorityofall thetracts Meanpercentage ofbelow-poverty ofall thetracts 40 25 20 30 Count 20 10 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 Percentageofbelow-poverty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Percentage of minority 40 Meanpercentageofunemployedinthe increasedpopulation Mean percentage of no-high-school-diploma in the increased population Mean percentageof unemployed ofall the tracts 3oMean percentage of no-high-school-diploma of all the tracts 30 25 25 15 10 10 5 5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='35 Percentageofunemployed Percentage of no-high-school-diploma (c) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The histogram of the proportion of the (1) minority population, (2) below-poverty population, (3) unemployed population and (4) population without high school diploma among the increased population that are under threat, among the census tracts threatened by the pollutant dispersion from the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 Regions along the north coastline in Texas are subject to higher threats The threatened population investigated in this study are under the compound influence of the spatial distribution of population, flood hazards and industrial and toxic site facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' For example, this study reveals that certain regions along the North coastline in Texas, such as Houston, Beaumont, Nederland and Port Arthur, are especially threatened by the pollutant dispersion from the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' However, the underlying factors behind respective regions’ threat extent are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' As shown in Figure 9, for regions in Beaumont, Nederland and Port Arthur, their threats mainly derive from the high concentration of industrial and toxic facilities along the coastline, which has put nearly all the residents in these regions under threat (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' While for Houston, the risk mainly derives from the high population density in the risky areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Although the flooded facilities in Houston are much less compared to Beaumont, Nederland and Port Arthur, the high population density around the flooded facilities, still put the coastal areas in Houston under great threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Such difference among different cities reveals the heterogeneous cause for the threats, underneath the seemingly similar phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Mean percentage of below-poverty in the increased population Mean percentage of minority in the increased population 40Meanpercentageofbelow-povertyofallthetracts Mean percentage of pminority of all the tracts 50 30 40 unt 20 10- 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 0 Percentageofbelow-poverty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='0 Percentage of minority Mean percentage of unemployed in the increased population Mean percentage of no-high-school-diploma in the increased population Meanpercentageofunemployedofallthetracts 10 Mean percentage of no-high-school-diploma of all the tracts 35 30 30, Count 20 15 10 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='35 Percentageof unemployed Percentage of no-high-school-diploma Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The flooded facilities in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' There are 230 flooded facilities in total: 113 flooded TRI facilities, 2 flooded NPL facilities, 115 flooded industrial facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The detailed information of the flooded sites can be found in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='2 Socially vulnerable populations disproportionately suffer from the threats and will suffer more in the future Our study draws a different conclusion from the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The previous findings conclude that non-vulnerable population are associated with higher coastal flood threats (Chakraborty, Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' On the contrary, our study concludes that the socially vulnerable communities have a greater threat exposure from the coastal flood threats and the associated pollutant dispersion from flooded industrial and toxic facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Such difference originates from that our study defines the threat of coastal flood from a new perspective: previous works mainly identify the threat based on the ‘pre-flood’ risk (100-year flood zones);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' while our study derives the coastal flood with reference to the inundation due to the sea level rise, and identifies the threat from potential pollutant dispersion from the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Such perspective for viewing additional dimensions of threats of coastal flood has long been understudied by the previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Contrary to the conclusions of previous works, our finding points out that, due to their social and economic status, vulnerable populations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', minority, low-income/below-poverty, unemployed, without high-school diploma) are more likely to live closer to the industrial and toxic facilities, and actually suffer more from the threat of the pollutant dispersion, than other communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Such threat is twofold, which means not only that the threatened population under current scenario have a higher proportion of socially vulnerable people, but also that, the socially vulnerable populations will still constitute a large proportion of the increased population that will be exposed to such threats in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The threat of pollutant dispersion from flooded facilities may potentially put risk on the health of the disadvantageous communities, whose living and health condition will be further aggravated given their unfavorable economic status and insufficient coverage of medical insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' These findings provide Flooded TRI facilities Flooded NPL facilities : Flooded Industrial facilitiesimportant and novel complementary insights to the existing knowledge of social inequality caused by the flooding and environmental hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='3 Practical implications The threat of flood-induced pollutant dispersion, is a compound outcome affected by the spatial distribution of flood hazards, population and industrial/toxic site facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Hence, policies needed to alleviate the threats of flood-induced pollutant dispersion should account for the interactions among these three determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Based on the results from this study, we identified the following strategies that call for the actions from public officials, regulators, and decision makers to mitigate the threat from the pollutant dispersion and improve environmental justice in flood context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The first strategy is relocating industrial and toxic facilities away the regions that are susceptible to the flood inundation along the coastline as well as populous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Relocation of communities is a strategy commonly discussed in view of the flood threats (Sipe and Vella 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The future sea level rise inevitably erodes the coastline and posits greater threat on the facilities along the coastal areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' We argue that a straightforward solution to this issue is to relocate the facilities away from the coastal areas, or compromisingly in coastal areas with less population concentrations, which is particularly important for the future practice of urban planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Second, it is important to emphasize the necessity for adopting measures of coastal flood protection, especially for the facilities that are infeasible to relocate within short term due to historical or economic reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Coastal protection measures are effective in reducing the flood threats (Hallegatte, Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' This may be more economically friendly than relocation of facilities, in the short term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Yet considering the long-term trend of sea level rise, the coastal flood protection may need to be strengthened or rebuilt to deal with the increasing threat, which may eventually result in larger costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Besides, the local environmental factors such as elevation, imperviousness, distance to the streamline/coastline should be considered for customized protection that fits the local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Finally, we call for greater attention to equity in policies and regulations related to environmental impacts of industrial and toxic sites, urban development plans, and flood risk reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Our analysis has shown that socially vulnerable populations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', low income, ethnically minority, unemployed, without high- school diploma) are disproportionately exposed to a greater threat of flood-induced pollutant dispersion and the subsequent health impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The rising sea levels will further worsen this threat exposure, and cause more profound negative effects on the health and wellbeing of these sub-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Urban development plans, environmental regulations, and flood risk reduction investments should be designed and implemented in ways to equitably reduce the exposure to these threats for socially vulnerable populations (Maantay and Maroko 2009, Montgomery and Chakraborty 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='4 Limitation and future directions This study and its findings contribute to a better understanding of the intersection of environmental justice issues and flood hazards under sea level rise in coastal areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Nevertheless, the analysis in this study has some limitations and further efforts are needed for more robust results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' First, the socioeconomic census dataset used in this study can be updated in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Due to the data availability issue, we use the census data in current scenario to quantify the future population threatened by the pollutant dispersion from the flooded facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' This may introduce imprecision into our results as the fluctuation of demography in the long term are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Future studies are needed to examine forecasted census information in examining future threats and risks to populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Second, the types of facilities used in this study can be expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Again, due to the data availability, this study includes the NPL and TRI at the toxic facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Some other types of facilities such as Risk Management Plan Sites (RMP) and Treatment Storage and Disposal Sites (TSD) may be further integrated into the facilities to provide a more comprehensive picture for the toxic facilities that are potentially threatened by the seal level rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Third, the accuracy of the flood map in this study can be refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Deltares global flood map (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='1) utilizes a series of models to produce the inundation maps of flood depth, yet lacking in consideration of certain factors such as the implementation of future coastal flood protection measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' As updated flood inundation predictions become available, similar analyses can be repeated to refresh the insights obtained in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Concluding remarks Pollutant dispersion caused by the coastal floods is an increasing risk to public health and wellbeing, especially with the growing impacts of climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' However, most of the previous work on flood risk primarily focused on risks from floodplains, with insufficient attention to flood-induced pollutant dispersion exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' In addition, limited number of studies have examined the effects of future sea level rise on flood-induced pollutant dispersion in coastal areas and the extent to which this risk affects the overall societal environmental justice by threating different socio-demographic subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' To address this gap, our study examined the coastal areas of Texas as a case study and utilized the flood inundation under current and future scenarios, to investigate the threats of coastal pollutant dispersion and its environmental justice implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Our analysis shows that socially vulnerable communities are disproportionately exposed to the threats currently, and will continue to be inequitably exposed to flood- induced pollutant dispersion in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Specifically, socially vulnerable communities with populations of minority, low-income, unemployed, and without high-school diploma, have disparate threat exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' To mitigate such threat exposures at the intersection of environmental injustice and sea level rise, we suggest that policies and plans should consider mitigation strategies, such as facility relocation, coastal protection, and home buyout programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' The analysis from this study provides new insights regarding ways coastal flooding endangers public health as well as our overall societal inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' This research bridges the gap at the intersection of environmental justice and climate change impacts by investigating the extent to which pollutant dispersion due to the sea level rise threatens different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Acknowledgements The authors would like to acknowledge funding support from the Texas A&M X-Grant Presidential Excellence Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Attached as a separate file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Reference: Alipour, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' "Leveraging machine learning for predicting flash flood damage in the Southeast US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='" Environmental Research Letters 15(2): 024011.' 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Infrastructure Engineering 35(7): 668-684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' Du, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' "Health impacts of floods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='" Prehospital and disaster medicine 25(3): 265-272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content=' EPA (2022).' metadata={'source': 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health impacts of flooding: a systematic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} +page_content='" Science of the total environment 626: 165-194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfd_hU/content/2301.00312v1.pdf'} diff --git a/BNE0T4oBgHgl3EQfxwK1/content/tmp_files/2301.02651v1.pdf.txt b/BNE0T4oBgHgl3EQfxwK1/content/tmp_files/2301.02651v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc2e0a9ccf3d2d56a5ce5ebc45c7c03db41ddef6 --- /dev/null +++ b/BNE0T4oBgHgl3EQfxwK1/content/tmp_files/2301.02651v1.pdf.txt @@ -0,0 +1,2221 @@ +IEEE TRANSACTIONS ON POWER SYSTEMS +1 +A Robust Data-driven Process Modeling Applied to +Time-series Stochastic Power Flow +Pooja Algikar, Member, IEEE,, Yijun Xu, Senior Member, IEEE, Somayeh Yarahmadi, Member, IEEE, Lamine +Mili, Life Fellow, IEEE. +Abstract—In this paper, we propose a robust data-driven pro- +cess model whose hyperparameters are robustly estimated using +the Schweppe-type generalized maximum likelihood estimator. +The proposed model is trained on recorded time-series data of +voltage phasors and power injections to perform a time-series +stochastic power flow calculation. Power system data are often +corrupted with outliers caused by large errors, fault conditions, +power outages, and extreme weather, to name a few. The +proposed model downweights vertical outliers and bad leverage +points in the measurements of the training dataset. The weights +used to bound the influence of the outliers are calculated using +projection statistics, which are a robust version of Mahalanobis +distances of the time series data points. The proposed method +is demonstrated on the IEEE 33-Bus power distribution system +and a real-world unbalanced 240-bus power distribution system +heavily integrated with renewable energy sources. Our simulation +results show that the proposed robust model can handle up to +25% of outliers in the training data set. +Index Terms—Time-series Stochastic Power Flow; Robust +Process Modeling; Robust Mahalanobis Distances; Generalized +Maximum Likelihood Estimator; Outlier Detection and Identifi- +cation. +I. INTRODUCTION +A power system, as it stands currently, involves real-time +operational and control actions based on the information +provided by the state estimator. The latter processes a set +of measurements at periodic time intervals consisting of real +and reactive power flows and power injections and voltage +magnitudes at selected lines and buses. They are collected +from metered devices such as SCADA measurements, phasor +measurement units (PMUs), and intelligent electronic devices +(IEDs), among others [1]. They facilitate the time-series power +flow analysis to forecast load duration curves and hence, to +determine overload conditions in power distribution systems. +It is well known that these measurements are often corrupted +with outliers. For instance, during fault conditions, the in- +terference of inrush current in switchgear temporarily causes +errors in the measurements. The communication methods used +in power distribution systems are often exposed to heavy +electromagnetic interference, resulting in corrupted data [2]. +Furthermore, asynchronous sample time of PMUs [3], [4] and +magnetic saturation and hysteresis in potential and current +This work is supported, in part, by NSF 1917308 and by the Research +Startup Fund of Southeast University in China under Grant 3216002206A1. +P. Algikar, L. Mili are with the Electrical Engineering Department, Vir- +ginia Tech, Falls Church, VA 22043, USA. (e-mail:{apooja19, syarahmadi, +lmili}@vt.edu). +Y. Xu is with the Southeast University, Nanjing, Jiangsu, China. (e- +mail:yijunxu@seu.edu.cn). +transformers cause measurement errors in current and voltage +phasors [5]. Under these conditions, the state estimations based +on the weighted least squares method suffer from masking +and smearing effects, thus yielding inaccurate results. The +situation exacerbates under heavy penetration of renewable +energy sources (RES) and distributed generations (DGs) due to +the stochastic dynamics that they introduce in the power grid. +{For a large-scale power system, performing classical Monte +Carlo (MC) simulations of the thousands of realizations for +uncertainty quantification requires high computational power. +Therefore, developing robust and computationally efficient +models and tools that process real measurements to analyze +the stochastic dynamics of a power system is of paramount +importance. +In the literature, several stochastic power flow methods have +been proposed to carry out sensitivity analysis and uncertainty +quantification [6]. Among them, the most popular methods are +the MC simulations and meta-models. As discussed earlier, +MC methods turn out to be computationally inefficient when +thousands of simulation runs are needed to achieve meaningful +statistical results in uncertainty quantification. +Meta-models, also known as emulators, surrogates, or re- +sponse surfaces, only statistically represent the deterministic +power flow simulator. Those based on Gaussian processes are +non-parametric reduced-ordered models in which the model +output realizations are assumed to follow a Gaussian distribu- +tion [7]. Other types of meta-models extensively developed in +the literature are based on polynomial chaos [8]–[15]. Ni et al. +[9] developed a sparse polynomial chaos expansion to tackle a +large number of random input variables considering correlation +among them. Wu et al. [10] proposed generalized polynomial +chaos (gPC) with rectangular formulations to preserve the +non-linearity of power flow. Wang et al. [11] extended the +gPC to the data-driven gPC to better deal with the dependent +correlated uncertainties among input variables. Xu et al. [12] +developed hierarchical polynomial chaos analysis of variance +(ANOVA) for efficient extension of the gPC to large-scale +systems without falling prey to the curse of dimension. A few +methods using neural networks include deep neural network +models [16], [17], graph convolutional network models [18], +and graph neural network [19] to overcome the computational +challenge. +Once developed, the meta-model, which statistically repre- +sents the power flow simulator, is run while considering thou- +sands of new input variables to perform sensitivity analysis +and uncertainty quantification. {However, none of them are +robust enough to be trained on real-time data, which makes +arXiv:2301.02651v1 [eess.SY] 6 Jan 2023 + +IEEE TRANSACTIONS ON POWER SYSTEMS +2 +them unsuitable to perform time-series stochastic power flow +analysis. +The accuracy of the results obtained from a meta-model +is highly dependent on the quality of the training data. A +well-trained meta-model requires data points that fill the input +design space. As a result, the data points are sampled from +an assumed probability distribution in the design space of +stochastic input variables. The assumed probability distribu- +tions typically are the Gaussian distribution for the load, +the Weibull distribution for the wind speed, and the Beta +distribution for the solar irradiance, among others. However, +in practice, these distributions may not represent the actual +data [20], [21], yielding inaccurate uncertainty quantification +results. The conventional meta-modeling methods are not +designed to handle the misrepresentations of power curve +distribution, yielding biased results. +Consequently, modern data-driven models are introduced in +the literature, which make use of the raw observational data +as a consequence of the proliferation of sensing and metering +devices. This approach captures the natural stochasticity of +the underlying process. For example, Wang +et al. in [22] +developed a data-driven emulator using polynomial chaos that +estimates the statistics of the voltage phasors while Xu et al. +in [23] proposed a fully non-parametric approach to avoid +assuming a parametric distribution for the Gaussian process +meta-model. However, all these methods are relying on raw +data without considering outliers. It is well known that wind +generation (WG) time series data are frequently contaminated +with communication errors, wind turbine outages, and curtail- +ments [20] while PV time series data are contaminated with +large signal noise, sensor failures, communication equipment +failures, maximum power tracking abnormalities, array shut- +downs, and power limitations, to name a few [21]. +As a consequence, various data preprocessing techniques +have been proposed to account for these abnormal tendencies. +For instance, Long et al. [24] developed an algorithm based on +the mathematical morphology operation of wind power curve +image for detecting and cleaning the wind turbine abnormal +data. In [25], a method for filtering out the outliers in raw +wind data considering the degree of similarity between the in- +dividual objects is developed. These data-cleaning algorithms +combined with non-robust data-driven models such as [22], +[23] will be time-consuming in real-time stochastic analysis +for increasing power system size with limited computational +power and databases for newer wind or solar farms. Their +suitability in real-time statistical analysis is arguable. Unlike +the proposed method, they are not effectively integrated with +the estimation process and are not robust against all types of +arising outliers in the time frame of the operation. +In this paper, we develop a real-time data-driven time-series +stochastic power flow analysis based on a robust process model +(RPM). The proposed RPM makes use of the Schweppe-type +generalized maximum likelihood estimator (SHGM) that can +handle up to 25% of outliers in the training data set. Recall that +outliers may be either vertical outliers or bad leverage points. +The other estimators proposed in robust statistics for linear +regression are M-estimators, which are not robust against bad +leverage points [26], [27]. In power systems, leverage points +are power flows on relatively short lines or power injection +on buses with relatively many incident lines [28]. We assess +the robustness of the RPM only theoretically by using three +statistical concepts, namely, influence function, finite-sample +breakdown point, and asymptotic maximum bias curve. The +robustness of the RPM is experimentally demonstrated on the +radial IEEE-33 bus power distribution system and a real-world +240-bus power distribution system located in the Midwest U.S. +for which the training data are manually added with vertical +outliers and bad leverage points up to 25%. +This paper is organized as follows. Section II discusses +the conventional GPM followed by theoretical background +on statistical robustness concepts that are used to access the +robustness of the RPM. Section III presents the development +of the proposed methodology. Section IV demonstrates the +performance of the RPM on the IEEE 33-bus distribution +system and a real-world 240-bus distribution system. Section +V concludes the paper and outlines future work. +II. BACKGROUND +A. Formulation of the Time-series Stochastic Power Flow in +the Gaussian Process Framework +The power flow simulator, represented by f(·), is assumed +to be a function of active and reactive power injection mea- +surements at all the p buses, xti ∈ R2p in training interval +t = [1, . . . , n]. The output variables are yti = f(xti) + ϵti are +voltage magnitude or phase angle measurement at a bus with +an independent and identically distributed additive Gaussian +measurement noise, ϵti ∼ N(0, σ2 +0). In the Gaussian process +modeling framework, the uncertainty about the power flow +simulator output is characterized as a Gaussian process having +a specific mean function m(·), and a covariance (kernel) +function k(·, ·). As a Gaussian process distribution, the sim- +ulator output variables f(xt1), f(xt2), . . . , f(xtn) follow a +multivariate normal distribution. To incorporate our belief +about the simulator, the prior distribution is formulated as a +Gaussian process with mean function m0(·) and covariance +function k0(·, ·). Formally, we have +f(·)| β, l, τ 2 ∼ GP(m0(·), k0(·, ·)), +(1) +where the mean function m0(·) takes the form +m0(x) = h(x)T β. +(2) +Here, h(xti) +: +R2p +→ +Rq +denotes the basis func- +tion that can be chosen to model the assumed degree +of non-linearity of the power system, that is, h(xti) = +[1, xti, x2 +ti, x3 +ti, . . .]T . +For +example, +the +constant, +linear, +and quadratic basis functions are respectively given by +h(xti) = [1], h(xti) = [1, xti1, . . . , xti2p]T , and h(x) = +[1, xti1, . . . , xti2p, x2 +ti1, . . . , x2 +ti2p]T ∈ Rq, q = 4p + 1, i = +1, 2, . . . , n. The kernel function k0(xti, xtj) denotes the co- +variance between corresponding output points (yti, ytj). A +commonly used covariance function is the radial basis function +given by +k0(xti, xtj|l) = τ 2exp +� +− +2p +� +k=1 +(xtik − xtjk)2 +2lk2 +� +, +(3) + +IEEE TRANSACTIONS ON POWER SYSTEMS +3 +where l = (l1, . . . , l2p) denotes the characteristic length-scale, +which models the rapidity of the process; i, j = 1, . . . , n. +Some other covariance functions are listed in Table I. +Let us gather n input and output measurements into the ma- +trix X = [xt1, . . . , xtn]T and the vector y = [yt1, . . . , ytn]T , +respectively. The matrix of of basis functions is then repre- +sented as H(X) = [h(xt1), . . . , h(xtn)]T . The distribution of +the output vector y of the power system according to (1) is a +multivariate normal random vector having a covariance func- +tion diagonally additive with noise elements ϵ ∼ N(0, σ2 +nIn). +Formally, we have +y|X, β, l, τ 2, σ2 +n ∼ N (H(X)β, Σ(X)) , +(4) +where Σ(X) = k0(X, X) + σ2 +nIn. The noise elements ϵ with +zero mean and variance σ2 +n, also called ”nugget”, account for +model uncertainty and numerical stability. The training data +set is constituted by (y, X). +For the stochastic power flow analysis, let us consider that +we draw K samples of the input test variable xt∗ at instances +t∗ = [1, . . . , n∗] in the prediction interval. In time-series +analysis, the assumption of stationarity and ergodicity for a +specific range of time is often made. The latter means that the +sample average, commonly known as the ensemble average, +is equal to the time average. The assumption of ergodicity +allows us to model a stochastic time-series power flow using +a single real-time measurement per instance. Let us group the +K sampled test predictors for instance t∗ +i denoted as x(k) +t∗ +i +for +k in [1, K] into X∗ +i = [x(1) +t∗ +i , . . . , x(K) +t∗ +i +]T . Using a hierarchical +formulation, the model output variables y∗ obtained through +the power flow simulator f(·) at the test points X∗ +i together +with the training output variables follow a joint multivariate +Gaussian distribution given by +� +y +y∗|X∗ +i +� +∼ N +�� m0(X) +m0(X∗ +i ) +� +, +� Σ(X) +C(X∗ +i ) +CT (X∗ +i ) +V(X∗ +i ) +�� +, +(5) +where C(X∗ +i ) = k0(X, X∗ +i ), CT (X∗ +i ) = k0(X∗ +i , X) and +V(X∗ +i ) = k0(X∗ +i , X∗ +i ). The covariance matrix, Σ(X), is rep- +resented by Σ hereafter. Furthermore, we assume an a priori +Gaussian probability distribution for the simulator output at +the test points, f(X∗ +i )|X∗ +i , that is, +f(X∗ +i )|X∗ +i ∼ GP (m0(X∗), V(X∗ +i )) . +(6) +Upon conditioning and using the standard techniques in mul- +tivariate distributions, we get +f(X∗ +i )|X∗ +i , y, X, β, l, τ 2, σ2 +n ∼ GP (µ∗(X), Σ∗(X)) , +(7) +where the estimated mean function �µ∗(X∗ +i ) is given by +�µ∗(X∗ +i ) = � +m0(X∗ +i ) + �CT (X∗ +i )�Σ−1r, +(8) +and the estimated covariance function �Σ∗(X∗) is expressed as +�Σ∗(X∗ +i ) = �V(X∗ +i ) − �CT (X∗ +i )�Σ−1 �C(X∗ +i ), +(9) +for all the instances in the predictive interval, i = 1, . . . , n∗. +The estimate of the mean function given by (8) acts as a +computationally efficient surrogate model that captures the +behavior of the power flow while the covariance matrix +estimate given by (9) quantifies the associated uncertainty. +Let us apply a weak prior for (β, τ 2), p(β, τ 2) ∝ +1 +τ 2 , +combining with (4) and using Bayes’ theorem yields a poste- +rior distribution for (β, τ 2), which is normal inverse-gamma +distribution given by +β|y, X, l, τ 2, σ2 +n ∼ N(ˆβ, τ 2(HT Σ−1H)−1), +(10) +where ˆβ is the weighted least squares estimate given by ˆβ = +(HT Σ−1H)−1HT Σ−1y, and +τ 2|y, X, l, σ2 +n ∼ InvGamma +�n − q +2 +, (n − q − 2)ˆτ 2 +2 +� +, (11) +where ˆτ 2 = yT (Σ−1−Σ−1H(HT Σ−1H)−1HT Σ−1)y +(n−q−2) +. +B. Smearing and Masking Effects in Conventional GPM De- +sign +The residual vector is defined as the difference between the +observation vector, y, and the estimated vector, �y. Formally, +we have +r = y − �y, +(12) +where +ˆy = Sy, +(13) +and +where +S +is +the +hat +matrix +given +by +S += +H(HT Σ−1H)−1HT Σ−1. Substituting (13) into (12) yields +r = y − Sy, +(14) += (I − S)y, +(15) += Wy. +(16) +Substituting the expression y = m(X) + e; e ∼ N(0, Σ) +into (16) yields +r = We. +(17) +Here, W is called the residual sensitivity matrix as it expresses +the sensitivity of the residuals to the errors. For the WLS +estimator, the outlier detection and identification statistical +tests suffer from the smearing and masking effect of the +outliers on the residuals as shown next. +• Smearing effect +Let us assume that e1 ̸= 0 and ei = 0 for i = 2, 3, . . . , n. +From (14), we have +r1 = W11e1 ̸= 0, +(18) +r2 = W21e1 ̸= 0, +(19) +rn = Wn1e1 ̸= 0. +(20) +This is known as the smearing effect of one outlier on the +residuals, which makes the identification of that outlier +using the residual statistical test difficult to achieve. +• Masking effect +Let us assume that the e1 ̸= 0 and e2 ̸= 0 while ei = 0 +for i = 3, 4, . . . , n. The associated residuals are expressed +as +r1 = W11e1 + W12e2, +(21) +r2 = W21e1 + W22e2. +(22) +Therefore, there exist e1 and e2 so that the residuals r1 ≈ +0 and r2 ≈ 0. This is known as the masking effect of the +outliers on the residuals, which results in the failure of +the outlier residual identification test. + +IEEE TRANSACTIONS ON POWER SYSTEMS +4 +TABLE I: Commonly used Kernel Functions +Type +Expression +Exponential +kE(xti, xtj) +τ 2exp +� +− �2p +k=1 +|xtik −xtjk | +lk +� +Matern +3/2 +kM(xti, xtj) +τ 2 +� +1 + �2p +k=1 +√ +3(xtik −xtjk ) +lk +� +exp +� +− �2p +k=1 +√ +3|xtik −xtjk | +lk +� +Rational +Quadratic +kRQ(xti, xtj) +τ 2 +� +1 + exp +� +− �2p +k=1 +(xtik −xtjk )2 +2l2 +kα +��−α +C. Robustness Concepts +In this subsection, we briefly review the definition of +the statistical efficiency of an estimator and the robustness +concepts developed in robust statistics, namely, the asymptotic +influence function, the breakdown point, and the asymptotic +maximum bias curve. +1) Asymptotic Influence Function: For an ϵ-contaminated +model G(r) = (1−ϵ)Φ(r)+ϵF(r), where Φ is the cumulative +Gaussian distribution function and F is the unknown distribu- +tion function of residuals. The influence function is based on +the Gateaux derivative that quantifies the local sensitivity of +an estimator T(G) to an arbitrary infinitesimal contamination +H = ∆r. It is expressed as +IF(ri, hi; Φ) = lim +ϵ→0 +T((1 − ϵ)Φ + ϵ∆r) − T(Φ) +ϵ +. +(23) +The total influence function, IF(r; Φ), of an M-estimator for a +linear regression model is equal to the product of the scalar- +valued influence of residuals, IR(ri; Φ), and the vector-valued +influence of position, IP(hi; Φ). Formally, we have +IF(r; Φ) = IR(ri; Φ)IP(hi; Φ). +(24) +They are given by IR(ri; Φ) = +ψ( ri +s ) +E[ψ′( +ri +s )] and IP(hi; Φ) = +(HT H)−1hi. For an M-estimator, the IR(·) is bounded if the +ψ(·) is bounded while the IP(·) is always unbounded, revealing +its non-robustness to bad leverage points. +2) Finite-Sample Breakdown Point: The maximum value of +ϵ, denoted as ϵ∗, for which the maximum bias of an estimator +is finite is called the finite-sample breakdown point of that +estimator. Formally we have ϵ∗ = max{ϵ; bmax(ϵ) < ∞}. +[29] and [30] showed that the maximum finite-sample break- +down point of any regression equivariant estimator under the +assumption of general position is given by +� n−q +2 +� +/n [31]. +3) +Asymptotic maximum bias curve: The asymptotic max- +imum bias curve is the curve of the upper bound of a +bias of an estimator for an increasing level of contamination +0 ≤ ϵ < ϵ∗. The asymptotic maximum bias curve of any Fisher +consistent estimator, ˆθ, in its functional form, T , at any ϵ +contaminated model is defined as bmax(ϵ) = sup +H +|T (G)−θ|. In +the location case, Huber [30] showed that the sample median +has the smallest possible asymptotic maximum bias curve +among all location equivariant estimators. In linear regression, +the estimator that has the minimum asymptotic bias curve is +unknown. +4) Statistical Efficiency: The minimum possible variance +that any estimator of location, ˆθn, is able to attend at an +assumed probability distribution, F, is given by the Cramer- +Rao lower bound, which is defined as the inverse of the Fisher +information, If. Formally, we have +V ar(√nˆθn; F) ≥ 1 +If +; ∀n. +(25) +The ratio of the Cramer-Rao lower bound and the variance +of an estimator is called the efficiency of that estimator. An +asymptotically efficient estimator is one whose variance attains +the Cramer-Rao lower bound for n tending to infinity. +III. ROBUST DATA-DRIVEN PROCESS EMULATOR +In this section, we discuss the development of the proposed +RPM model. We rewrite (4) in the form of a regression +problem in terms of the mean function hyperparameter given +by +y(X) = H(X)β + e, +(26) +The hyperparameters of the mean and covariance function +incorporating the maximum likelihood estimation method are +obtained by solving +�θ = +arg max +(β,l,τ 2,σ2n)∈RqR2pR+∗R+∗ log L +� +y|X, β, l, τ 2, σ2 +n +� +, (27) +where θ represents a vector of hyperparameters (β, l, τ 2, σ2 +n). +A. Schweppe-type Generalized Maximum Likelihood Estima- +tor +We propose to estimate β in a robust manner using the +SHGM estimator. The SHGM estimator minimizes a weighted +loss function of residuals ri given by +J(β) = min +ˆβ +n +� +i=1 +w2 +i ρ +� ri +wis +� +, +(28) +where ρ(·) is a non-linear loss function of the standardized +residuals, rSi = +ri +wis. The residual scale s is robustly estimated +by s∗ = 1.4826 bm median|r| when there is a little to none +knowledge about the error covariance. +s = 1.48261 + +5 +n − q median|r|. +(29) +We choose the Huber loss function because of its convexity +and its quadratic characteristic at its center. It is defined as +ρ(ri) = +� +r2 +i +2 +for ri < c, +c|ri| − c2 +2 +for ri ≥ c. +(30) + +IEEE TRANSACTIONS ON POWER SYSTEMS +5 +The threshold parameter c is typically chosen to be equal to +1.5, which offers a good compromise between a high statistical +efficiency at the Gaussian distribution and good robustness +against outliers. +B. Weights Based on Projection Statistics +The weights are calculated using the projection statistics, +which are a robust version of the Mahalanobis distances. +Formally, they are defined as the maximum of the standardized +projection distances obtained by projecting the point cloud in +the directions that originate from the coordinate-wise median +and that pass through each of the data points [28]). Let hT (xi) +be represented by hT +i . Formally, we have +PSi = max +||v||=1 +hT +i v − med +j (hT +j v) +1.4826 med +k +|hT +k v − med +j (hT +j v)|, +(31) +where vj = +uj +||uj||; uj = hi − M; j = 1, . . . , n. Here, M +denotes the coordinatewise median given by +M = { med +j=1,...,n hj1, . . . , +med +j=1,...,n hjq}. +The weights are calculated as +w(hi) = +� +1, +PS2 +i ≤ b; +b +PS2 +i , +otherwise. +(32) +The data point hi is considered as a leverage point when the +associated PS2 +i is greater than b. The weights downweight the +bad leverage point and vertical outliers while retaining the +good leverage points. +C. Robust Estimation of the Mean Function Hyperparameter +We estimate the hyperparameter of the mean by setting the +gradient of the objective function J(β) with respect to β to +zero, which is given by +n +� +i=1 +wihi +∂ρ(rSi) +∂rSi += 0. +(33) +Let us define the psi-function as ψ(rSi) = ∂ρ(rSi) +∂rSi . (33) now +becomes +n +� +i=1 +wihiψ(rSi). +(34) +Dividing (34) by the standardized residuals rSi, we get +m +� +i=1 +q +� ri +wis +� +hiri = 0, +(35) +where q(rSi) = +ψ(rSi) +rSi +is called a weight function. For the +case of Huber ρ-function, it is defined as +q(rSi) = +� +1, +ri ≤ c +b sign(rSi) +rSi +, +otherwise . +(36) +Substituting the expression of the residuals, ri, and rewriting +(36) in matrix form yields +HT Q(y − Hβ) = 0 +(37) +β=(HT QΣ−1H)−1HT Q(k)Σ−1y, +(38) +where Q = diag(q(rSi)). Since β is a function of Q, we +solve for β in an iterative manner by incorporating iterative +re-weighted least squares (IRLS) algorithm. Formally, we have +β(k+1) = (HT Q(k)Σ−1H)−1HT Q(k)Σ−1y. +(39) +D. Iterative Procedure for the Hyperparameter Estimation +In this subsection, the robust estimation of the hyperparam- +eters (l, τ 2, σ2 +n) of the RPM associated with the covariance +function is discussed. With the observation set available from +the MC simulation of the code, (y, X), we estimate the hyper- +parameter �β using the algorithm given by (39). The maximum +likelihood estimate of the remaining hyperparameters, namely, +(l, τ 2, σ2 +n), is formulated as +(�l, �τ 2, �σ2 +n) = arg max +l,τ 2,σ2n +log L +� +Y|X, �β, l, τ 2, σ2 +n +� +. +(40) +Let us define the resulting log L function by +Γ(l, τ 2, σ2 +n) = log |k(X, X|l, τ 2) + σ2 +nIn|. +(41) +Consequently, the maximum likelihood estimate of (l, τ 2, σ2 +n) +reduces to +(�l, �τ 2, �σ2 +n) = arg min +l,τ 2,σ2 +n +Γ(l, τ 2, σ2 +n). +(42) +The hyperparameters, (�l, �τ 2, �σ2 +n), are estimated by utilizing +a gradient-based optimizer as described in [32]. We can +then update �β as ��β = �β(�l, �τ 2, �σ2 +n). The algorithm used for +Algorithm 1 Algorithm for Estimating the RDP Hyperparam- +eters +1: Develop the power-system simulator in the case where the +measurements are unavailable; +2: Run the simulator at required input power measurements +to obtain the voltage magnitude and the voltage phase +angle at each load bus, which constitutes the training data- +set (X, y); +3: Construct H using a suitable basis function; +4: Calculate the projection statistics of the row vectors of H +given by (31); +5: Calculate the weights w based on the PS given by (32); +6: Initialize β using the weighted least squares solution as +β0 = (HT Σ−1H)−1HT Σ−1y; +7: Update β by executing the IRLS algorithm given by +(39) until convergence while setting the hyperparameters +(l, τ 2, σ2 +n) at their initial values to obtain �β; +8: Update (l, τ 2, σ2 +n) while setting β = �β; +9: Iterate Steps 7 and 8 until convergence, e.g. ||r|| ≤ 0.001, +to obtain the final hyperparameter estimates, (��β, �l, �τ 2�σ2 +n). +estimating the hyperparameters is summarized in Algorithm +1. Once all the hyperparameters of the RPM, (β, l, τ 2, σ2 +n), +are estimated, we use (8) as a robust computationally efficient +surrogate and (9) to quantify its variance. + +IEEE TRANSACTIONS ON POWER SYSTEMS +6 +E. Robustness of the RPM +The influence function of the SHGM estimator is given by +IF(rSi, hi; Φ) = +ψ(rSi) +EΦ[ψ +′(rSi)](HT H)−1hiwi, +(43) +For the SHGM estimator, one can notice that the influence of +position, IP(hi, Φ) = (HT H)−1hiwi, is bounded thanks to +the weights calculated using the projection statistics (see Sec- +tion III), whose breakdown point attains the maximum given +by [(n−q−1)/2] +n +[31]. Note that the SHGM estimator reduces to +an ℓ2-norm estimator for small standardized residuals and to +the ℓ1-norm estimator for larger ones. Therefore, it has a high +statistical efficiency at the Gaussian distribution while being +robust to outliers. +IV. CASE STUDIES +In this section, we compare the performance of the proposed +model RPM to that of the Gaussian process model (GPM) +when applied to a standard IEEE 33-bus system (Case A) +and to a real-world 240-bus distribution system located in +the Midwest U.S. with high penetration of RESs and DGs +(Case B). We add vertical outliers, i.e., outliers in y, bad +leverage points, i.e., outliers in X, and good leverage points, +i.e., outliers in both (X, y) up to 25% in the training data. +To demonstrate the good performance of the RPM for non- +Gaussian distribution noises, we assume that the noise follows +the Student’s t distribution with 10 degrees of freedom. This +distribution is chosen because it has heavier tails for low +degrees of freedom, producing sampling values that may fall +far from its median. We compare the performances of the GPM +and the RPM using mean absolute error and the root mean +square index for each of the cases. +A. IEEE 33-Bus System +The RPM is applied to a standard IEEE 33-bus system, +to which are attached four RES, namely, a PV (PG24) to +Bus 24, and three WGs (PG13, PG14, PG26) to Buses 13, 14, +and 26 of capacity 1 kW, 50 kW, 10kW, 10kW, respectively. +The time-series data considered for the RES power outputs +and loads are the real measurements with a resolution of 1s. +We run the power flow simulator at n = 150 input data +points X = [xt1, . . . , xt150]T to obtain the corresponding +voltage magnitude and angle values y = [yt1, . . . , yt150]T that +constitutes the training data. Trained on (y, X), the RPM and +the conventional GPM are used to make predictions for the +next n∗ = 60 data points constituted as the validation data +set at instances t∗ = [t151, . . . , t210]. To perform stochastic +analysis, Latin hypercube sampling is employed to generate +7000 samples of the input variables at each instance in +the validation data set following the Weibull distribution for +WGs (Pt∗ +i ,G13, Pt∗ +i ,G14, Pt∗ +i ,G24 ∼ Weibull(2.06, 7.1)) and the +Beta distribution for the PV (Pt∗ +i ,G26 ∼ Beta(2.06, 2.5)); +i = 151, . . . , 210. The results obtained from the Monte +Carlo (MC) simulations performed at these samples stand +as reference values for comparing the results obtained from +the RPM and the GPM. The robustness of the RPM is +demonstrated by the addition of 25% outliers as shown in +Fig. 1 (a) in the training data set. To be precise, we impose +a worst-case scenario by adding bad leverage points to the +input data points [xt1, . . . , xt37], i.e., to the measurements +(PG13, PG14, PG24, PG26) and to the load consumption of the +load buses, {PL1, PL2, . . . , PL33}. Similarly, vertical outliers +are added to the output data points [yt1, . . . , yt37], i.e., to +the measurements of voltage phasors. We observe from the +weights displayed in Fig. 1 (b) that the SHGM estimator +downweights the bad leverage points and vertical outliers. The +prediction results of the voltage magnitude and angle for Bus +19 with the percentage of outliers up to 25 in the training data +constitute a benchmark for this study. The mean and standard +deviation values (indicated as error bars) of the prediction +results for the voltage angle of Bus 19 are displayed in Fig. 2 +(a), where the error bars represent standard deviation values. +Fig. 2 (c) depicts the data fit for the training duration [t1−t150] +obtained for the RPM. Fig. 2 (b) compares the probability +density of the voltage angle of Bus 19 calculated from the 7000 +realizations at the next instance t∗ +151 obtained from the RPM +to the MC simulation output. Fig. 3 (a) displays the predicted +values of the voltage magnitude at Bus 19 obtained from +the RPM and from the GPM. We observe that the predicted +values from the GPM deviate largely from the true values. +This is due to the fact that the estimate of the mean function +hyperparameter of the conventional GPM is centered at the +basic weighted least squares estimate. Therefore, it fails to +represent the simulator in presence of outliers while the RPM +succeeds. Also, the prediction accuracy is displayed in Fig. +3 (b) using root mean square error (RMSE) values when +the training data set is added with an increasing percentage +of outliers up to 25%. We notice that the RPM consistently +exhibits low RMSE values. The RMSE and mean absolute +error (MAE) values for the forecast of the voltage phasors at +Bus 19 are listed in Table II for the cases of training data added +with and without the outliers for both the linear and quadratic +basis function. We observe that the prediction results are more +accurate for the quadratic basis function in the case of added +outliers. Therefore, by the principle of parsimony, we choose +a quadratic basis to obtain the results for the voltage phasors +for all the 33-buses in the network plotted in Fig. 4. +B. Real-World 240-Bus System +We integrate the 240-bus radial distribution system [33] +with RES, namely 35 PVs and 35 WGs distributed across the +network. Their locations in the network are displayed in Fig. +6. Please note that the RES are connected to each phase of +the indicated buses in the network. +The training data set (X, y) is obtained by running the +three-phase power flow simulator for the hourly spaced load +and active and reactive power injection measurements for +7 days i.e. a total of n = 168 data points constitute a +training data set for the design of the GPM and the RPM. +Both the GPM and RPM are analyzed using the prediction +results obtained for the n∗ = 24 test data points, which +are the validation data points. For the probabilistic analysis, +7000 samples are drawn using Latin hypercube sampling +from input variables of load following Gaussian distribution + +IEEE TRANSACTIONS ON POWER SYSTEMS +7 +(a) +(b) +Fig. 1: Outliers corrupting the training data set; (a) QQ-plot of the measurements corrupted with 25% of outliers; (b) plot of +the weights using the PSs vs. the outlier magnitudes. +TABLE II: The RMSE and MAE for the Bus 19 of the IEEE 33-bus system +Quadratic Basis +Linear Basis +With 25% outliers +Without outliers +With 25% outliers +Without outliers +RPM +GPM +RPM +GPM +RPM +GPM +RPM +GPM +Measure +V +A +V +A +V +A +V +A +V +A +V +A +V +A +V +A +RMSE +0.0034 9.7810e−4 1.0274 0.1527 0.0931 +0.0058 +0.0657 0.1468 0.0264 +0.01264 +0.7995 0.0206 0.01005 0.01747 0.00838 0.01645 +MAE +0.003 8.2815e−4 7.3087 0.1287 0.0672 2.5516e−4 0.0334 0.1669 0.0047 9.7962e−4 4.2355 0.0029 0.00058 0.00176 0.00042 0.0018 +(a) +(b) +(c) +Fig. 2: RPM results for the voltage phase angle at Bus 19: (a) +prediction at the test points; (b) probability density at the test +points; and (c) fitted values over the training data points. +Pt∗ +i ,L ∼ N(Pt∗ +i ,L, 0.05Pt∗ +i ,L), i = 169, . . . , 192, the WGs’ +output following the Weibull distribution, and the PVs’ output +following the Beta distribution with the shape and scale +parameters same as mentioned in Section 4.A. Voltage phasors +predictions at Bus 2003.2 for a day ahead forecast constitute +as a benchmark for this study. We display the mean of the +prediction results obtained from the GPM and the RPM of the +voltage magnitude at Bus 2003.2 using linear and quadratic +basis functions in Figs. 9 (b) and (d), respectively, where the +error bars represent the standard deviations as showing their +Fig. 3: Comparison between the performance of the RPM and +the GPM: (a) voltage magnitude at Bus 19; (b) RMSE values. +probability distributions individually will be inconvenient. +Similarly, the prediction results of the voltage angle at Bus +2003.2 are plotted in Figs. 10 (b) and (d). +We now add up to 25% of outliers in the power injection +measurements and voltage magnitudes and phase angles, i.e., +to the first 42 input and output data points in the training +dataset. To best demonstrate the robustness of the RPM, we +choose the outlier distribution to be Student’s t with 10 degrees +of freedom because of heavier tails as displayed in Fig. 5. +Similarly, the same percentage of outliers is included in the +measurements of active and reactive load power. We display +in Figs. 7 (b) and (d) the probability density function of the +voltage magnitude at Bus 2003.2 for the test input at instance +t∗ +169 obtained using linear and quadratic basis function. Sim- +ilarly, the probability density results of the voltage angle at + +10000 +P +S +G13 +e +8000 +Quantil +p +G14 +p +6000 +G24 +P +leasurement +G26 +4000 +2000 ++++ +0 +-2000 +-2 +-3 +-1 +0 +1 +2 +30.8 +ghts +90.6 +Wei +0.4 +0.2 +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +P +(kw) +G130.03 +--MC +IPrediction (RPM) +0.028 +0.026 +0.022 +0.02 +150 +160 +170 +180 +190 +200 +210 +Bus Numbel1000 +--- MC Output + - RPM Output +800 +Probability Density +600 +400 +200 +0 +0.022 +0.024 +0.026 +0.028 +Angle (rad)2 +--- Training Data + - Data fit (RPM) +1.5 +Angle (rad) +0.5 +0 +0 +50 +100 +150 +Time (s)10 +I MC +IPrediction (RPM) +5 +Voltage (pu) +IPrediction (GPM) +0 +5 +.10 +-15 +150 +160 +170 +180 +190 +200 +210 +Time (s)15 +- -RPM +----GPM +10 +RMSE +5 +5 +10 +15 +20 +25 +0 +Percentage of OutliersIEEE TRANSACTIONS ON POWER SYSTEMS +8 +(a) +(b) +Fig. 4: RPM predictions for the IEEE 33-bus system with +the training data set added with 25% of outliers: (a) voltage +magnitudes; (b) voltage phase angles. +(a) +(b) +Fig. 5: QQ plot of 25% outliers added in (a) power injection +measurements; (b) voltage magnitude measurements. +Bus 2003.2 are plotted in Figs. 7 (a) and (b). We compare +the prediction results of the voltage magnitude at Bus 2003.2 +obtained from the GPM and the RPM with basis function as +linear and quadratic in Figs. 9 (a) and (c), respectively. The +voltage angle results are displayed in Figs. 10 (a) and (c). The +RMSE and MAE values of the prediction results for the cases +of addition of outliers in the training data set and without the +addition of outliers, both with the linear and basis function, are +listed in Table III. We observe that the RMSE and the MAE +values obtained from the GPM and the RPM for the voltage +magnitudes’ predictions are lesser with linear basis functions +than the ones with the quadratic basis functions for both +cases. As for the voltage angles, both models’ performance +is better with the quadratic basis function for the case with +outliers in the training data set. The linear basis function yields +better performance for the case without outliers. Therefore, we +choose the linear basis to plot the voltage magnitudes and the +quadratic basis function for the voltage angles to obtain further +predictions at all the buses in the system for the case with +outliers in the training data set. The mean values (indicated as +dots) and standard deviations (indicated as error bars) of both +the models’ prediction results for the voltage magnitudes (see, +Fig. 12) and phase angles (see, Fig. 13) at the buses of the +240−bus system are compared with the MC simulation results. +We observe that the results obtained from the comparable +GPM deviate significantly from the true values in both the +mean and standard deviation. The performance of the RPM is +comparably accurate on account of the trade-off between the +accuracy and robustness of the SHGM estimator. Conventional +GPM is strongly biased towards outliers, thus the prediction +results deviate further away from the MC results. The bias is +particularly significant in phase C results because of the high +variance of voltage phasors due to the large power flow in +lines. For a large magnitude of outliers, the resulting bias is +the worst-case scenario that can be imposed on power system +measurements. The proposed RPM keeps this bias finite as +long as the added outliers are added without exceeding the +breakdown point, whereas the bias is unbounded for the results +of conventional GPM. +The RMSE of the predicted values for the voltage magnitude +and the angle at Bus 2003.2 with an increasing percentage of +outliers added in the training data are plotted in Fig.11 (a) and +(b), respectively. We observe that the RMSE values obtained +from the GPM are higher than the ones obtained from the +RPM which provides consistently low RMSE results. +Remark 1: Note that, because of the lack of availability of +the real measurements of the voltage phasors (output variables +y), they are obtained by running the power flow simulator us- +ing real measurements of active and reactive power injections +(input variables X). +V. CONCLUSION AND FUTURE WORK +In this paper, we propose a robust process model to perform +stochastic power flow calculations using time-series measure- +ments of power injections and voltage phasors. The proposed +model captures the natural stochastic dynamics introduced in +the power grid by the RES and DGs. This is accomplished by +training the RPM on recorded time series data set containing +the measurements of active and reactive power injections from +the RES and DGs and nodal voltage phasors. We demonstrate +the RPM on the standard IEEE 33-bus system and a real-world +240-bus system. We show that the proposed methodology can +handle 25% of outliers i.e. bad leverage points and vertical +outliers in the training data set using the root mean square +errors and maximum absolute errors of the predicted values +for voltage phasors. +In future work, we will focus on extending the applications +of the RPM for optimal power flow calculations. We will also +investigate the performance of other robust estimators with +high breakdown points for handling outliers in the training +data of more than 25%. + +Quantiles of Power Injection Sample +×104 +4 +2 +0 +2 +4 +-6 +-2 +0 +2 +Standard Normal Quantiles3 +2 +0 +-2 +3 +-2 +0 +1 +2 +Standard Normal Quantiles1.05 +Voltage (pu) +0.95 +-I·MC +IPrediction (RPM) +0.9 +0.85 +0.8 +0 +5 +10 +15 +20 +25 +30 +35 +Bus Number2 +--MC +IPrediction (RPM) +0.5 +0 +0 +5 +10 +15 +20 +25 +30 +35 +Bus NumberIEEE TRANSACTIONS ON POWER SYSTEMS +9 +Fig. 6: The online diagram of the 240 bus system integrated with RES. Blue and red squares indicate the PVs and WGs, +respectively +TABLE III: The RMSE and MAE for the Bus 2003.2 of the 240-bus system +Quadratic Basis +Linear Basis +With 25% outliers +Without outliers +With 25% outliers +Without outliers +RPM +GPM +RPM +GPM +RPM +GPM +RPM +GPM +Measure +V +A +V +A +V +A +V +A +V +A +V +A +V +A +V +A +RMSE +0.4222 3.8734 0.4242 5.5007 0.1888 4.9835 2.2246 11.1942 0.1921 8.8619 4.0814 53.4306 0.1183 4.4287 0.0637 6.1481 +MAE +0.3845 3.1791 0.3164 4.5781 0.1544 3.8252 1.9426 8.5223 0.1859 3.3976 3.0779 34.3391 0.1026 1.5122 0.0513 4.9164 +REFERENCES +[1] T. D. Liacco, “The role of state estimation in power system operation,” +IFAC Proceedings Volumes, vol. 15, no. 4, pp. 1531–1533, 1982. +[2] P. Zhang, F. Li, N. B. I. T. o. S. Grid, and u. 2010, “Next-generation +monitoring, analysis, and control for the future smart control center,” +ieeexplore.ieee.org. +[3] A. Allen, S. Santoso, and E. 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Source +THO +ESOE +3056 8 +30578305883059B3060830618 +3007 +3039 +3055 +Mswx +1009 +1008 +4041 +1006 +1005 +1004 +1002 +1082 +63H1 +LA +2057 +O +O +7 +3012A +03021c +3029A +900m +3038 +CB_102 +3054 +32# ! +3004 +3062B +3063 B +3064B3065B3066B3067B +208 +11 +OHS +. +CB_101[ +cB_301 +3011A +3020 c +3028A +3037 +1011A +1007 AB +1003 +10010 +3001 +LOEZ +LOT +2k5 +245 +DH +20G7 +20 +cB_201 Feeder C +3010A +3019 c +3027 A +3036 +3045c +■ +Feeder A +2001 +OH5 +OHS +1012 B +3009A +3018 c +3026 A +3035 +3047 +3044 c +3002 +OHS +3022l +OHS +10148 +1015 8 +OH1 +3040 +OH1 +3046 +1013 +3008 +3015 +OHS +2H5 +OHS +3013 B +3016 B3023A +3031 +3041c +1016 c +251 +3048 +3049 +3050 +3051 +3052 +245 +255 +3032 +3042c +i U2 +30148 +3017 B +3024A +3033 +1017 c +■3043 c +iFeeder B + 3025 A +3034 +3097A +2009 A +3069 +OH +OH1 +3075 cB_302 3076 oH1 3077 oH1 3078 +■ +HO +O +3096A +3068 +3106 c +3074 +3073 +UG2 + 2007 +138# i +2008 AO +3070 +3095A + 3079 +3105c +2011H +25 +OHI +2005 +2004 +THO +2003 +OH1 +2002 +604 +O3094A +- +03080 +12 +3104c +2006 +■ +3072 +VE60EO +2012 +UG2 ! +3101c +3102c3103c +NB.202 +2021 +2026H12027 +031318 +OH2032 +OHI +3091A +3090 A +3089A +3088A +3087A +3086A 338 +3084A +3083A +092UG2 +3005 +OO +O +izen +■ +UG2i +CB_204。 +2013 +642! +2USRi +CB_203 +3098 : +31306 +2028 A +3107 ±28, +31228 +2022c +. +CAP_201 ) +2038 +UG21 +izon +31296 +2034 +-328 +..-276 +-,3099 8 +291 +2029 +2046 B +3121: +2G2: +3128 B +20168 +2023 c +CB_303 +2030 +3120 8 +232i +142 : +2048B UG2 +31178 +31168 +31278 +2042 +2017b +2024c +usi +2031A +31148 +UG2! +3118 .- +49 +332 +32; +UG2! +3119 +31238 31248 31258 31268 +139h +2043 +02018 8 +02025 c +2044! +31158 +■ +3132A +2052 +2 i +. +Transformer and Load +262 +3136A + Capacitor Bank +131403141c +3133 +3155c +3154c3153c +3152c +3151c3150c +3149c +3148c +3142c +3143 +3-phaseUG +3137 A +Circuit Breaker +352; +UG2 +A +3138A +209R +to CB_102 +3162c3161c +3160c +3159c +3158c +3157c +31563144c +3145c3146c3147c +UG2 ! +Normally Open +3139 A +N.o.IEEE TRANSACTIONS ON POWER SYSTEMS +10 +(a) +(b) +(c) +(d) +Fig. 7: Comparison between the GPM and the RPM probabil- +ity density results for the voltage magnitude of Bus 2003.2 in +the 240−bus network when (a) the training data set is added +with 25% of outliers ; (b) training data set is not added with +outliers for linear basis; (c) the training data set is added with +25% of outliers ; (d) training data set is not added with outliers +for quadratic basis. +[14] J. 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Kimber, “A time- +series distribution test system based on real utility data,” in 2019 North +American Power Symposium (NAPS), pp. 1–6, IEEE, 2019. + +70 +---- MC +- - RPM +60 +Probability Density +50 +40 +30 +20 +10 +0 +0.3 +0.4 +0.5 +0.6 +Voltage (pu)70 +---- MC +H +- - RPM +60 +Probability Density +50 +40 +30 +20 +10 +0 +0.3 +0.35 +0.4 +0.45 +0.5 +Voltage (pu)70 +----MC +- - RPM +60 +Probability Density +50 +40 +30 +20 +10 +0 +0.3 +0.4 +0.5 +0.6 +0.7 +Voltage (pu)70 +--MC +- RPM +60 +Probability Density +50 +40 +30 +20 +10 +0 +0 +0.2 +0.4 +0.6 +Voltage (pu)0.3 +----MC +- - RPM +0.25 +Density +0.2 +Probability +0.15 +0.1 +0.05 +0 +06- +-80 +-70 +-60 +Angle (rad)0.3 +---- MC +- - RPM +0.25 +Density +0.2 +Probability +0.15 +0.1 +0.05 +1 +0 +-90 +-80 +-70 +-60 +Angle (rad)0.3 +---- MC +- - RPM +0.25 +Density +0.2 +- +Probability +0.15 +1 +1 +0.1 +- +1 +1 +0.05 +0 +-90 +-85 +-80 +-75 +-70 +Angle (rad)0.3 +----MC +- - RPM +0.25 +Density +0.2 +0.15 +Probabili +0.1 +0.05 +0 +06- +-85 +-80 +-75 +-70 +Angle (rad)IEEE TRANSACTIONS ON POWER SYSTEMS +11 +(a) +(b) +(c) +(d) +Fig. 9: Comparison between the GPM and the RPM forecast results for the voltage magnitude of Bus 2003.2 in the 240−bus +network when (a) the training data set is added with 25% of outliers ; (b) training data set is not added with outliers for linear +basis; (c) the training data set is added with 25% of outliers ; (d) training data set is not added with outliers for quadratic +basis. +(a) +(b) +(c) +(d) +Fig. 10: Comparison between the GPM and the RPM forecast results for the voltage angle of Bus 2003.2 in the 240−bus +network when (a) the training data set is added with 25% of outliers ; (b) training data set is not added with outliers for linear +basis; (c) the training data set is added with 25% of outliers ; (d) training data set is not added with outliers for quadratic +basis. + +10 +(nd) +5 +/oltage ( +0 +-5 +---MC +IPredicted Values (RPM) +-10 +IPredicted Values (GPM) +-15 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)0.6 +(nd) +Itage +Vol +0.2 +I--MC +IPredicted Values (RPM) +Predicted Values (GPM) +0 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)1.5 +-}--MC +IPredicted Values (RPM) +IPredicted Values (GPM) +0.5 +Vol +0 +-0.5 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)4 +I--MC +(nd) +IPredicted Values (RPM) +IPredicted Values (GPM) +2 +Vol +0 +/ +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)200 +(rad) +0 +-}--MC +Ang +IPredicted Values (RPM) +IPredicted Values (GPM) +-400 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)0 +-{--MC +Angle (rad) +IPredicted Values (RPM) +IPredicted Values (GPM) +50 +-100 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h).20 +Angle (rad) +-40 +-60 +-{--MC +-80 +IPredicted Values (RPM) +IPredicted Values (GPM) +-100 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)-20 +-}-MC +Angle (rad) +-40 +IPredicted Values (RPM) +IPredicted Values (GPM) +-60 +-80 +-100 +170 +172 +174 +176 +178 +180 +182 +184 +186 +188 +190 +192 +Time (h)IEEE TRANSACTIONS ON POWER SYSTEMS +12 +(a) +(b) +Fig. 11: +RMSE vs the percentage of outliers added in training data for the prediction results at Bus 2003.2 (a) voltage +magnitude; (b) voltage angle. +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 12: The obtained prediction results of voltage magnitude from the RPM compared with those obtained from the GPM +of 240−bus system with 25% of outliers added in training data. (a) The results obtained from the RPM for phase a; (b) +comparison between the GPM and the RPM for phase a; (c) The results obtained from the RPM for phase b; (d) comparison +between the GPM and the RPM for phase b; (e) The results obtained from the RPM for phase c; (f) comparison between the +GPM and the RPM for phase c. + +8 + -RPM +----GPM +. +RMSE +4 +2 +5 +10 +15 +20 +25 +Fraction of Outliers14 +- -RPM +---- GPM +12 +10 +RMSE +8 +6 +2 +0 +5 +10 +15 +20 +251.4 +-}-MC (Phase A) +IPredicted Values (Phase A) (RPM) +1.2 +0.8 +Voltage +0.6 +0.4 +0.2 +Bus Number3.5 +}-. True Data (Phase A) +IPredicted Values (Phase A) (RPM) +3 +IPredicted Values (Phase A) (GPM) +2.5 +Voltage (pu) +2 +1.5 +0.5 +0 +-0.5 +Bus number1.4 +{- MC (Phase B) +IPredicted Values (Phase B) (RPM) +1.2 +Voltage (pu) +0.8 +0.6 +0.4 +0.2 +0 +Bus Number2 +1.5 +Voltage (pu) +0.5 +amm +H +i +0 +-0.5 +I-- MC (Phase B) +IPredicted Values (Phase B) (RPM) +IPredicted Values (Phase B) (GPM) +Bus number1.5 +-{-MC (Phase C) +Predicted Values (Phase C) (RPM) +1.4 +1.3 +(nd) +Voltage +0.9 +0.8 +& +Bus Number-{-MC (Phase C) +1.5 +IPredicted Values (Phase C) (RPM) +↑Predicted Values (Phase C) (GPM) +06006000000 +(nd) +Voltage ( +0.5 +0 +690 +5 +9 +9 +30 +3107 +Bus NumberIEEE TRANSACTIONS ON POWER SYSTEMS +13 +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 13: The obtained prediction results of voltage angle from the RPM compared with those obtained from the conventional +GPM of 240−bus system with 25% of outliers added in training data. (a) The results obtained from the RPM for phase a; (b) +comparison between the GPM and the RPM for phase a; (c) The results obtained from the RPM for phase b; (d) comparison +between the GPM and the RPM for phase b; (e) The results obtained from the RPM for phase c; (f) comparison between the +GPM and the RPM for phase c. + +200 +1 MC +IPredicted Values (Phase A) (RPM) +150 +100 +50 +0 +-50 +Bus Number200 +1 MC +IPredicted Values (Phase A) (RPM) +Predicted Values (Phase A) (GPM) +150 +100 +50 +0 +50 +Bus Number150 +100 +50 +Angle (rad) +0 +-50 +T +-100 +MC (Phase B) +IPredicted Values (Phase B) (RPM) +-150 +Bus Number400 +300 +200 +100 +F 开 + (rad) +Angle +0 +-100 +-200 +I--MC (Phase B) +-300 +IPredicted Values (Phase B) (RPM) +Predicted Values (Phase B) (GPM) +400 +Bus Number350 +300 +250 +200 +150 +100 +-}-MC (Phase C) +IPredicted Values (Phase C) (RPM) +50 +& +Bus Number700 +600 +500 +400 +Angle (rad) +300 +200 +100 +0 +-100 +{--MC (Phase C) +-200 +IPredicted Values (Phase C) (RPM) +IPredicted Values (Phase C) (GPM) +-300 +5 +3 +9 +30 +Bus Number \ No newline at end of file diff --git a/BNE0T4oBgHgl3EQfxwK1/content/tmp_files/load_file.txt b/BNE0T4oBgHgl3EQfxwK1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dfde996ecf660188a74f45ce8a7f4061b4aebd4 --- /dev/null +++ b/BNE0T4oBgHgl3EQfxwK1/content/tmp_files/load_file.txt @@ -0,0 +1,1215 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf,len=1214 +page_content='IEEE TRANSACTIONS ON POWER SYSTEMS 1 A Robust Data-driven Process Modeling Applied to Time-series Stochastic Power Flow Pooja Algikar, Member, IEEE,, Yijun Xu, Senior Member, IEEE, Somayeh Yarahmadi, Member, IEEE, Lamine Mili, Life Fellow, IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Abstract—In this paper, we propose a robust data-driven pro- cess model whose hyperparameters are robustly estimated using the Schweppe-type generalized maximum likelihood estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The proposed model is trained on recorded time-series data of voltage phasors and power injections to perform a time-series stochastic power flow calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Power system data are often corrupted with outliers caused by large errors, fault conditions, power outages, and extreme weather, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The proposed model downweights vertical outliers and bad leverage points in the measurements of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The weights used to bound the influence of the outliers are calculated using projection statistics, which are a robust version of Mahalanobis distances of the time series data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The proposed method is demonstrated on the IEEE 33-Bus power distribution system and a real-world unbalanced 240-bus power distribution system heavily integrated with renewable energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Our simulation results show that the proposed robust model can handle up to 25% of outliers in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Index Terms—Time-series Stochastic Power Flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Robust Process Modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Robust Mahalanobis Distances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Generalized Maximum Likelihood Estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Outlier Detection and Identifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' INTRODUCTION A power system, as it stands currently, involves real-time operational and control actions based on the information provided by the state estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The latter processes a set of measurements at periodic time intervals consisting of real and reactive power flows and power injections and voltage magnitudes at selected lines and buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' They are collected from metered devices such as SCADA measurements, phasor measurement units (PMUs), and intelligent electronic devices (IEDs), among others [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' They facilitate the time-series power flow analysis to forecast load duration curves and hence, to determine overload conditions in power distribution systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' It is well known that these measurements are often corrupted with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For instance, during fault conditions, the in- terference of inrush current in switchgear temporarily causes errors in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The communication methods used in power distribution systems are often exposed to heavy electromagnetic interference, resulting in corrupted data [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Furthermore, asynchronous sample time of PMUs [3], [4] and magnetic saturation and hysteresis in potential and current This work is supported, in part, by NSF 1917308 and by the Research Startup Fund of Southeast University in China under Grant 3216002206A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Algikar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Mili are with the Electrical Engineering Department, Vir- ginia Tech, Falls Church, VA 22043, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (e-mail:{apooja19, syarahmadi, lmili}@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Xu is with the Southeast University, Nanjing, Jiangsu, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (e- mail:yijunxu@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' transformers cause measurement errors in current and voltage phasors [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Under these conditions, the state estimations based on the weighted least squares method suffer from masking and smearing effects, thus yielding inaccurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The situation exacerbates under heavy penetration of renewable energy sources (RES) and distributed generations (DGs) due to the stochastic dynamics that they introduce in the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' {For a large-scale power system, performing classical Monte Carlo (MC) simulations of the thousands of realizations for uncertainty quantification requires high computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Therefore, developing robust and computationally efficient models and tools that process real measurements to analyze the stochastic dynamics of a power system is of paramount importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In the literature, several stochastic power flow methods have been proposed to carry out sensitivity analysis and uncertainty quantification [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Among them, the most popular methods are the MC simulations and meta-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' As discussed earlier, MC methods turn out to be computationally inefficient when thousands of simulation runs are needed to achieve meaningful statistical results in uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Meta-models, also known as emulators, surrogates, or re- sponse surfaces, only statistically represent the deterministic power flow simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Those based on Gaussian processes are non-parametric reduced-ordered models in which the model output realizations are assumed to follow a Gaussian distribu- tion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Other types of meta-models extensively developed in the literature are based on polynomial chaos [8]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [9] developed a sparse polynomial chaos expansion to tackle a large number of random input variables considering correlation among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [10] proposed generalized polynomial chaos (gPC) with rectangular formulations to preserve the non-linearity of power flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [11] extended the gPC to the data-driven gPC to better deal with the dependent correlated uncertainties among input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [12] developed hierarchical polynomial chaos analysis of variance (ANOVA) for efficient extension of the gPC to large-scale systems without falling prey to the curse of dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' A few methods using neural networks include deep neural network models [16], [17], graph convolutional network models [18], and graph neural network [19] to overcome the computational challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Once developed, the meta-model, which statistically repre- sents the power flow simulator, is run while considering thou- sands of new input variables to perform sensitivity analysis and uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' {However, none of them are robust enough to be trained on real-time data, which makes arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='02651v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='SY] 6 Jan 2023 IEEE TRANSACTIONS ON POWER SYSTEMS 2 them unsuitable to perform time-series stochastic power flow analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The accuracy of the results obtained from a meta-model is highly dependent on the quality of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' A well-trained meta-model requires data points that fill the input design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' As a result, the data points are sampled from an assumed probability distribution in the design space of stochastic input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The assumed probability distribu- tions typically are the Gaussian distribution for the load, the Weibull distribution for the wind speed, and the Beta distribution for the solar irradiance, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' However, in practice, these distributions may not represent the actual data [20], [21], yielding inaccurate uncertainty quantification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The conventional meta-modeling methods are not designed to handle the misrepresentations of power curve distribution, yielding biased results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Consequently, modern data-driven models are introduced in the literature, which make use of the raw observational data as a consequence of the proliferation of sensing and metering devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' This approach captures the natural stochasticity of the underlying process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' in [22] developed a data-driven emulator using polynomial chaos that estimates the statistics of the voltage phasors while Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' in [23] proposed a fully non-parametric approach to avoid assuming a parametric distribution for the Gaussian process meta-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' However, all these methods are relying on raw data without considering outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' It is well known that wind generation (WG) time series data are frequently contaminated with communication errors, wind turbine outages, and curtail- ments [20] while PV time series data are contaminated with large signal noise, sensor failures, communication equipment failures, maximum power tracking abnormalities, array shut- downs, and power limitations, to name a few [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' As a consequence, various data preprocessing techniques have been proposed to account for these abnormal tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For instance, Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [24] developed an algorithm based on the mathematical morphology operation of wind power curve image for detecting and cleaning the wind turbine abnormal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In [25], a method for filtering out the outliers in raw wind data considering the degree of similarity between the in- dividual objects is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' These data-cleaning algorithms combined with non-robust data-driven models such as [22], [23] will be time-consuming in real-time stochastic analysis for increasing power system size with limited computational power and databases for newer wind or solar farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Their suitability in real-time statistical analysis is arguable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Unlike the proposed method, they are not effectively integrated with the estimation process and are not robust against all types of arising outliers in the time frame of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In this paper, we develop a real-time data-driven time-series stochastic power flow analysis based on a robust process model (RPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The proposed RPM makes use of the Schweppe-type generalized maximum likelihood estimator (SHGM) that can handle up to 25% of outliers in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Recall that outliers may be either vertical outliers or bad leverage points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The other estimators proposed in robust statistics for linear regression are M-estimators, which are not robust against bad leverage points [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In power systems, leverage points are power flows on relatively short lines or power injection on buses with relatively many incident lines [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We assess the robustness of the RPM only theoretically by using three statistical concepts, namely, influence function, finite-sample breakdown point, and asymptotic maximum bias curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The robustness of the RPM is experimentally demonstrated on the radial IEEE-33 bus power distribution system and a real-world 240-bus power distribution system located in the Midwest U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' for which the training data are manually added with vertical outliers and bad leverage points up to 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Section II discusses the conventional GPM followed by theoretical background on statistical robustness concepts that are used to access the robustness of the RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Section III presents the development of the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Section IV demonstrates the performance of the RPM on the IEEE 33-bus distribution system and a real-world 240-bus distribution system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Section V concludes the paper and outlines future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formulation of the Time-series Stochastic Power Flow in the Gaussian Process Framework The power flow simulator, represented by f(·), is assumed to be a function of active and reactive power injection mea- surements at all the p buses, xti ∈ R2p in training interval t = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The output variables are yti = f(xti) + ϵti are voltage magnitude or phase angle measurement at a bus with an independent and identically distributed additive Gaussian measurement noise, ϵti ∼ N(0, σ2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In the Gaussian process modeling framework, the uncertainty about the power flow simulator output is characterized as a Gaussian process having a specific mean function m(·), and a covariance (kernel) function k(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' As a Gaussian process distribution, the sim- ulator output variables f(xt1), f(xt2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , f(xtn) follow a multivariate normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' To incorporate our belief about the simulator, the prior distribution is formulated as a Gaussian process with mean function m0(·) and covariance function k0(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have f(·)| β, l, τ 2 ∼ GP(m0(·), k0(·, ·)), (1) where the mean function m0(·) takes the form m0(x) = h(x)T β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (2) Here, h(xti) : R2p → Rq denotes the basis func- tion that can be chosen to model the assumed degree of non-linearity of the power system, that is, h(xti) = [1, xti, x2 ti, x3 ti, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For example, the constant, linear, and quadratic basis functions are respectively given by h(xti) = [1], h(xti) = [1, xti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , xti2p]T , and h(x) = [1, xti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , xti2p, x2 ti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , x2 ti2p]T ∈ Rq, q = 4p + 1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The kernel function k0(xti, xtj) denotes the co- variance between corresponding output points (yti, ytj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' A commonly used covariance function is the radial basis function given by k0(xti, xtj|l) = τ 2exp � − 2p � k=1 (xtik − xtjk)2 2lk2 � , (3) IEEE TRANSACTIONS ON POWER SYSTEMS 3 where l = (l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , l2p) denotes the characteristic length-scale, which models the rapidity of the process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Some other covariance functions are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Let us gather n input and output measurements into the ma- trix X = [xt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , xtn]T and the vector y = [yt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , ytn]T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The matrix of of basis functions is then repre- sented as H(X) = [h(xt1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , h(xtn)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The distribution of the output vector y of the power system according to (1) is a multivariate normal random vector having a covariance func- tion diagonally additive with noise elements ϵ ∼ N(0, σ2 nIn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have y|X, β, l, τ 2, σ2 n ∼ N (H(X)β, Σ(X)) , (4) where Σ(X) = k0(X, X) + σ2 nIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The noise elements ϵ with zero mean and variance σ2 n, also called ”nugget”, account for model uncertainty and numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The training data set is constituted by (y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For the stochastic power flow analysis, let us consider that we draw K samples of the input test variable xt∗ at instances t∗ = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n∗] in the prediction interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In time-series analysis, the assumption of stationarity and ergodicity for a specific range of time is often made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The latter means that the sample average, commonly known as the ensemble average, is equal to the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The assumption of ergodicity allows us to model a stochastic time-series power flow using a single real-time measurement per instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Let us group the K sampled test predictors for instance t∗ i denoted as x(k) t∗ i for k in [1, K] into X∗ i = [x(1) t∗ i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , x(K) t∗ i ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Using a hierarchical formulation, the model output variables y∗ obtained through the power flow simulator f(·) at the test points X∗ i together with the training output variables follow a joint multivariate Gaussian distribution given by � y y∗|X∗ i � ∼ N �� m0(X) m0(X∗ i ) � , � Σ(X) C(X∗ i ) CT (X∗ i ) V(X∗ i ) �� , (5) where C(X∗ i ) = k0(X, X∗ i ), CT (X∗ i ) = k0(X∗ i , X) and V(X∗ i ) = k0(X∗ i , X∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The covariance matrix, Σ(X), is rep- resented by Σ hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Furthermore, we assume an a priori Gaussian probability distribution for the simulator output at the test points, f(X∗ i )|X∗ i , that is, f(X∗ i )|X∗ i ∼ GP (m0(X∗), V(X∗ i )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (6) Upon conditioning and using the standard techniques in mul- tivariate distributions, we get f(X∗ i )|X∗ i , y, X, β, l, τ 2, σ2 n ∼ GP (µ∗(X), Σ∗(X)) , (7) where the estimated mean function �µ∗(X∗ i ) is given by �µ∗(X∗ i ) = � m0(X∗ i ) + �CT (X∗ i )�Σ−1r, (8) and the estimated covariance function �Σ∗(X∗) is expressed as �Σ∗(X∗ i ) = �V(X∗ i ) − �CT (X∗ i )�Σ−1 �C(X∗ i ), (9) for all the instances in the predictive interval, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The estimate of the mean function given by (8) acts as a computationally efficient surrogate model that captures the behavior of the power flow while the covariance matrix estimate given by (9) quantifies the associated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Let us apply a weak prior for (β, τ 2), p(β, τ 2) ∝ 1 τ 2 , combining with (4) and using Bayes’ theorem yields a poste- rior distribution for (β, τ 2), which is normal inverse-gamma distribution given by β|y, X, l, τ 2, σ2 n ∼ N(ˆβ, τ 2(HT Σ−1H)−1), (10) where ˆβ is the weighted least squares estimate given by ˆβ = (HT Σ−1H)−1HT Σ−1y, and τ 2|y, X, l, σ2 n ∼ InvGamma �n − q 2 , (n − q − 2)ˆτ 2 2 � , (11) where ˆτ 2 = yT (Σ−1−Σ−1H(HT Σ−1H)−1HT Σ−1)y (n−q−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Smearing and Masking Effects in Conventional GPM De- sign The residual vector is defined as the difference between the observation vector, y, and the estimated vector, �y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have r = y − �y, (12) where ˆy = Sy, (13) and where S is the hat matrix given by S = H(HT Σ−1H)−1HT Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Substituting (13) into (12) yields r = y − Sy, (14) = (I − S)y, (15) = Wy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (16) Substituting the expression y = m(X) + e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' e ∼ N(0, Σ) into (16) yields r = We.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (17) Here, W is called the residual sensitivity matrix as it expresses the sensitivity of the residuals to the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For the WLS estimator, the outlier detection and identification statistical tests suffer from the smearing and masking effect of the outliers on the residuals as shown next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Smearing effect Let us assume that e1 ̸= 0 and ei = 0 for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' From (14), we have r1 = W11e1 ̸= 0, (18) r2 = W21e1 ̸= 0, (19) rn = Wn1e1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (20) This is known as the smearing effect of one outlier on the residuals, which makes the identification of that outlier using the residual statistical test difficult to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Masking effect Let us assume that the e1 ̸= 0 and e2 ̸= 0 while ei = 0 for i = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The associated residuals are expressed as r1 = W11e1 + W12e2, (21) r2 = W21e1 + W22e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (22) Therefore, there exist e1 and e2 so that the residuals r1 ≈ 0 and r2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' This is known as the masking effect of the outliers on the residuals, which results in the failure of the outlier residual identification test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' IEEE TRANSACTIONS ON POWER SYSTEMS 4 TABLE I: Commonly used Kernel Functions Type Expression Exponential kE(xti, xtj) τ 2exp � − �2p k=1 |xtik −xtjk | lk � Matern 3/2 kM(xti, xtj) τ 2 � 1 + �2p k=1 √ 3(xtik −xtjk ) lk � exp � − �2p k=1 √ 3|xtik −xtjk | lk � Rational Quadratic kRQ(xti, xtj) τ 2 � 1 + exp � − �2p k=1 (xtik −xtjk )2 2l2 kα ��−α C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Robustness Concepts In this subsection, we briefly review the definition of the statistical efficiency of an estimator and the robustness concepts developed in robust statistics, namely, the asymptotic influence function, the breakdown point, and the asymptotic maximum bias curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 1) Asymptotic Influence Function: For an ϵ-contaminated model G(r) = (1−ϵ)Φ(r)+ϵF(r), where Φ is the cumulative Gaussian distribution function and F is the unknown distribu- tion function of residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The influence function is based on the Gateaux derivative that quantifies the local sensitivity of an estimator T(G) to an arbitrary infinitesimal contamination H = ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' It is expressed as IF(ri, hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ) = lim ϵ→0 T((1 − ϵ)Φ + ϵ∆r) − T(Φ) ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (23) The total influence function, IF(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ), of an M-estimator for a linear regression model is equal to the product of the scalar- valued influence of residuals, IR(ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ), and the vector-valued influence of position, IP(hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have IF(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ) = IR(ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ)IP(hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (24) They are given by IR(ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ) = ψ( ri s ) E[ψ′( ri s )] and IP(hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ) = (HT H)−1hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For an M-estimator, the IR(·) is bounded if the ψ(·) is bounded while the IP(·) is always unbounded, revealing its non-robustness to bad leverage points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2) Finite-Sample Breakdown Point: The maximum value of ϵ, denoted as ϵ∗, for which the maximum bias of an estimator is finite is called the finite-sample breakdown point of that estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally we have ϵ∗ = max{ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' bmax(ϵ) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [29] and [30] showed that the maximum finite-sample break- down point of any regression equivariant estimator under the assumption of general position is given by � n−q 2 � /n [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3) Asymptotic maximum bias curve: The asymptotic max- imum bias curve is the curve of the upper bound of a bias of an estimator for an increasing level of contamination 0 ≤ ϵ < ϵ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The asymptotic maximum bias curve of any Fisher consistent estimator, ˆθ, in its functional form, T , at any ϵ contaminated model is defined as bmax(ϵ) = sup H |T (G)−θ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In the location case, Huber [30] showed that the sample median has the smallest possible asymptotic maximum bias curve among all location equivariant estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In linear regression, the estimator that has the minimum asymptotic bias curve is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 4) Statistical Efficiency: The minimum possible variance that any estimator of location, ˆθn, is able to attend at an assumed probability distribution, F, is given by the Cramer- Rao lower bound, which is defined as the inverse of the Fisher information, If.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have V ar(√nˆθn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' F) ≥ 1 If ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (25) The ratio of the Cramer-Rao lower bound and the variance of an estimator is called the efficiency of that estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' An asymptotically efficient estimator is one whose variance attains the Cramer-Rao lower bound for n tending to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' ROBUST DATA-DRIVEN PROCESS EMULATOR In this section, we discuss the development of the proposed RPM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We rewrite (4) in the form of a regression problem in terms of the mean function hyperparameter given by y(X) = H(X)β + e, (26) The hyperparameters of the mean and covariance function incorporating the maximum likelihood estimation method are obtained by solving �θ = arg max (β,l,τ 2,σ2n)∈RqR2pR+∗R+∗ log L � y|X, β, l, τ 2, σ2 n � , (27) where θ represents a vector of hyperparameters (β, l, τ 2, σ2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Schweppe-type Generalized Maximum Likelihood Estima- tor We propose to estimate β in a robust manner using the SHGM estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The SHGM estimator minimizes a weighted loss function of residuals ri given by J(β) = min ˆβ n � i=1 w2 i ρ � ri wis � , (28) where ρ(·) is a non-linear loss function of the standardized residuals, rSi = ri wis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The residual scale s is robustly estimated by s∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4826 bm median|r| when there is a little to none knowledge about the error covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='48261 + 5 n − q median|r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (29) We choose the Huber loss function because of its convexity and its quadratic characteristic at its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' It is defined as ρ(ri) = � r2 i 2 for ri < c, c|ri| − c2 2 for ri ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (30) IEEE TRANSACTIONS ON POWER SYSTEMS 5 The threshold parameter c is typically chosen to be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5, which offers a good compromise between a high statistical efficiency at the Gaussian distribution and good robustness against outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Weights Based on Projection Statistics The weights are calculated using the projection statistics, which are a robust version of the Mahalanobis distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, they are defined as the maximum of the standardized projection distances obtained by projecting the point cloud in the directions that originate from the coordinate-wise median and that pass through each of the data points [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Let hT (xi) be represented by hT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have PSi = max ||v||=1 hT i v − med j (hT j v) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4826 med k |hT k v − med j (hT j v)|, (31) where vj = uj ||uj||;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' uj = hi − M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Here, M denotes the coordinatewise median given by M = { med j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=',n hj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , med j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=',n hjq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The weights are calculated as w(hi) = � 1, PS2 i ≤ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' b PS2 i , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (32) The data point hi is considered as a leverage point when the associated PS2 i is greater than b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The weights downweight the bad leverage point and vertical outliers while retaining the good leverage points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Robust Estimation of the Mean Function Hyperparameter We estimate the hyperparameter of the mean by setting the gradient of the objective function J(β) with respect to β to zero, which is given by n � i=1 wihi ∂ρ(rSi) ∂rSi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (33) Let us define the psi-function as ψ(rSi) = ∂ρ(rSi) ∂rSi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (33) now becomes n � i=1 wihiψ(rSi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (34) Dividing (34) by the standardized residuals rSi, we get m � i=1 q � ri wis � hiri = 0, (35) where q(rSi) = ψ(rSi) rSi is called a weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For the case of Huber ρ-function, it is defined as q(rSi) = � 1, ri ≤ c b sign(rSi) rSi , otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (36) Substituting the expression of the residuals, ri, and rewriting (36) in matrix form yields HT Q(y − Hβ) = 0 (37) β=(HT QΣ−1H)−1HT Q(k)Σ−1y, (38) where Q = diag(q(rSi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Since β is a function of Q, we solve for β in an iterative manner by incorporating iterative re-weighted least squares (IRLS) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Formally, we have β(k+1) = (HT Q(k)Σ−1H)−1HT Q(k)Σ−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (39) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Iterative Procedure for the Hyperparameter Estimation In this subsection, the robust estimation of the hyperparam- eters (l, τ 2, σ2 n) of the RPM associated with the covariance function is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' With the observation set available from the MC simulation of the code, (y, X), we estimate the hyper- parameter �β using the algorithm given by (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The maximum likelihood estimate of the remaining hyperparameters, namely, (l, τ 2, σ2 n), is formulated as (�l, �τ 2, �σ2 n) = arg max l,τ 2,σ2n log L � Y|X, �β, l, τ 2, σ2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (40) Let us define the resulting log L function by Γ(l, τ 2, σ2 n) = log |k(X, X|l, τ 2) + σ2 nIn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (41) Consequently, the maximum likelihood estimate of (l, τ 2, σ2 n) reduces to (�l, �τ 2, �σ2 n) = arg min l,τ 2,σ2 n Γ(l, τ 2, σ2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (42) The hyperparameters, (�l, �τ 2, �σ2 n), are estimated by utilizing a gradient-based optimizer as described in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We can then update �β as ��β = �β(�l, �τ 2, �σ2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The algorithm used for Algorithm 1 Algorithm for Estimating the RDP Hyperparam- eters 1: Develop the power-system simulator in the case where the measurements are unavailable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2: Run the simulator at required input power measurements to obtain the voltage magnitude and the voltage phase angle at each load bus, which constitutes the training data- set (X, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3: Construct H using a suitable basis function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 4: Calculate the projection statistics of the row vectors of H given by (31);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 5: Calculate the weights w based on the PS given by (32);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 6: Initialize β using the weighted least squares solution as β0 = (HT Σ−1H)−1HT Σ−1y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 7: Update β by executing the IRLS algorithm given by (39) until convergence while setting the hyperparameters (l, τ 2, σ2 n) at their initial values to obtain �β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 8: Update (l, τ 2, σ2 n) while setting β = �β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 9: Iterate Steps 7 and 8 until convergence, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' ||r|| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='001, to obtain the final hyperparameter estimates, (��β, �l, �τ 2�σ2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' estimating the hyperparameters is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Once all the hyperparameters of the RPM, (β, l, τ 2, σ2 n), are estimated, we use (8) as a robust computationally efficient surrogate and (9) to quantify its variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' IEEE TRANSACTIONS ON POWER SYSTEMS 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Robustness of the RPM The influence function of the SHGM estimator is given by IF(rSi, hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Φ) = ψ(rSi) EΦ[ψ ′(rSi)](HT H)−1hiwi, (43) For the SHGM estimator, one can notice that the influence of position, IP(hi, Φ) = (HT H)−1hiwi, is bounded thanks to the weights calculated using the projection statistics (see Sec- tion III), whose breakdown point attains the maximum given by [(n−q−1)/2] n [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Note that the SHGM estimator reduces to an ℓ2-norm estimator for small standardized residuals and to the ℓ1-norm estimator for larger ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Therefore, it has a high statistical efficiency at the Gaussian distribution while being robust to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' CASE STUDIES In this section, we compare the performance of the proposed model RPM to that of the Gaussian process model (GPM) when applied to a standard IEEE 33-bus system (Case A) and to a real-world 240-bus distribution system located in the Midwest U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' with high penetration of RESs and DGs (Case B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We add vertical outliers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=', outliers in y, bad leverage points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=', outliers in X, and good leverage points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=', outliers in both (X, y) up to 25% in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' To demonstrate the good performance of the RPM for non- Gaussian distribution noises, we assume that the noise follows the Student’s t distribution with 10 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' This distribution is chosen because it has heavier tails for low degrees of freedom, producing sampling values that may fall far from its median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We compare the performances of the GPM and the RPM using mean absolute error and the root mean square index for each of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' IEEE 33-Bus System The RPM is applied to a standard IEEE 33-bus system, to which are attached four RES, namely, a PV (PG24) to Bus 24, and three WGs (PG13, PG14, PG26) to Buses 13, 14, and 26 of capacity 1 kW, 50 kW, 10kW, 10kW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The time-series data considered for the RES power outputs and loads are the real measurements with a resolution of 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We run the power flow simulator at n = 150 input data points X = [xt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , xt150]T to obtain the corresponding voltage magnitude and angle values y = [yt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , yt150]T that constitutes the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Trained on (y, X), the RPM and the conventional GPM are used to make predictions for the next n∗ = 60 data points constituted as the validation data set at instances t∗ = [t151, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , t210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' To perform stochastic analysis, Latin hypercube sampling is employed to generate 7000 samples of the input variables at each instance in the validation data set following the Weibull distribution for WGs (Pt∗ i ,G13, Pt∗ i ,G14, Pt∗ i ,G24 ∼ Weibull(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='06, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1)) and the Beta distribution for the PV (Pt∗ i ,G26 ∼ Beta(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='06, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' i = 151, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The results obtained from the Monte Carlo (MC) simulations performed at these samples stand as reference values for comparing the results obtained from the RPM and the GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The robustness of the RPM is demonstrated by the addition of 25% outliers as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 1 (a) in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' To be precise, we impose a worst-case scenario by adding bad leverage points to the input data points [xt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , xt37], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=', to the measurements (PG13, PG14, PG24, PG26) and to the load consumption of the load buses, {PL1, PL2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , PL33}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Similarly, vertical outliers are added to the output data points [yt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , yt37], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=', to the measurements of voltage phasors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We observe from the weights displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 1 (b) that the SHGM estimator downweights the bad leverage points and vertical outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The prediction results of the voltage magnitude and angle for Bus 19 with the percentage of outliers up to 25 in the training data constitute a benchmark for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The mean and standard deviation values (indicated as error bars) of the prediction results for the voltage angle of Bus 19 are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2 (a), where the error bars represent standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2 (c) depicts the data fit for the training duration [t1−t150] obtained for the RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2 (b) compares the probability density of the voltage angle of Bus 19 calculated from the 7000 realizations at the next instance t∗ 151 obtained from the RPM to the MC simulation output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3 (a) displays the predicted values of the voltage magnitude at Bus 19 obtained from the RPM and from the GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We observe that the predicted values from the GPM deviate largely from the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' This is due to the fact that the estimate of the mean function hyperparameter of the conventional GPM is centered at the basic weighted least squares estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Therefore, it fails to represent the simulator in presence of outliers while the RPM succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Also, the prediction accuracy is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3 (b) using root mean square error (RMSE) values when the training data set is added with an increasing percentage of outliers up to 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We notice that the RPM consistently exhibits low RMSE values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The RMSE and mean absolute error (MAE) values for the forecast of the voltage phasors at Bus 19 are listed in Table II for the cases of training data added with and without the outliers for both the linear and quadratic basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We observe that the prediction results are more accurate for the quadratic basis function in the case of added outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Therefore, by the principle of parsimony, we choose a quadratic basis to obtain the results for the voltage phasors for all the 33-buses in the network plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Real-World 240-Bus System We integrate the 240-bus radial distribution system [33] with RES, namely 35 PVs and 35 WGs distributed across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Their locations in the network are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Please note that the RES are connected to each phase of the indicated buses in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The training data set (X, y) is obtained by running the three-phase power flow simulator for the hourly spaced load and active and reactive power injection measurements for 7 days i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' a total of n = 168 data points constitute a training data set for the design of the GPM and the RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Both the GPM and RPM are analyzed using the prediction results obtained for the n∗ = 24 test data points, which are the validation data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For the probabilistic analysis, 7000 samples are drawn using Latin hypercube sampling from input variables of load following Gaussian distribution IEEE TRANSACTIONS ON POWER SYSTEMS 7 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 1: Outliers corrupting the training data set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (a) QQ-plot of the measurements corrupted with 25% of outliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) plot of the weights using the PSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' the outlier magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' TABLE II: The RMSE and MAE for the Bus 19 of the IEEE 33-bus system Quadratic Basis Linear Basis With 25% outliers Without outliers With 25% outliers Without outliers RPM GPM RPM GPM RPM GPM RPM GPM Measure V A V A V A V A V A V A V A V A RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0034 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='7810e−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='01264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='7995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='01005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='01747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='00838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='01645 MAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2815e−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0672 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5516e−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0047 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='7962e−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='00058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='00176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='00042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0018 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2: RPM results for the voltage phase angle at Bus 19: (a) prediction at the test points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) probability density at the test points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' and (c) fitted values over the training data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Pt∗ i ,L ∼ N(Pt∗ i ,L, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='05Pt∗ i ,L), i = 169, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' , 192, the WGs’ output following the Weibull distribution, and the PVs’ output following the Beta distribution with the shape and scale parameters same as mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Voltage phasors predictions at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 for a day ahead forecast constitute as a benchmark for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We display the mean of the prediction results obtained from the GPM and the RPM of the voltage magnitude at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 using linear and quadratic basis functions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 9 (b) and (d), respectively, where the error bars represent the standard deviations as showing their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3: Comparison between the performance of the RPM and the GPM: (a) voltage magnitude at Bus 19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) RMSE values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' probability distributions individually will be inconvenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Similarly, the prediction results of the voltage angle at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 10 (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We now add up to 25% of outliers in the power injection measurements and voltage magnitudes and phase angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=', to the first 42 input and output data points in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' To best demonstrate the robustness of the RPM, we choose the outlier distribution to be Student’s t with 10 degrees of freedom because of heavier tails as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Similarly, the same percentage of outliers is included in the measurements of active and reactive load power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We display in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 7 (b) and (d) the probability density function of the voltage magnitude at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 for the test input at instance t∗ 169 obtained using linear and quadratic basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Sim- ilarly, the probability density results of the voltage angle at 10000 P S G13 e 8000 Quantil p G14 p 6000 G24 P leasurement G26 4000 2000 +++ 0 2000 2 3 1 0 1 2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='8 ghts 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='6 Wei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 P (kw) G130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='03 --MC IPrediction (RPM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='02 150 160 170 180 190 200 210 Bus Numbel1000 --- MC Output RPM Output 800 Probability Density 600 400 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='028 Angle (rad)2 --- Training Data Data fit (RPM) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 Angle (rad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 0 0 50 100 150 Time (s)10 I MC IPrediction (RPM) 5 Voltage (pu) IPrediction (GPM) 0 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='10 15 150 160 170 180 190 200 210 Time (s)15 -RPM ----GPM 10 RMSE 5 5 10 15 20 25 0 Percentage of OutliersIEEE TRANSACTIONS ON POWER SYSTEMS 8 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 4: RPM predictions for the IEEE 33-bus system with the training data set added with 25% of outliers: (a) voltage magnitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) voltage phase angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 5: QQ plot of 25% outliers added in (a) power injection measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) voltage magnitude measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 7 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We compare the prediction results of the voltage magnitude at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 obtained from the GPM and the RPM with basis function as linear and quadratic in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 9 (a) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The voltage angle results are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 10 (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The RMSE and MAE values of the prediction results for the cases of addition of outliers in the training data set and without the addition of outliers, both with the linear and basis function, are listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We observe that the RMSE and the MAE values obtained from the GPM and the RPM for the voltage magnitudes’ predictions are lesser with linear basis functions than the ones with the quadratic basis functions for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' As for the voltage angles, both models’ performance is better with the quadratic basis function for the case with outliers in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The linear basis function yields better performance for the case without outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Therefore, we choose the linear basis to plot the voltage magnitudes and the quadratic basis function for the voltage angles to obtain further predictions at all the buses in the system for the case with outliers in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The mean values (indicated as dots) and standard deviations (indicated as error bars) of both the models’ prediction results for the voltage magnitudes (see, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 12) and phase angles (see, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 13) at the buses of the 240−bus system are compared with the MC simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We observe that the results obtained from the comparable GPM deviate significantly from the true values in both the mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The performance of the RPM is comparably accurate on account of the trade-off between the accuracy and robustness of the SHGM estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Conventional GPM is strongly biased towards outliers, thus the prediction results deviate further away from the MC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The bias is particularly significant in phase C results because of the high variance of voltage phasors due to the large power flow in lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' For a large magnitude of outliers, the resulting bias is the worst-case scenario that can be imposed on power system measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The proposed RPM keeps this bias finite as long as the added outliers are added without exceeding the breakdown point, whereas the bias is unbounded for the results of conventional GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The RMSE of the predicted values for the voltage magnitude and the angle at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 with an increasing percentage of outliers added in the training data are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='11 (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We observe that the RMSE values obtained from the GPM are higher than the ones obtained from the RPM which provides consistently low RMSE results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Remark 1: Note that, because of the lack of availability of the real measurements of the voltage phasors (output variables y), they are obtained by running the power flow simulator us- ing real measurements of active and reactive power injections (input variables X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this paper, we propose a robust process model to perform stochastic power flow calculations using time-series measure- ments of power injections and voltage phasors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' The proposed model captures the natural stochastic dynamics introduced in the power grid by the RES and DGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' This is accomplished by training the RPM on recorded time series data set containing the measurements of active and reactive power injections from the RES and DGs and nodal voltage phasors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We demonstrate the RPM on the standard IEEE 33-bus system and a real-world 240-bus system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We show that the proposed methodology can handle 25% of outliers i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' bad leverage points and vertical outliers in the training data set using the root mean square errors and maximum absolute errors of the predicted values for voltage phasors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' In future work, we will focus on extending the applications of the RPM for optimal power flow calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' We will also investigate the performance of other robust estimators with high breakdown points for handling outliers in the training data of more than 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Quantiles of Power Injection Sample ×104 4 2 0 2 4 6 2 0 2 Standard Normal Quantiles3 2 0 2 3 2 0 1 2 Standard Normal Quantiles1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='05 Voltage (pu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='95 I·MC IPrediction (RPM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='8 0 5 10 15 20 25 30 35 Bus Number2 --MC IPrediction (RPM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 0 0 5 10 15 20 25 30 35 Bus NumberIEEE TRANSACTIONS ON POWER SYSTEMS 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 6: The online diagram of the 240 bus system integrated with RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Blue and red squares indicate the PVs and WGs, respectively TABLE III: The RMSE and MAE for the Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 of the 240-bus system Quadratic Basis Linear Basis With 25% outliers Without outliers With 25% outliers Without outliers RPM GPM RPM GPM RPM GPM RPM GPM Measure V A V A V A V A V A V A V A V A RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4222 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='8734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4242 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1888 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 820–821, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Xin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Wu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Ju, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Jiang, “Data-driven 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Zhao, “Probabilistic power flow calculation and variance analysis based on hierarchical adaptive polynomial chaos-anova method,” IEEE Transactions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3316– 3325, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Laowanitwattana and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Uatrongjit, “Probabilistic power flow analysis based on arbitrary polynomial chaos expansion for networks with uncer- tain renewable sources,” IEEJ transactions on Electrical and Electronic Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 1754–1759, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Source THO ESOE 3056 8 30578305883059B3060830618 3007 3039 3055 Mswx 1009 1008 4041 1006 1005 1004 1002 1082 63H1 LA 2057 O O 7 3012A 03021c 3029A 900m 3038 CB_102 3054 32# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3004 3062B 3063 B 3064B3065B3066B3067B 208 11 OHS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='CB_101[ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3072 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='VE60EO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2012 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='UG2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3101c 3102c3103c NB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='202 2021 2026H12027 031318 OH2032 OHI 3091A 3090 A 3089A 3088A 3087A 3086A 338 3084A 3083A 092UG2 3005 OO O izen ■ UG2i CB_204。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2013 642!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 2USRi CB_203 3098 : 31306 2028 A 3107 ±28, 31228 2022c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' CAP_201 ) 2038 UG21 izon 31296 2034 328 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='.-276 ,3099 8 291 2029 2046 B 3121: 2G2: 3128 B 20168 2023 c CB_303 2030 3120 8 232i 142 : 2048B UG2 31178 31168 31278 2042 2017b 2024c usi 2031A 31148 UG2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3118 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='- 49 332 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' UG2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 3119 31238 31248 31258 31268 139h 2043 02018 8 02025 c 2044!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 31158 ■ 3132A 2052 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Transformer and Load 262 3136A Capacitor Bank 131403141c 3133 3155c 3154c3153c 3152c 3151c3150c 3149c 3148c 3142c 3143 3-phaseUG 3137 A Circuit Breaker 352;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' UG2 A 3138A 209R to CB_102 3162c3161c 3160c 3159c 3158c 3157c 31563144c 3145c3146c3147c UG2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' Normally Open 3139 A N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IEEE TRANSACTIONS ON POWER SYSTEMS 10 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 7: Comparison between the GPM and the RPM probabil- ity density results for the voltage magnitude of 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distribution test system based on real utility data,” in 2019 North American Power Symposium (NAPS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 1–6, IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 70 ---- MC - RPM 60 Probability Density 50 40 30 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='6 Voltage (pu)70 ---- MC H - RPM 60 Probability Density 50 40 30 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 Voltage (pu)70 ----MC - RPM 60 Probability Density 50 40 30 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3 0.' metadata={'source': 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+page_content='3 ---- MC - RPM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='25 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='15 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='05 0 90 85 80 75 70 Angle (rad)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3 ----MC - RPM 0.' metadata={'source': 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2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 in the 240−bus network when (a) the training data set is added with 25% of outliers ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) training data set is not added with outliers for linear basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (c) the training data set is added with 25% of outliers ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (d) training data set is not added with outliers for quadratic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 10: Comparison between the GPM and the RPM forecast results for the voltage angle of Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 in the 240−bus network when (a) the training data set is added with 25% of outliers ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) training data set is not added with outliers for linear basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (c) the training data set is added with 25% of outliers ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (d) training data set is not added with outliers for quadratic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 10 (nd) 5 /oltage ( 0 5 ---MC IPredicted Values (RPM) 10 IPredicted Values (GPM) 15 170 172 174 176 178 180 182 184 186 188 190 192 Time (h)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='6 (nd) Itage Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 I--MC IPredicted Values (RPM) Predicted Values (GPM) 0 170 172 174 176 178 180 182 184 186 188 190 192 Time (h)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 }--MC IPredicted Values (RPM) IPredicted Values (GPM) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Time (h)200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='(rad) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='}--MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Ang ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IPredicted Values (RPM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IPredicted Values (GPM) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='184 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='186 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='188 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='190 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Time (h)0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='{--MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Angle (rad) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IPredicted Values (RPM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IPredicted Values (GPM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='172 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='176 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='178 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='184 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='186 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='188 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='190 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Time (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='20 Angle (rad) 40 60 {--MC 80 IPredicted Values (RPM) IPredicted Values (GPM) 100 170 172 174 176 178 180 182 184 186 188 190 192 Time (h)-20 }-MC Angle (rad) 40 IPredicted Values (RPM) IPredicted Values (GPM) 60 80 100 170 172 174 176 178 180 182 184 186 188 190 192 Time (h)IEEE TRANSACTIONS ON POWER SYSTEMS 12 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 11: RMSE vs the percentage of outliers added in training data for the prediction results at Bus 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 (a) voltage magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) voltage angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 12: The obtained prediction results of voltage magnitude from the RPM compared with those obtained from the GPM of 240−bus system with 25% of outliers added in training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (a) The results obtained from the RPM for phase a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) comparison between the GPM and the RPM for phase a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (c) The results obtained from the RPM for phase b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (d) comparison between the GPM and the RPM for phase b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (e) The results obtained from the RPM for phase c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (f) comparison between the GPM and the RPM for phase c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 8 RPM ----GPM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' RMSE 4 2 5 10 15 20 25 Fraction of Outliers14 -RPM ---- GPM 12 10 RMSE 8 6 2 0 5 10 15 20 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 }-MC (Phase A) IPredicted Values (Phase A) (RPM) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='8 Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 Bus Number3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 }-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' True Data (Phase A) IPredicted Values (Phase A) (RPM) 3 IPredicted Values (Phase A) (GPM) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 Voltage (pu) 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 Bus number1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 {- MC (Phase B) IPredicted Values (Phase B) (RPM) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 Voltage (pu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='2 0 Bus Number2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 Voltage (pu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 amm H i 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 I-- MC (Phase B) IPredicted Values (Phase B) (RPM) IPredicted Values (Phase B) (GPM) Bus number1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 {-MC (Phase C) Predicted Values (Phase C) (RPM) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='3 (nd) Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='8 & Bus Number-{-MC (Phase C) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 IPredicted Values (Phase C) (RPM) ↑Predicted Values (Phase C) (GPM) 06006000000 (nd) Voltage ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='5 0 690 5 9 9 30 3107 Bus NumberIEEE TRANSACTIONS ON POWER SYSTEMS 13 (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' 13: The obtained prediction results of voltage angle from the RPM compared with those obtained from the conventional GPM of 240−bus system with 25% of outliers added in training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (a) The results obtained from the RPM for phase a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (b) comparison between the GPM and the RPM for phase a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (c) The results obtained from the RPM for phase b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (d) comparison between the GPM and the RPM for phase b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (e) The results obtained from the RPM for phase c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' (f) comparison between the GPM and the RPM for phase c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1 MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IPredicted Values (Phase A) (RPM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Bus Number200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='1 MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='IPredicted Values (Phase A) (RPM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='Predicted Values (Phase A) (GPM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfxwK1/content/2301.02651v1.pdf'} 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Adaptive Tensor Parallelism for Foundation Models +Shenggan Cheng +School of Computing +National University of Singapore +Ziming Liu +School of Computing +National University of Singapore +Jiangsu Du +School of Computer Science and Engineering +Sun Yat-sen University +Yang You +School of Computing +National University of Singapore +Abstract +Foundation models have impressive performance and gener- +alization capabilities across a wide range of applications. The +increasing size of the models introduces great challenges for +the training. Tensor parallelism is a critical technique that is +currently used in almost all foundation model training and has +a significant impact on overall training performance. However, +current tensor parallelism in machine learning frameworks +misses optimization opportunities in fitting various intercon- +nection topologies. In this work, we present ATP, an adaptive +tensor parallelism framework for foundation models, which +can automatically select the optimal parallel strategy on dif- +ferent interconnections. We propose column- and row-first +tensor parallelism based on 2D device meshes and construct a +search space. Combined with the hierarchical communication +matrix, ATP can identify the optimal strategy in the search +space. We also propose chunk-based overlapping to reduce +communication overhead. Our evaluations show ATP consis- +tently outperforms the state-of-the-art approaches for various +model sizes and interconnects, achieving end-to-end train- +ing performance improvements of up to 37-64% on specific +interconnects. Based on our theoretical model, the communi- +cation overhead of ATP decreases with scaling, indicating a +qualitative leap forward. +1 +Introduction +The term foundation models [2] refers to pretrained models +with large-scale parameters that can be adapted to a wide +range of downstream tasks. These models have impressive +performance with generalization capabilities and have led to +state-of-the-art results on benchmarks of natural language +processing (BERT [5], GPT-3 [3]) and computer vision (CLIP +[20], Florence [35]). The most obvious trend in recent years +is the rapid increase in the size of foundation models: from +100 billion [3,31,36] to over 500 billion [4,26]. +Training foundation models, such as GPT-3 (which has +175 billion parameters), can be challenging due to the huge +computational costs and memory capacity bottlenecks. For +example, it would take about 288 years to train GPT-3 using +a single NVIDIA V100, and the model’s parameters would +not fit in the main memory of even the most advanced GPUs +(such as NVIDIA 80 GB A100). To address these issues, +various parallelism techniques have been proposed, including +tensor (intra-layer) parallelism [15,25], pipeline (inter-layer) +parallelism [7, 14] and ZeRO redundant optimizer [21, 22]. +These approaches aim to efficiently parallelize computation +and distribute parameters among multiple devices. +Among these, tensor parallelism is the most important train- +ing technique. This approach evenly distributes both compu- +tations and parameters across multiple devices. In the training +of foundation models such as GPT-3 [3], OPT [36], Yuan [31], +Megatron-Turing [26], tensor parallelism and pipeline paral- +lelism are often used in combination. Tensor parallelism splits +the matrix multiplications across different devices and is typi- +cally used at the intra-node level, while pipeline parallelism +is used at the inter-node level. Empirical results have shown +that tensor parallelism should be used as much as possible +without introducing cross-node communication, meaning that +tensor parallelism should generally be used up to degree-N +when using N-GPU servers [15]. In automatic parallelism +approaches, tensor parallelism is a key design factor for the +search space. For example, in Alpa [37], the search space +includes both inter-operator and intra-operator parallelism, +where intra-operator parallelism represents tensor parallelism. +Tensor parallelism is not only used for training, but also for +inference to reduce latency and provide additional memory +capacity across GPUs to fit parameters [18]. +However, existing approaches has several drawbacks. 1) +they only consider simple interconnect topologies, such as +NVSwitch [16], and cannot adapt to more complicated topolo- +gies. For example, when performing tensor parallelism on +multi-node systems, the all-reduce operation is limited by the +slowest link in the topology, preventing it from exploiting the +high-bandwidth NVLink within the server. 2) tensor paral- +lelism relies heavily on high-bandwidth interconnects, such +as NVLink. On servers that are not equipped with NVLink, +1 +arXiv:2301.08658v1 [cs.DC] 20 Jan 2023 + +the communication costs introduced by tensor parallelism +can become a significant bottleneck for training. 3) unlike +data parallelism, the communication introduced by tensor +parallelism is synchronous, which makes it more difficult to +overlap with computations, leading to more overhead. As a +result, tensor parallelism is not as scalable as data parallelism. +To address these limitations, we present Adaptive Tensor +Parallelism (ATP) for foundation models. Unlike existing ten- +sor parallelism approaches that have a fixed communication +pattern, ATP can select different communication patterns on +different topological interconnections. We propose column- +and row-first tensor parallelism based on 2D device meshes. +Different 2D device meshes form the search space, and we +use the hierarchical communication matrix to estimate the +communication cost of different strategies. This allows ATP +to identify the best strategy in the search space, making it a +topo-aware approach. In addition, we use chunk-based over- +lapping to reduce communication overhead. ATP not only +outperforms state-of-the-art approaches but also has a simple +API for users. +In summary, we make the following contributions: +• We propose column- and row-first tensor parallelism +based on two-dimensional device meshes and construct +a novel search space for tensor parallelism. +• We design hierarchical communication matrix to de- +scribe the communication characteristics of complicated +interconnects and identify the optimal strategy in the +search space. +• We implement ATP with some communication optimiza- +tions, including chunk-based overlapping. +• ATP achieves up to 37-64% improvement over the state- +of-the-art approaches in specific interconnects. In the- +oretical analysis, the communication cost of ATP de- +creases with scaling on some topologies, representing a +significant improvement. +2 +Understanding Tensor Parallelism +2.1 +Tensor Parallelism on Transformer +Transformer has become the bedrock of foundation models +due to their excellent ability to model sequences. Taking a +language model as an example, the input tokens are first fed +into an embedding layer and then passed through a single- +stack transformer encoder or decoder with the L layers (see +Figure 1). The input of transformer layer is a 3D tensor of +size [b,s,h] where b,s,h are batch, sequence and hidden-size +dimensions. Each transformer layer consists of a multi-head +attention (MHA) block followed by a feed-forward block. In +the MHA block, the sequence data is fed into three different +MLP layers to obtain Query(Q), Key(K), Value(V). Then di- +vided into a heads, each with a hidden size of d = h/a. For +each head, the attention score is calculated using Formula 1. +Self Attention +Linear +Dropout +Transformer Layer +x L +LayerNorm +Self Attention +Linear +Dropout +GPU 1 +GPU 2 +f +LayerNorm +Linear +GeLU +Linear +Dropout +LayerNorm +Linear +GeLU +Linear +Dropout +f +f +f +LayerNorm +Figure 1: Transformer layer with tensor parallelism on two +GPUs. f and f are conjugate communication operators. In +the forward pass, f represents an all-reduce operation, while +in the backward pass f represents an all-reduce operation. +The feed-forward block has two layers of multi-layer percep- +tron (MLP). The first layer increases the hidden size to 4h +and the second layer reduces it back to h. +Att(Q,K,V) = softmax(QKT +√ +d +)V +Q,K,V ∈ Rs×d +(1) +Tensor Parallelism, which is commonly used for training +foundation models, was proposed by Megatron-LM [15]. As +shown in Figure 1, tensor parallelism parallelizes the MHA +and feed-forward blocks. In the feed-forward block, there are +two MLP layers and an activation function (GeLU): +Y = GeLU(XA), +Z = YB +where A and B are the weight of two MLP layers. We +can split A by columns and B by rows and parallelize the +computation into two parts: +A = [A1,A2], +B = +� +B1 +B2 +� +[Y1,Y2] = [GeLU(XA1),GeLU(XA2)] +Z = reduce(Y1B1,Y2,B2) +MHA blocks can be parallelized in a similar way by divid- +ing the weight matrix of Q,K,V by columns and the output +linear layer by rows. +This approach introduces two all-reduce operations in both +the forward and backward passes of each layer, distributing +computation and memory footprints across devices. Inside the +MHA and feed-forward blocks, tensor parallelism parallelizes +model parameters, optimizer state, and activations. The Lay- +erNorm, including the input/output activations, is duplicated +in each tensor-parallel worker. +Several recent works, such as 2D tensor parallelism [32] +use the Scalable Universal Matrix Multiplication Algorithm +2 + +(SUMMA) [29]. We regard 2.5D tensor parallelism as an +extension of 2D tensor parallelism since it can be seen as a +combination of 2D Tensor Parallelism and Data Parallelism. +2D Tensor Parallelism has two main issues: 1) broadcast of +the weight matrix is expensive because the size of the weights +is much larger than the activation of the giant model, and 2) +multiple broadcasts in a single layer result in high overhead +and low bandwidth utilization. +2.2 +Modern Accelerator Interconnect +Due to the high demand on communication in distributed train- +ing, a number of accelerator-specific interconnect hardware +has been developed and more complicated topologies have +been introduced. Multi-GPU servers dominate the training +hardware, so the interconnect architecture of GPUs is gener- +ally divided into two hierarchical levels, intra-node level and +inter-node level. +Intra-node Level. At this hierarchy, training scales within +a single node, where the interconnections between multiple +GPUs are PCIe (64 GB/s for PCIe 4.0) or NVLink (600 GB/s +for NVLink-v3). Figure 2 shows the examples of these archi- +tectures. The PCIe network forms a balanced tree structure, +where each GPU is connected to a PCIe switch, which is fur- +ther connected to a CPU socket. The sockets are then bridged +by GMI Link for AMD EPYC or QPI [38] for Intel XEON. +NVLink is a GPU-oriented interconnect technology proposed +by NVIDIA with various configuration options. For exam- +ple, in Figure 2(a), all GPUs are connected to the NVSwitch +via NVLink, improving all-to-all communication capabilities. +In Figure 2(b), the eight GPUs are divided into four groups, +with two GPUs in each group connected via NVLink. In this +case, the GPU interconnect has a clear NUMA effect (closer +proximity leads to stronger communication capabilities). +Inter-node Level. In data centers, the main interconnec- +tion technologies across nodes are Ethernet and InfiniBand [8]. +Cloud servers or supercomputers used for training are typi- +cally have at least 50 Gbps or 200 Gbps of communication +bandwidth. There are many different configurations for topolo- +gies at this hierarchy, as shown in Figure 3, including fat tree, +torus, and dragonfly. There are also many accelerator-specific +direct interconnects at this level, such as the 2D Torus Inter- +Core Interconnect (ICI) links in TPU pods that directly con- +nects up to 1024 TPUv3 cores [12] and NVIDIA’s upcoming +Nvlink-Network Switch [9], which can also connect hundreds +of GPUs directly via high-bandwidth NVLink [9]. The de- +velopment of new interconnected hardware and topologies +presents opportunities and challenges for software-hardware +co-design [6]. +2.3 +Communication Analysis +Communication Pattern. Similar to data parallelism, tensor- +parallel communication requires all-reduce operations. How- +CPU 0 +CPU 1 +GMI Link +PCIe +Switch +PCIe +Switch +PCIe +Switch +PCIe +Switch +GPU +GPU +GPU +GPU +GPU +GPU +GPU +GPU +NVSwitch +PCIe +NVLink +(a) Server with NVSwitch +CPU 0 +CPU 1 +GMI Link +PCIe +Switch +PCIe +Switch +PCIe +Switch +PCIe +Switch +GPU +GPU +GPU +GPU +GPU +GPU +GPU +GPU +PCIe +NVLink +(b) Server with NVLink +Figure 2: PCIe and NVLink topology for intra-node intercon- +nect. (a) and (b) show the multi-gpu server with full intercon- +nection and dual card interconnection. +(a) Fat Tree +(b) Torus +(c) Dragonfly +Figure 3: Inter-node interconnect topology. +ever, because the computations are interdependent with +the communication, the all-reduce here must be syn- +chronous. Tensor-parallel communication relies more on high- +bandwidth communication because it cannot overlap with +backward computation as data-parallel communication does. +Communication Cost. As mentioned in Section 2.1, the +size of the tensor for each all-reduce is [b,s,h], where b,s,h +are batch, sequence and hidden-size dimensions. In a model +with L layers, a total of 4L all-reduce operations are re- +quired in the forward and backward passes. Assuming an +all-reduce bandwidth of B, the communication cost for one +step is 4Lbsq/B. In training foundation models, synchronous +all-reduce of dozens of gigabytes of large tensors is required +in one step, posing a high demand on interconnect bandwidth. +All-reduce Performance. Various vendors provide collec- +tive communications library (CCL) to deliver intra-node and +inter-node level communication capabilities for deep learn- +ing frameworks, such as NCCL on the NVIDIA platform. +These high-performance all-reduce implementations are pri- +3 + +marily based on the ring algorithm [24]. The performance of +ring all-reduce depends on the bandwidth of the slowest link +on the ring, which may result in some local high-bandwidth +interconnects being wasted. For example, in cross-server all- +reduce communication, the all-reduce performance is limited +by the bandwidth of the cross-server interconnect, and the +high bandwidth of the NVLink inside the server is wasted. +3 +Adaptive Tensor Parallelism +3.1 +Sharding Notion +Many research and machine learning frameworks that use the +concept of sharding to describe how tensors are distributed +across multiple devices in a parallel strategy, particularly in +the context of auto-parallelism. In this paper, we will be focus- +ing on the sharding notion from PyTorch Distributed Tensor. +Device Mesh. In the training of foundation models, it is of- +ten necessary to use a combination of different parallelization +strategies, which involves communication between certain +ranks. To manage these communication groups, we group the +devices into multidimensional device meshes. A device mesh +with N devices can be expressed as DeviceMesh(d1,d2,...dn). +A device mesh can be thought of as having n levels, with the +devices in the current group being divided into di sub-groups +at the i-th level. The total number of devices in the mesh is +equal to N = d1 × d2 × .. × dn. For example, a device mesh +with four devices can be represented in four different ways: +1D: [4], 2D: [1,4], [2,2], [4,1]. DeviceMesh(d1,4/d1) means +that the four devices are divided into d1 groups, with each +group having 4/d1 devices for sharding. Figure 4 shows a +device mesh with d1 = 2. +Sharding Spec. We use the sharding spec to describe the +strategy for distributing the global tensor across the devices +in a device mesh. Like Alpa [37] and OneFlow [34], we +have chosen to use three types of placement strategies: Shard, +Replicate, and Partial: +• Shard(d): split the tensor along the d-th dimension +across devices. +• Replicate: replicate the tensor across devices. +• Partial(op): partial the tensor across devices, requiring +all-reduce communication to obtain the global tensor. +Op represents the ReduceOp, such as SUM or MAX. +From the perspective of a device mesh, the sharding of a +tensor can be thought of as the selection of a placement strat- +egy at each level of the device mesh. Therefore, the sharding +spec can be viewed as a sequence of placement strategies, the +length of which is equal to the number of dimensions of the +device mesh. For an N-dimensional device mesh, the shard- +ing spec of the global tensor can be defined as [P1,P2,...PN], +where Pi ∈ {Shard(d),Replicate,Partial(op)} represents the +placement strategy for the i-th dimension of the device mesh. +Figure 4 illustrates a valid sharding spec for a 2D tensor on a +2D device mesh. In the sharding spec [Shard(1),Shard(0)], +Shard(1) means that the tensor is column-wise partitioned +into two parts for rank-[0,1] and rank-[2,3], and then Shard(0) +means that each of these partitioned tensors is further row- +wise partitioned for each pair of ranks. If the sharding spec is +[Replicate,Shard(0)], the sharded tensor for rank-0 and rank- +2 will be [1,2,3,4] and rank-1 and rank-3 will be [5,6,7,8]. +Unlike the sharding spec in Alpa, which binds the placement +strategy to the tensor dimensions, we have chosen to bind the +strategy to the dimensions of the device mesh. This approach +is more concise and easier to understand. +DeviceMesh(2,2) +Sharding Spec: +[Shard(1), Shard(0)] +1 +2 +3 +4 +5 +6 +7 +8 +Rank 0 +Rank 1 +1 +2 +5 +6 +Rank 2 +Rank 3 +3 +4 +7 +8 +Column-wise +Shading +Row-wise +Sharding +Row-wise +Sharding +Global Tensor +Rank 0 +Rank 1 +Rank 2 +Rank 3 +Figure 4: Sharding a 2D tensor on a 2D device mesh. +The parallelism strategy can be represented as a sharding +spec on a 1D device mesh. Table 3.1 illustrates the sharding +specs of the input, weight, and output in data parallelism (DP) +and tensor parallelism in Megatron-LM (Column TP and Row +TP). For the combination of DP and TP, a 2D device mesh is +typically used, with the first dimension of the sharding spec +corresponding to DP (as shown in the first row of the table) +and the second dimension corresponding to TP (as shown in +the second or third row of the table). +Table 1: Sharding specs for MLP layer on 1D device mesh. +Parallelism +Input +Weight +Output +DP +[Split(0)] +[Replicate] +[Split(0)] +Column TP +[Replicate] +[Split(1)] +[Split(1)] +Row TP +[Split(1)] +[Split(0)] +[Partial(SUM)] +3.2 +Search Space for Tensor Parallelism +With the definition of device mesh and sharding spec, we +can express more complex parallelization strategies. In this +section, we will introduce two types of tensor parallelism +algorithms on 2D device meshes, and demonstrate how they +can be applied to transformer models. Finally, we will discuss +the search space for tensor parallelism. +In Megatron-LM’s tensor parallelism, the weight matrix +(W) is split row-wise or column-wise on a 1D device mesh. +When using a 2D device mesh, W is split at two levels. +4 + +1 +5 +1 +2 +4 +3 +1 +2 +1 +1 +2 +1 +2 +Rank 0 +X: +X += +4 +5 +1 +20 +4 +Rank 2 +X += +Rank 1 +Rank 3 +21 +6 +17 +7 +21 +6 +AllReduce #1 +X: [Shard(1), Replicate] +W: [Shard(0), Shard(1)] +Y: [Partial(SUM), Shard(1)] +W: +Y: +AllReduce #2 +Row-First Tensor Parallelism +Column-First Tensor Parallelism +X: [Replicate, Shard(1)] +W: [Shard(1), Shard(0)] +Y: [Shard(1), Partial(SUM)] +AllReduce #1 +AllReduce #2 +3 +1 +1 +2 +2 +3 +2 +4 +15 +3 +17 +7 +3 +1 +2 +3 +6 +X += +1 +1 +2 +1 +5 +1 +5 +1 +X += +2 +10 +2 +8 +7 +11 +4 +8 +7 +11 +4 +1 +1 +2 +1 +2 +Rank 0 +X += +Rank 2 +Rank 1 +Rank 3 +2 +2 +4 +1 +5 +1 +5 +1 +X += +2 +10 +2 +3 +1 +2 +3 +6 +X += +1 +1 +2 +4 +5 +1 +20 +4 +X += +3 +15 +3 +AllReduce #1 +AllReduce #1 +21 +6 +17 +7 +8 +7 +11 +4 +Figure 5: Row-First and Column-First Tensor Parallelism for Y = XW on DeviceMesh(2,2). The green matrix represents the +global tensor and the purple matrix represents sharded tensor. These two approaches have different priorities in the sharding of +the weight matrix (W). A grouped all-reduce communication is required to obtain the global tensor of Y. +As shown in Figure 5, there are two sharding specs for W, +[Shard(0),Shard(1)] and [Shard(1),Shard(0)]. In the first +spec, the matrix is split row-wise at the first level of the de- +vice mesh, and then column-wise at the second level. The +other spec splits column-wise at the first level and then row- +wise at the second level. +Then we consider how to perform the distributed GEMM +Y = XW on DeviceMesh(d1,d2). Based on the two sharding +specs for the weight matrix, two tensor parallelism approaches +can be proposed: row-first tensor parallelism and column-first +tensor parallelism. As shown in the left half of Figure 5, in +row-first tensor parallelism, the sharding specs of W and X +are [Shard(0),Shard(1)] and [Shard(1),Replicate]. The lo- +cal GEMM operation at each rank results in a sharded Y with +the sharding spec [Partial(SUM),Shard(1)]. To obtain the +global tensor result of Y, it is necessary to perform an all- +reduce operation on the first dimension of the device mesh +and then an all-gather operation on the second dimension. +Column-first tensor parallelism uses a different sharding strat- +egy for the weight matrix, and the dimensions of the device +mesh are transposed in the communication, resulting in a dif- +ferent communication pattern compared to row-first tensor +parallelism. +For row-first tensor parallelism, when the size of the global +tensor X is [b,h1], the size of W is [h1,h2], and the size of Y is +[b,h2], the sharded X has a size of [b,h1/d1], the sharded W +has a size of [h1/d1,h2/d2], and the sharded Y has a size of +[b,h2/d2]. The GEMM on global tensors requires b×h1 ×h2 +fused multiply-add (FMA) operations, while each rank re- +quires b×h1 ×h2/(d1 ×d2) FMA operations. The computa- +tions are evenly distributed across the ranks, requiring commu- +nication on a tensor of size [b,h2/d2]. In contrast, for column- +first tensor parallelism, the size of the tensor for communica- +tion is [b,h2/d1]. +3.2.1 +Transformer +As introduced in Section 2.1, the transformer layer consists of +an attention block followed by a feed-forward block. We will +describe how to use row- and column-first tensor parallelism +in these two blocks. +Feed-forward. There are two MLP layers and an acti- +vation function in feed-forward (Y = GeLU(XA),Z = YB). +Megatron-LM uses column parallelism for the first MLP layer +and row parallelism for the second. Similarly, in our approach, +we use column-first tensor parallelism for the first MLP layer +and row-first for the second, as shown in Figure 6(b). The +sharded feed-forward block can be expressed as follows: +Yshard = f3(XshardAshard) +Zshard = f4(GeLU(Yshard)Bshard) +The sharding specs of weight matrix Ashard and Bshard are +[Shard(1),Shard(0)] and [Shard(0),Shard(1)], respectively. +The input(Xshard) and output tensor(Zshard) have the same +sharding spec: [Replicate,Shard(1)]. f3 and f4 are two con- +jugate communication operations: f3 performs an all-reduce +on the first dimension of the device mesh in the forward pass +and on the second dimension in the backward pass, while f4 +is the reverse. +Attention block. In this block, we use a similar strategy to +the feed-forward block, using column-first tensor parallelism +in the QKV Linear and row-first in the Output Linear, as +shown in Figure 6(a). The main difference is that we want +the computation of the Attention Core to be fully sharded, +so the sharding spec of the input tensor should not include +Replicate. To achieve this, we scatter the input tensor before +the Attention Core, and gather the output tensor before the +Output Linear. f1 and f2 are two conjugate communication +operations, similar to f3 and f4 in the feed-forward block. +5 + +f2 +f1 +QKV +Linear +h/d2 +3h/d1 +Linear + h/d2 +Attention +Core +Column-First Tensor Parallelism +Row-First Tensor Parallelism +(a) Attention block +GeLU +f4 +f3 +MLP 1 +h/d2 +4h/d1 +MLP 2 + h/d2 +(b) Feed-forward +Figure 6: Parallel transformer layer with row- and column-first +tensor parallelism. f1,3 and f2,4 are conjugate operations: f1,3 +performs an all-reduce on the first dimension of the device +mesh in the forward pass and on the second dimension in the +backward pass, while f2,4 is the reverse. +f1 has a smaller amount of communication compared to f3 +because the output hidden size of the QKV Linear is smaller. +Different device meshes correspond to different tensor sizes +and communication groups for row- and column-first tensor +parallelism, which can result in different optimal tensor par- +allelism strategies for different interconnect topologies. For +example, if we have 2n devices, then we will have n+1 kinds +of 2D device meshes. This forms a search space in which we +need to find the device mesh with the optimal performance +for tensor parallelism in a specific topology. +Under the DeviceMesh(N,1) device mesh, our approach is +similar to the tensor parallelism in Megatron-LM [15]. Google +proposed a novel tensor parallelism algorithm [18] to reduce +the inference latency of the giant transformer model on TPU +clusters. On q2 TPUs with a 2D torus topology, Google’s +algorithm is similar to our proposed tensor parallelism on +DeviceMesh(q/2,2q). These tensor parallelism methods are +all contained in our search space, which demonstrates the +potency of our search space and guarantees that the final +performance will not be worse than these traditional tensor +parallelism methods. +3.3 +Theoretical Analysis +To analyze the communication cost theoretically, we assume +that B1 is the all-reduce bandwidth on the first dimension +of the device mesh and B2 is the bandwidth on the second +dimension. For an L-layer transformer model with an input +size of [b,s,h], where b, s, and h are the batch, sequence, and +hidden size dimensions, the communication time of the all- +reduce introduced by tensor parallelism is: +Tcomm = 2Lbs×(Tf1 +Tf2 +Tf3 +Tf4) += 2Lbs×( 3h +d1B2 ++ +h +d2B1 ++ 4h +d1B2 ++ +h +d2B1 +) += 2Lbs×( 7h +d1B2 ++ 2h +d2B1 +) +(2) +It is worth noting that as the number of tensor-parallel +workers (d1 ×d2) increases, the communication time (Tcomm) +decreases. This communication characteristic helps to extend +tensor parallelism to a larger scale. +From the perspective of the communication group, +Megatron-LM requires the participation of all tensor-parallel +workers. On a 2D device mesh, communication is grouped, +meaning that each communication requires only one row or +column of workers on the device mesh. This communica- +tion pattern has the potential to better exploit complicated +interconnect topologies. +3.4 +Hierarchical Communication Matrix +As previously mentioned, the goal is to find the optimal device +mesh that provides the best performance in a specific topology. +Currently, many performance analysis and automatic paral- +lelism frameworks for model training rely on communication +volume to measure communication cost. However, to more +effectively find the optimal device mesh, we utilize hierar- +chical communication matrix to account for communication +costs in complex topologies. +One of the main features of modern accelerator intercon- +nects is their hierarchical nature. Within nodes, multiple GPUs +are connected using PCIe or NVLink, and different nodes +are connected using InfiniBand or Ethernet. The traditional +method of describing the communication capabilities of a +topology is through a P2P communication matrix, which is a +N ×N matrix that represents the bandwidth of the interconnec- +tions between the N ranks. However, we have introduced the +concepts of hierarchy and group bandwidth into the communi- +cation matrix to more accurately represent the communication +capabilities of the topology. Figure 7 shows some examples +of hierarchical communication matrix. +The hierarchical communication matrix has multiple lay- +ers, each containing a P2P communication matrix and group +bandwidth. At higher layers, each rank may represent a group +of devices. For example, in Figure 7(a), there are two layers, +with each rank representing a node (consisting of 4 devices) +at the higher level and a single device at the lower level. The +P2P communication matrix is the aggregate bandwidth be- +tween the two ranks, which at a higher level is the two groups +of devices. The group bandwidth for each rank represents +the aggregate bandwidth of that group of devices to the out- +side world. In Figure 7(a), we can see that there are 4 GPUs +6 + +25 +25 +25 +25 +25 +25 +0 +1 +2 +3 +0 +1 +2 +3 +200 +200 +200 +200 +200 +200 +0 +1 +2 +3 +0 +1 +2 +3 +Group Bandwidth：600 GB/s +Group Bandwidth：25 GB/s +(a) Interconnect Topology 1 +100 +100 +100 +100 +100 +100 +100 +100 +0 +1 +2 +3 +0 +1 +2 +3 +25 +25 +25 +25 +25 +25 +25 +25 +0 +1 +2 +3 +0 +1 +2 +3 +Group Bandwidth：50 GB/s +Group Bandwidth：200 GB/s +(b) Interconnect Topology 2 +Figure 7: Examples of hierarchical communication matrix. +(a) is a cluster of four nodes connected via 200 Gbps HDR, +with four GPUs per node interconnected using NVLinkv3. (b) +is 4×4 devices with 2D Torus interconnect. +interconnected by 4 NVLinks, each providing 50 GB/s of +bandwidth. Therefore, the P2P communication bandwidth is +200 GB/s, while the group bandwidth for each GPU is 600 +GB/s. Figure 7(b) shows a 4x4 device mesh with a 2D Torus +interconnect. Assuming that the bandwidth of each link is 25 +GB/s, the second level of the matrix shows a torus ring in +which devices i and i + 1, i − 1 are connected. For the first +level, there are 4 links between rings (one for each device), +resulting in a P2P bandwidth of 100 GB/s. The links in both +directions on the ring make the group bandwidth twice as +large as P2P. +3.5 +Adaptive Tensor Parallelism +By combining the search space (described in Section 3.2) and +the hierarchical communication matrix (described in Section +3.4), we can propose Adaptive Tensor Parallelism (ATP) for +automatically selecting the optimal tensor parallelism strategy. +In the search space, we can estimate the communication time +for each device mesh and identify the optimal mesh with the +minimum communication time. +First, we estimate the bandwidth of all-reduce communica- +tion for each device mesh. As discussed in Section 3.3, on the +device mesh, all-reduce is required for one column or one row +of workers, whose bandwidths correspond to B1 and B2, re- +spectively. We use the hierarchical communication matrix to +estimate the bandwidth that can be provided by the intercon- +nect link under all-reduce, resulting in B +′ +1 and B +′ +2, respectively. +Assuming that the hierarchical communication matrix has l +layers with Ri ranks per layer, and denoting the group band- +width of layer j as GroupBWj, we can compute B′ +1 and B′ +2 +as follows: when considering DeviceMesh(d1,d2), the first +dimension involves layers 1 to i, and the second dimension is +i(+1) to l (i+1 when d1 ̸= ∏j:1→i Rj). The performance of +all-reduce depends on the bandwidth of the bottleneck, so B′ +1 +and B′ +2 can be calculated using the following formula: +B′ +1 = min(GroupBW1→i/d2) +B′ +2 = min(GroupBWi+(1)→l) +(3) +When performing all-reduce in the first dimension of the +device mesh, there are d2 groups of all-reduce, each with d1 +devices. Since d2 groups of all-reduce share the interconnec- +tion bandwidth, so we need to divide the bandwidth by d2 +when calculating B +′ +1. The P2P communication matrix can be +used to correct the group bandwidth when the all-reduce al- +gorithm cannot utilize the full group bandwidth. For example, +considering DeviceMesh(8,2) in Figure 7(a), B +′ +2 corresponds +to the bandwidth between the two GPUs, which is equal to +200 GB/s. Note that B +′ +2 is lower than the group bandwidth +(600 GB/s) due to the constraints of the P2P communication +matrix. B +′ +2 corresponds to 8-GPU inter-node communication, +and since the two groups of all-reduce will share the inter- +node bandwidth, B +′ +1 is equal to 12.5 GB/s. +We assume that the cost of all-reduce follows the cost +model of Rabenseifner’s algorithm [27]. Ignoring the latency +part, we can calculate the algorithm bandwidth B1 and B2 for +DeviceMesh(d1,d2) using the following formula: +B1 = +d1 +2(d1 −1)B +′ +1 +B2 = +d2 +2(d2 −1)B +′ +2 +(4) +After obtaining B1, B2 from the hierarchical communica- +tion matrix, we can calculate the tensor parallel communica- +tion time based on Equation 2. ATP selects the device mesh +with the smallest Tcomm out of all the possible device meshes. +4 +Implementation +4.1 +Chunk-based Overlapping +Unlike data parallelism, which introduces all-reduce commu- +nication that can overlap with backward computation, tensor +parallelism introduces synchronous all-reduce communica- +tion that can cause more severe bottlenecks. To reduce over- +head, we propose a chunk-based overlapping technique in +which the tensor is computed sequentially in chunks, creating +overlapping opportunities +Since there is no dependency between the computation +and communication of different samples in a batch, we can +chunk on the batch dimension to reduce overhead. Figure 8 +shows a chunk size of 2. For each block, the data from two +chunks can be processed independently, with communication +7 + +QKV +Linear +MLP1 +MLP2 +Attention +Linear +Attention +Core +f1 +f2 +f3 +f4 +Transformer Layer +MLP 2 +f3 +f4 +QKV +Linear +f1 +f1 #1 +f1 #2 +f2 #1 +f2 #2 +f3 #1 +f3 #2 +f4 #1 +f4 #2 +QKV Linear +#1 +QKV Linear +#2 +AttCore +#1 +AttCore +#2 +AttLinear +#1 +AttLinear +#2 +MLP1 +#1 +MLP1 +#2 +MLP2 +#1 +MLP2 +#2 +Chunk = 2 +Figure 8: Chunk-based overlapping for one transformer layer. +This example shows chunk sizes of 2. The lighter color is for +computation and communication of the first chunk, and the +darker color is for the second. +triggered after the first chunk is processed and the second +chunk is processed simultaneously. While a larger chunk size +allows for more overlapping, it can also lead to inefficient +computation and communication. Therefore, in practice, it is +usually sufficient to set the chunk size to 2 or 4. +Unlike some previous work that fused all-reduce with +GEMM [10], chunk-based overlapping combines the trans- +former model structure with overlapping on a larger scope. +4.2 +Other Details +For linear layers, the chunk-based technique is used to overlap +computation and communication in the forward pass. How- +ever, in the backward pass, the gradient of the input and the +weight are two matrix products that do not depend on each +other. The back-propagation equations for the matrix-matrix +product Y = XW are as follows: +∂L +∂X = ∂L +∂Y W T, +∂L +∂W = XT ∂L +∂Y +After computing the gradient of X, an all-reduce com- +munication is required, which can be overlapped with the +computation of the gradient of W. To prevent the commu- +nication kernel from being delayed and reducing the effec- +tiveness of overlapping, we set the environment variable +CUDA_DEVICE_MAX_CONNECTIONS to 1. +To reduce the development cost of distributed models, ATP +offers easy-to-use APIs for higher-level frameworks. Fig- +ure 9 shows an example code of column-first tensor par- +allelism on DeviceMesh(2,2). To use ATP, users simply +call the init_mesh function and use the ATPLinear mod- +ule instead of the Linear module in PyTorch. The com- +munication groups in the device mesh and the placement +strategy are based on the PyTorch SPMD implementation +(https://github.com/pytorch/tau/tree/89700fd). +5 +Evaluation +The evaluation aim to answer the following questions: +import atp +atp.distributed.init_mesh((2, 2)) +shard_strategy = [atp.spmd.Shard(1), +                  atp.spmd.Shard(0)] +atp_linear = atp.nn.ATPLinear(256, 1024, shard_strategy) +output_ = atp_linear(torch.randn(4, 512, 128)) +Figure 9: An example code snippet for ATP’s API. In +this example, the user creates a DeviceMesh(2,2) using the +init_mesh and specifies the column-first tensor parallelism +ATPLinear. +• Q1: How does ATP compare with the state-of-the-art +tensor parallelism approaches? +• Q2: What is the impact of the overlapping optimization? +• Q3: Does hierarchical communication matrix help ATP +to select to the optimal strategy? +• Q4: How effective is ATP on different interconnect +topologies, and how should future interconnect topolo- +gies be designed? +In our experiments, we use three different hardware envi- +ronments: two single-node machines (Machine A and Ma- +chine B) and a multi-node cluster (Cluster C). Both Machine +A and Machine B are equipped with 8 NVIDIA A100 80 GB +GPUs. Machine A is equipped with NVSwitch and Machine +B is equipped with dual-GPU NVLink. The interconnection +topology of these machines can be seen in Figures 2(a) and +2(b). Cluster C, which is a multi-node cluster comprised of +NVIDIA A100 GPUs, has a inter-node bandwidth of 200 +Gbps. +In these three hardware environments, we conduct experi- +ments using four different interconnects: (1) Machine A with- +out NVLink (IC1), (2) Machine B with NVLink (IC2), (3) Ma- +chine A with NVLink (IC3), and (4) Cluster C with InfiniBand +(IC4). We evaluate the performance of PCIe on Machine A by +setting the NCCL environment variable NCCL_P2P_DISABLE +to 1, which disables NVLink. +We use appropriately sized GPT models with half-precision +(FP16). There are four sizes of GPT models (M1, M2, M3, +M4) as listed in Table 5. We use the metric of achieved ter- +aFlOP/s per GPU to evaluate performance. To calculate the +number of floating-point operations (FLOPs) in a transformer +layer, we use the formula from Megatron-LM [15]. We con- +sider FLOPs in both the forward and backward passes, but +do not consider activation checkpointing and therefore do not +need to multiply the forward FLOPs by 2 as in Megatron-LM. +By default, we set the batch size to 4 and the sequence length +to 2048. +8 + +M1 +M2 +M3 +M4 +0.0 +2.5 +5.0 +7.5 +10.0 +Achieved +teraFLOP/s per GPU ++64% ++37% ++53% ++58% ++70% ++25% ++4% +-21% +Megatron-LM +2D/2.5D +ATP +(a) IC1 +M1 +M2 +M3 +M4 +0 +25 +50 +75 +100 +Achieved +teraFLOP/s per GPU ++0% ++5% ++9% ++10% +(b) IC2 +M1 +M2 +M3 +M4 +0 +50 +100 +150 +Achieved +teraFLOP/s per GPU ++9% ++10% ++4% ++4% +(c) IC3 +M2 +M3 +M4 +0 +5 +10 +15 +Achieved +teraFLOP/s per GPU +-16% ++2% ++8% +(d) IC4 +Figure 10: Comparison with the SOTA Methods. The percentages on the bars represent the performance improvement of ATP +over Megatron-LM. +Table 2: Four sizes of GPT models for evaluation. +hidden size +heads +#billion params +per layer +#TFLOPs +per layer +M1 +2048 +16 +0.048 +2.625 +M2 +4096 +32 +0.192 +9.75 +M3 +8192 +64 +0.768 +37.5 +M4 +12288 +96 +1.728 +83.25 +5.1 +Comparison with the SOTA Methods +To answer Q1, we compare ATP with Megatron-LM [15] +and 2D/2.5D tensor parallelism [30,32]. In order to ensure +fairness in the comparison, we do not use optimized kernels in +any of the implementations. As baseline implementations, we +use Megatron-LM public code v2.6 and used ColossalAI [1] +v0.1.12 for 2D/2.5D tensor parallelism. For each of the four +interconnects (IC1,2,3,4), we scale the training to the maximum +number of GPUs: 8 GPUs for IC1,2,3 and 16 GPUs for IC4. +We treat 2D/2.5D tensor parallelism as a single baseline, using +2D for 16 GPUs and 2.5D for 8 GPUs (2D tensor parallelism +is only suitable for devices with square numbers). +As shown in Figure 10, ATP consistently outperform +Megatron-LM and 2D/2.5D tensor parallelism across vari- +ous topologies and model sizes. The optimization effect of +ATP varies significantly depending on the interconnect topol- +ogy. On PCIe-based IC1, ATP demonstrated the greatest per- +formance improvement, ranging from 37% to 64%. This is +mainly due to the use of DeviceMesh(2,4) in ATP, which +significantly reduced communication cost. On NVLink-based +IC2,3, the performance improvement is smaller, with a max- +imum improvement of about 10%. For these interconnects, +ATP and Megatron-LM have similar communication costs +because ATP select the optimal DeviceMesh(N,1). The per- +formance improvement in ATP is mainly due to better com- +putation and communication overlap. On InfiniBand-based +IC4 (16 devices), ATP shows a 4% improvement for larger +models. A more detailed analysis can be found in Section 5.3. +2D/2.5D tensor parallelism performs significantly worse +than Megatron-LM and ATP on IC2,3,4 due to the reasons +discussed in Section 3.3. On IC1, ATP performs better than +2D/2.5D except for the smallest model size (M1). As the +model size increased, the performance advantage of 2D/2.5D +relative to Megatron-LM decreases, and it performed poorly +with M4. +5.2 +Overlapping Optimization +To investigate Q2, we compare ATP with different chunk +sizes in Table 5.2. The table shows the training performance +of M2,3,4 on four interconnect topologies scaled to 8 GPUs. +The baseline represents the performance under the optimal +ATP strategy, and then we add chunk optimization by setting +the chunk size to 2 or 4. +Table 3: Overlapping Optimization for Different Interconnect +Topologies (TeraFLOP/s per GPU). +IC1 +IC2 +IC3 +IC4 +M2 +Baseline +3.45 +42.82 +97.45 +15.31 +Chunk=2 +3.46 +43.59 +100.38 +17.20 +Chunk=4 +3.51 +43.73 +98.20 +18.62 +M3 +Baseline +6.68 +73.61 +151.98 +22.47 +Chunk=2 +6.79 +75.27 +152.64 +26.09 +Chunk=4 +6.72 +75.95 +151.79 +26.18 +M4 +Baseline +10.71 +96.83 +178.19 +32.52 +Chunk=2 +10.70 +100.25 +178.17 +34.95 +Chunk=4 +10.70 +100.23 +177.78 +38.81 +9 + +We make the following observations. 1) The chunk-based +overlapping optimization generally improves performance +across different model sizes and interconnect topologies. 2) +the effect of chunk optimization is more significant on some +interconnect, such as IC4. On IC4, larger chunk sizes also +have greater performance improvements. For example, setting +the chunk size to 4 results in a 16% to 21% improvement +in end-to-end performance. 3) For intra-node interconnects +(IC1,2,3), chunk-based overlapping typically resultes in a 1% +to 3% improvement in end-to-end performance. 4) In some +cases, performance may degrade as the chunk size increases. +This can be due to reduced parallelism, leading to inefficient +hardware utilization for smaller GEMMs and other operators, +as well as increased overhead at the framework level. +5.3 +Optimal Strategy for ATP +To understand Q3 and better compare the performance of +ATP under different device mesh configurations, we evaluated +the performance of different device meshes without commu- +nication optimization. The results for different interconnect +topologies and model sizes are shown in Figure 11. ATP-i +refers to ATP with DeviceMesh(N/i,i) (N devices). +For fully-connected topologies IC3,4, the hierarchical com- +munication matrix has only one layer. Using equations 2 and +4, ATP selects ATP-1 when the number of devices is less +than 8 and ATP-2 when it is greater than 8. The optimal +ATP strategy is ATP-1 for IC3 with 8 GPUs and ATP-2 for +IC4 with 16 GPUs. As shown in Figure 11(c), these results +are consistent with the theoretical analysis. For IC4, ATP- +2 demonstrated a slight improvement on M4. However, the +performance degrades when the model is small due to the +overhead introduced. +For the dual-GPUs interconnect IC2, the hierarchical com- +munication matrix has two layers, the first layer is 4 dual- +GPUs interconnected by PCIe and the second layer con- +sists of 2 GPUs connected by NVLink. We mainly com- +pare the second item in formula 2, 2h/d2B1. Under ATP-1, +this item is 2h/GroupBW2, while under ATP-2/4, we can get +B1 = GroupBW2/d2 using formula 3, which is the same for +all ATP strategies. However, the first item in ATP-1 is 0, mak- +ing ATP-1 the most optimal strategy in this case. This trend +is also shown in Figure 11(b). +For IC1, we find that the all-reduce bandwidth could not be +accurately evaluated using the hierarchical communication +matrix. In this case, we need to calibrate the bandwidths B1 +and B2 using measured performance. For DeviceMesh(2,4), +B1 is calibrated to 1.20 GB/s and B2 to 4.95 GB/s. For +DeviceMesh(8,1), B1 is calibrated to 0.97 GB/s. Using Equa- +tion 2, the Tcomm of ATP-4 was theoretically 46% lower than +that of ATP-1. As shown in Figure 11(a), the actual end-to-end +training performance of ATP-4 was significantly improved, +consistent with the calibrated theoretical analysis. +5.4 +Interconnect Topologies +In the above experiments, all interconnects except IC1 demon- +strated relatively good performance under ATP-1. With the +common GPU interconnect topologies currently available, +ATP is optimal on DeviceMesh(N,1) and does not show the +advantage of ATP. +To investigate which interconnect topologies ATP can show +superiority, we analyze the following two interconnect topolo- +gies from a theoretical perspective: NVLink-Network Switch +(IC5) [9] and 2D Torus (IC6). For IC5, because all GPUs +are interconnected through NVLink-Network Switch, they +have the same communication bandwidth. Therefore, its hi- +erarchical communication matrix has only one layer, and we +can assume B +′ +1 = B +′ +2 = GroupBW. For IC6, the hierarchical +communication matrix is shown in 7(b), in which case both +B +′ +1 and B +′ +2 are also equal to GroupBW. Next, we combine +equations 2 and 4 to obtain the tensor-parallel communication +time for DeviceMesh(d1,d2): +Tcomm = +2Lbsh +GroupBW × 14d2 +4d1 −18 +d1d2 +Assuming that ∆ is equal to 2Lbsh/GroupBW, we visualize +the ATP under different device mesh in Figure 12. We observe +that the communication cost increases with scaling in ATP- +1, while the cost decreases in ATP-2 and ATP-4. ATP-OPT +corresponds to the communication overhead in the optimal +device mesh. The communication cost of Megatron-LM has +the same curve as ATP-1, with the communication cost rising +with scaling. The communication cost of ATP gradually de- +creases as it scales up, and ATP will have a more significant +advantage at larger scales. +In summary, in response to Q4, we suggest that future inter- +connect topologies should consider using accelerator-specific +systems to avoid the bottlenecks of current CPU-oriented +communication technologies. Additionally, fully-connected +and torus topologies are expected to have good performance +and scalability, with the latter being more cost-effective. +6 +Related Work +Foundation Models. Foundation models have been widely +adopted in many fields such as language [3, 5, 31], vision +[20,35], and reasoning [17,19]. To train foundation models on +ultra-large datasets and model sizes, custom software systems +such as Megatron-LM [15], DeepSpeed [23], ColossalAI [1] +have been developed. These systems are built on paralleliza- +tion techniques such as tensor parallelism [15], pipeline paral- +lelism [7,14] and ZeRO redundant optimizer [21,22]. Tensor +parallelism is critical for the efficiency of training giant foun- +dation models, but it has been rarely studied systematically. +ATP provides an adaptive tensor parallelism framework to +improve training efficiency on complicated interconnection. +10 + +M1 +M2 +M3 +M4 +0 +5 +10 +Achieved +teraFLOP/s per GPU +(a) IC1 +M1 +M2 +M3 +M4 +0 +25 +50 +75 +100 +Achieved +teraFLOP/s per GPU +(b) IC2 +M1 +M2 +M3 +M4 +0 +50 +100 +150 +Achieved +teraFLOP/s per GPU +(c) IC3 +M2 +M3 +M4 +0 +5 +10 +Achieved +teraFLOP/s per GPU +ATP-1 +ATP-2 +ATP-4 +ATP-8 +(d) IC4 +Figure 11: Performance comparison for ATP with different device meshes. ATP-i refers to ATP with DeviceMesh(N/i,i) (N +devices). For IC1,2,3 with 8 devices, we compare ATP-1/2/4. For IC4 with 16 devices, we compare ATP-1/2/4/8. +2 +4 +8 +16 +32 +64 +128 +The number of devices +2D +4D +7D +Tcomm +ATP-1 (Megatron-LM) +ATP-2 +ATP-4 +ATP-OPT +Figure 12: Communication time of IC5,6 with increasing num- +ber of devices. ATP-i refers to DeviceMesh(N/i,i) and ATP- +OPT refers to the optimal device mesh. +Tensor Parallelism. Megatron-LM introduced tensor par- +allelism for training GPT and BERT models, making it a +standard technique for training foundation models. Several +recent works, such as 2D tensor parallelism [32] and 2.5D +tensor parallelism [30], use the Scalable Universal Matrix +Multiplication Algorithm (SUMMA) [29]. 2D tensor paral- +lelism splits both activations and weights, and has a smaller +memory footprint. 2.5D tensor parallelism extends the 2D +approach to support an arbitrary number of GPUs, but still +suffers from high communication overhead. Other works such +as Sagemaker [13] and GSPMD [33] split activations along +the hidden dimension, but also have inefficient communica- +tion. However, none of these approaches consider complicated +GPU interconnects, leaving room for optimization. ATP ad- +dresses this by proposing a search space of tensor parallelism +strategies and using a hierarchical communication matrix +to estimate the communication cost of each strategy. ATP +is more adaptable and outperforms other tensor parallelism +methods in some environments with complicated GPU inter- +connects. +Automatic Parallelism. Recent work has proposed meth- +ods for optimizing distributed training through automatic +parallelism. These methods, such as Alpa [37], Unity [28], +GSPMD [33], and Whale [11], can find the optimal hybrid +parallelization strategy within a certain search space rather +than relying on manually-designed strategies. However, these +existing approaches only support regular tensor parallelism +and simple performance model of communication. ATP offers +a more general tensor Parallelism search space for automatic +parallelism and can also better characterize the communica- +tion ability of complicated interconnects through the use of +hierarchical communication matrix. This can improve the ac- +curacy of these automatic parallelism methods in estimating +communication cost. +7 +Conclusion +In this work, we present Adaptive Tensor Parallelism (ATP) +for foundation models. We propose column- and row-first +tensor parallelism based on 2D device meshes, and use the +hierarchical communication matrix to identify the optimal +strategy with minimal communication cost on different inter- +connections. Additionally, we use chunk-based overlapping to +reduce the communication overhead introduced by ATP. ATP +outperforms state-of-the-art approaches for various intercon- +nects and model sizes and shows theoretical improvement for +some topologies. ATP can be implemented as a replacement +for the current state-of-the-art tensor parallelism approach, +providing improved performance, scalability, and adaptability +to different (future) topologies. +11 + +References +[1] Zhengda Bian, Hongxin Liu, Boxiang Wang, Haichen +Huang, Yongbin Li, Chuan-Qing Wang, Fan Cui, and +Yang You. Colossal-ai: A unified deep learning system +for large-scale parallel training. ArXiv, abs/2110.14883, +2021. +[2] Rishi Bommasani, Drew A. Hudson, Ehsan Adeli, Russ +Altman, Simran Arora, Sydney von Arx, Michael S. +Bernstein, Jeannette Bohg, Antoine Bosselut, Emma +Brunskill, Erik Brynjolfsson, S. Buch, Dallas Card, Ro- +drigo Castellon, Niladri S. Chatterji, Annie S. Chen, +Kathleen A. Creel, Jared Davis, Dora Demszky, Chris +Donahue, Moussa Doumbouya, Esin Durmus, Stefano +Ermon, John Etchemendy, Kawin Ethayarajh, Li Fei- +Fei, Chelsea Finn, Trevor Gale, Lauren E. Gillespie, +Karan Goel, Noah D. 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Safranek. +Intel® quickpath intercon- +nect architectural features supporting scalable system +architectures. +2010 18th IEEE Symposium on High +Performance Interconnects, pages 1–6, 2010. +14 + diff --git a/BtFAT4oBgHgl3EQfsR7g/content/tmp_files/load_file.txt b/BtFAT4oBgHgl3EQfsR7g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39a78f687d46dcdda892fc17a2d748d72b834bb9 --- /dev/null +++ b/BtFAT4oBgHgl3EQfsR7g/content/tmp_files/load_file.txt @@ -0,0 +1,651 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf,len=650 +page_content='ATP: Adaptive Tensor Parallelism for Foundation Models Shenggan Cheng School of Computing National University of Singapore Ziming Liu School of Computing National University of Singapore Jiangsu Du School of Computer Science and Engineering Sun Yat-sen University Yang You School of Computing National University of Singapore Abstract Foundation models have impressive performance and gener- alization capabilities across a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The increasing size of the models introduces great challenges for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Tensor parallelism is a critical technique that is currently used in almost all foundation model training and has a significant impact on overall training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, current tensor parallelism in machine learning frameworks misses optimization opportunities in fitting various intercon- nection topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this work, we present ATP, an adaptive tensor parallelism framework for foundation models, which can automatically select the optimal parallel strategy on dif- ferent interconnections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We propose column- and row-first tensor parallelism based on 2D device meshes and construct a search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Combined with the hierarchical communication matrix, ATP can identify the optimal strategy in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We also propose chunk-based overlapping to reduce communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Our evaluations show ATP consis- tently outperforms the state-of-the-art approaches for various model sizes and interconnects, achieving end-to-end train- ing performance improvements of up to 37-64% on specific interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Based on our theoretical model, the communi- cation overhead of ATP decreases with scaling, indicating a qualitative leap forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 1 Introduction The term foundation models [2] refers to pretrained models with large-scale parameters that can be adapted to a wide range of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These models have impressive performance with generalization capabilities and have led to state-of-the-art results on benchmarks of natural language processing (BERT [5], GPT-3 [3]) and computer vision (CLIP [20], Florence [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The most obvious trend in recent years is the rapid increase in the size of foundation models: from 100 billion [3,31,36] to over 500 billion [4,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Training foundation models, such as GPT-3 (which has 175 billion parameters), can be challenging due to the huge computational costs and memory capacity bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, it would take about 288 years to train GPT-3 using a single NVIDIA V100, and the model’s parameters would not fit in the main memory of even the most advanced GPUs (such as NVIDIA 80 GB A100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To address these issues, various parallelism techniques have been proposed, including tensor (intra-layer) parallelism [15,25], pipeline (inter-layer) parallelism [7, 14] and ZeRO redundant optimizer [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These approaches aim to efficiently parallelize computation and distribute parameters among multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Among these, tensor parallelism is the most important train- ing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This approach evenly distributes both compu- tations and parameters across multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the training of foundation models such as GPT-3 [3], OPT [36], Yuan [31], Megatron-Turing [26], tensor parallelism and pipeline paral- lelism are often used in combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Tensor parallelism splits the matrix multiplications across different devices and is typi- cally used at the intra-node level, while pipeline parallelism is used at the inter-node level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Empirical results have shown that tensor parallelism should be used as much as possible without introducing cross-node communication, meaning that tensor parallelism should generally be used up to degree-N when using N-GPU servers [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In automatic parallelism approaches, tensor parallelism is a key design factor for the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, in Alpa [37], the search space includes both inter-operator and intra-operator parallelism, where intra-operator parallelism represents tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Tensor parallelism is not only used for training, but also for inference to reduce latency and provide additional memory capacity across GPUs to fit parameters [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, existing approaches has several drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 1) they only consider simple interconnect topologies, such as NVSwitch [16], and cannot adapt to more complicated topolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, when performing tensor parallelism on multi-node systems, the all-reduce operation is limited by the slowest link in the topology, preventing it from exploiting the high-bandwidth NVLink within the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2) tensor paral- lelism relies heavily on high-bandwidth interconnects, such as NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On servers that are not equipped with NVLink, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='08658v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='DC] 20 Jan 2023 the communication costs introduced by tensor parallelism can become a significant bottleneck for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3) unlike data parallelism, the communication introduced by tensor parallelism is synchronous, which makes it more difficult to overlap with computations, leading to more overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As a result, tensor parallelism is not as scalable as data parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To address these limitations, we present Adaptive Tensor Parallelism (ATP) for foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Unlike existing ten- sor parallelism approaches that have a fixed communication pattern, ATP can select different communication patterns on different topological interconnections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We propose column- and row-first tensor parallelism based on 2D device meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Different 2D device meshes form the search space, and we use the hierarchical communication matrix to estimate the communication cost of different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This allows ATP to identify the best strategy in the search space, making it a topo-aware approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In addition, we use chunk-based over- lapping to reduce communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP not only outperforms state-of-the-art approaches but also has a simple API for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In summary, we make the following contributions: We propose column- and row-first tensor parallelism based on two-dimensional device meshes and construct a novel search space for tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We design hierarchical communication matrix to de- scribe the communication characteristics of complicated interconnects and identify the optimal strategy in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We implement ATP with some communication optimiza- tions, including chunk-based overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP achieves up to 37-64% improvement over the state- of-the-art approaches in specific interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the- oretical analysis, the communication cost of ATP de- creases with scaling on some topologies, representing a significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2 Understanding Tensor Parallelism 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1 Tensor Parallelism on Transformer Transformer has become the bedrock of foundation models due to their excellent ability to model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Taking a language model as an example, the input tokens are first fed into an embedding layer and then passed through a single- stack transformer encoder or decoder with the L layers (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The input of transformer layer is a 3D tensor of size [b,s,h] where b,s,h are batch, sequence and hidden-size dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Each transformer layer consists of a multi-head attention (MHA) block followed by a feed-forward block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the MHA block, the sequence data is fed into three different MLP layers to obtain Query(Q), Key(K), Value(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Then di- vided into a heads, each with a hidden size of d = h/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For each head, the attention score is calculated using Formula 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Self Attention Linear Dropout Transformer Layer x L LayerNorm Self Attention Linear Dropout GPU 1 GPU 2 f LayerNorm Linear GeLU Linear Dropout LayerNorm Linear GeLU Linear Dropout f f f LayerNorm Figure 1: Transformer layer with tensor parallelism on two GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' f and f are conjugate communication operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the forward pass, f represents an all-reduce operation, while in the backward pass f represents an all-reduce operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The feed-forward block has two layers of multi-layer percep- tron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The first layer increases the hidden size to 4h and the second layer reduces it back to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Att(Q,K,V) = softmax(QKT √ d )V Q,K,V ∈ Rs×d (1) Tensor Parallelism, which is commonly used for training foundation models, was proposed by Megatron-LM [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As shown in Figure 1, tensor parallelism parallelizes the MHA and feed-forward blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the feed-forward block, there are two MLP layers and an activation function (GeLU): Y = GeLU(XA), Z = YB where A and B are the weight of two MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We can split A by columns and B by rows and parallelize the computation into two parts: A = [A1,A2], B = � B1 B2 � [Y1,Y2] = [GeLU(XA1),GeLU(XA2)] Z = reduce(Y1B1,Y2,B2) MHA blocks can be parallelized in a similar way by divid- ing the weight matrix of Q,K,V by columns and the output linear layer by rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This approach introduces two all-reduce operations in both the forward and backward passes of each layer, distributing computation and memory footprints across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Inside the MHA and feed-forward blocks, tensor parallelism parallelizes model parameters, optimizer state, and activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The Lay- erNorm, including the input/output activations, is duplicated in each tensor-parallel worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Several recent works, such as 2D tensor parallelism [32] use the Scalable Universal Matrix Multiplication Algorithm 2 (SUMMA) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We regard 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism as an extension of 2D tensor parallelism since it can be seen as a combination of 2D Tensor Parallelism and Data Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2D Tensor Parallelism has two main issues: 1) broadcast of the weight matrix is expensive because the size of the weights is much larger than the activation of the giant model, and 2) multiple broadcasts in a single layer result in high overhead and low bandwidth utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2 Modern Accelerator Interconnect Due to the high demand on communication in distributed train- ing, a number of accelerator-specific interconnect hardware has been developed and more complicated topologies have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Multi-GPU servers dominate the training hardware, so the interconnect architecture of GPUs is gener- ally divided into two hierarchical levels, intra-node level and inter-node level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Intra-node Level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' At this hierarchy, training scales within a single node, where the interconnections between multiple GPUs are PCIe (64 GB/s for PCIe 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='0) or NVLink (600 GB/s for NVLink-v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Figure 2 shows the examples of these archi- tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The PCIe network forms a balanced tree structure, where each GPU is connected to a PCIe switch, which is fur- ther connected to a CPU socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The sockets are then bridged by GMI Link for AMD EPYC or QPI [38] for Intel XEON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' NVLink is a GPU-oriented interconnect technology proposed by NVIDIA with various configuration options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For exam- ple, in Figure 2(a), all GPUs are connected to the NVSwitch via NVLink, improving all-to-all communication capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In Figure 2(b), the eight GPUs are divided into four groups, with two GPUs in each group connected via NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this case, the GPU interconnect has a clear NUMA effect (closer proximity leads to stronger communication capabilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Inter-node Level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In data centers, the main interconnec- tion technologies across nodes are Ethernet and InfiniBand [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Cloud servers or supercomputers used for training are typi- cally have at least 50 Gbps or 200 Gbps of communication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' There are many different configurations for topolo- gies at this hierarchy, as shown in Figure 3, including fat tree, torus, and dragonfly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' There are also many accelerator-specific direct interconnects at this level, such as the 2D Torus Inter- Core Interconnect (ICI) links in TPU pods that directly con- nects up to 1024 TPUv3 cores [12] and NVIDIA’s upcoming Nvlink-Network Switch [9], which can also connect hundreds of GPUs directly via high-bandwidth NVLink [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The de- velopment of new interconnected hardware and topologies presents opportunities and challenges for software-hardware co-design [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='3 Communication Analysis Communication Pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Similar to data parallelism, tensor- parallel communication requires all-reduce operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' How- CPU 0 CPU 1 GMI Link PCIe Switch PCIe Switch PCIe Switch PCIe Switch GPU GPU GPU GPU GPU GPU GPU GPU NVSwitch PCIe NVLink (a) Server with NVSwitch CPU 0 CPU 1 GMI Link PCIe Switch PCIe Switch PCIe Switch PCIe Switch GPU GPU GPU GPU GPU GPU GPU GPU PCIe NVLink (b) Server with NVLink Figure 2: PCIe and NVLink topology for intra-node intercon- nect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' (a) and (b) show the multi-gpu server with full intercon- nection and dual card interconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' (a) Fat Tree (b) Torus (c) Dragonfly Figure 3: Inter-node interconnect topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ever, because the computations are interdependent with the communication, the all-reduce here must be syn- chronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Tensor-parallel communication relies more on high- bandwidth communication because it cannot overlap with backward computation as data-parallel communication does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Communication Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1, the size of the tensor for each all-reduce is [b,s,h], where b,s,h are batch, sequence and hidden-size dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In a model with L layers, a total of 4L all-reduce operations are re- quired in the forward and backward passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Assuming an all-reduce bandwidth of B, the communication cost for one step is 4Lbsq/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In training foundation models, synchronous all-reduce of dozens of gigabytes of large tensors is required in one step, posing a high demand on interconnect bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' All-reduce Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Various vendors provide collec- tive communications library (CCL) to deliver intra-node and inter-node level communication capabilities for deep learn- ing frameworks, such as NCCL on the NVIDIA platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These high-performance all-reduce implementations are pri- 3 marily based on the ring algorithm [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The performance of ring all-reduce depends on the bandwidth of the slowest link on the ring, which may result in some local high-bandwidth interconnects being wasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, in cross-server all- reduce communication, the all-reduce performance is limited by the bandwidth of the cross-server interconnect, and the high bandwidth of the NVLink inside the server is wasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3 Adaptive Tensor Parallelism 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1 Sharding Notion Many research and machine learning frameworks that use the concept of sharding to describe how tensors are distributed across multiple devices in a parallel strategy, particularly in the context of auto-parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this paper, we will be focus- ing on the sharding notion from PyTorch Distributed Tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Device Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the training of foundation models, it is of- ten necessary to use a combination of different parallelization strategies, which involves communication between certain ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To manage these communication groups, we group the devices into multidimensional device meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' A device mesh with N devices can be expressed as DeviceMesh(d1,d2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' A device mesh can be thought of as having n levels, with the devices in the current group being divided into di sub-groups at the i-th level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The total number of devices in the mesh is equal to N = d1 × d2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='. × dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, a device mesh with four devices can be represented in four different ways: 1D: [4], 2D: [1,4], [2,2], [4,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' DeviceMesh(d1,4/d1) means that the four devices are divided into d1 groups, with each group having 4/d1 devices for sharding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Figure 4 shows a device mesh with d1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Sharding Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We use the sharding spec to describe the strategy for distributing the global tensor across the devices in a device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Like Alpa [37] and OneFlow [34], we have chosen to use three types of placement strategies: Shard, Replicate, and Partial: Shard(d): split the tensor along the d-th dimension across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Replicate: replicate the tensor across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Partial(op): partial the tensor across devices, requiring all-reduce communication to obtain the global tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Op represents the ReduceOp, such as SUM or MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' From the perspective of a device mesh, the sharding of a tensor can be thought of as the selection of a placement strat- egy at each level of the device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Therefore, the sharding spec can be viewed as a sequence of placement strategies, the length of which is equal to the number of dimensions of the device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For an N-dimensional device mesh, the shard- ing spec of the global tensor can be defined as [P1,P2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='PN], where Pi ∈ {Shard(d),Replicate,Partial(op)} represents the placement strategy for the i-th dimension of the device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Figure 4 illustrates a valid sharding spec for a 2D tensor on a 2D device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the sharding spec [Shard(1),Shard(0)], Shard(1) means that the tensor is column-wise partitioned into two parts for rank-[0,1] and rank-[2,3], and then Shard(0) means that each of these partitioned tensors is further row- wise partitioned for each pair of ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' If the sharding spec is [Replicate,Shard(0)], the sharded tensor for rank-0 and rank- 2 will be [1,2,3,4] and rank-1 and rank-3 will be [5,6,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Unlike the sharding spec in Alpa, which binds the placement strategy to the tensor dimensions, we have chosen to bind the strategy to the dimensions of the device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This approach is more concise and easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' DeviceMesh(2,2) Sharding Spec: [Shard(1), Shard(0)] 1 2 3 4 5 6 7 8 Rank 0 Rank 1 1 2 5 6 Rank 2 Rank 3 3 4 7 8 Column-wise Shading Row-wise Sharding Row-wise Sharding Global Tensor Rank 0 Rank 1 Rank 2 Rank 3 Figure 4: Sharding a 2D tensor on a 2D device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The parallelism strategy can be represented as a sharding spec on a 1D device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1 illustrates the sharding specs of the input, weight, and output in data parallelism (DP) and tensor parallelism in Megatron-LM (Column TP and Row TP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For the combination of DP and TP, a 2D device mesh is typically used, with the first dimension of the sharding spec corresponding to DP (as shown in the first row of the table) and the second dimension corresponding to TP (as shown in the second or third row of the table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Table 1: Sharding specs for MLP layer on 1D device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Parallelism Input Weight Output DP [Split(0)] [Replicate] [Split(0)] Column TP [Replicate] [Split(1)] [Split(1)] Row TP [Split(1)] [Split(0)] [Partial(SUM)] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2 Search Space for Tensor Parallelism With the definition of device mesh and sharding spec, we can express more complex parallelization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this section, we will introduce two types of tensor parallelism algorithms on 2D device meshes, and demonstrate how they can be applied to transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Finally, we will discuss the search space for tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In Megatron-LM’s tensor parallelism, the weight matrix (W) is split row-wise or column-wise on a 1D device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' When using a 2D device mesh, W is split at two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 4 1 5 1 2 4 3 1 2 1 1 2 1 2 Rank 0 X: X = 4 5 1 20 4 Rank 2 X = Rank 1 Rank 3 21 6 17 7 21 6 AllReduce #1 X: [Shard(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Replicate] W: [Shard(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Shard(1)] Y: [Partial(SUM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Shard(1)] W: Y: AllReduce #2 Row-First Tensor Parallelism Column-First Tensor Parallelism X: [Replicate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Shard(1)] W: [Shard(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Shard(0)] Y: [Shard(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Partial(SUM)] AllReduce #1 AllReduce #2 3 1 1 2 2 3 2 4 15 3 17 7 3 1 2 3 6 X = 1 1 2 1 5 1 5 1 X = 2 10 2 8 7 11 4 8 7 11 4 1 1 2 1 2 Rank 0 X = Rank 2 Rank 1 Rank 3 2 2 4 1 5 1 5 1 X = 2 10 2 3 1 2 3 6 X = 1 1 2 4 5 1 20 4 X = 3 15 3 AllReduce #1 AllReduce #1 21 6 17 7 8 7 11 4 Figure 5: Row-First and Column-First Tensor Parallelism for Y = XW on DeviceMesh(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The green matrix represents the global tensor and the purple matrix represents sharded tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These two approaches have different priorities in the sharding of the weight matrix (W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' A grouped all-reduce communication is required to obtain the global tensor of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As shown in Figure 5, there are two sharding specs for W, [Shard(0),Shard(1)] and [Shard(1),Shard(0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the first spec, the matrix is split row-wise at the first level of the de- vice mesh, and then column-wise at the second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The other spec splits column-wise at the first level and then row- wise at the second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Then we consider how to perform the distributed GEMM Y = XW on DeviceMesh(d1,d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Based on the two sharding specs for the weight matrix, two tensor parallelism approaches can be proposed: row-first tensor parallelism and column-first tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As shown in the left half of Figure 5, in row-first tensor parallelism, the sharding specs of W and X are [Shard(0),Shard(1)] and [Shard(1),Replicate].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The lo- cal GEMM operation at each rank results in a sharded Y with the sharding spec [Partial(SUM),Shard(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To obtain the global tensor result of Y, it is necessary to perform an all- reduce operation on the first dimension of the device mesh and then an all-gather operation on the second dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Column-first tensor parallelism uses a different sharding strat- egy for the weight matrix, and the dimensions of the device mesh are transposed in the communication, resulting in a dif- ferent communication pattern compared to row-first tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For row-first tensor parallelism, when the size of the global tensor X is [b,h1], the size of W is [h1,h2], and the size of Y is [b,h2], the sharded X has a size of [b,h1/d1], the sharded W has a size of [h1/d1,h2/d2], and the sharded Y has a size of [b,h2/d2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The GEMM on global tensors requires b×h1 ×h2 fused multiply-add (FMA) operations, while each rank re- quires b×h1 ×h2/(d1 ×d2) FMA operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The computa- tions are evenly distributed across the ranks, requiring commu- nication on a tensor of size [b,h2/d2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In contrast, for column- first tensor parallelism, the size of the tensor for communica- tion is [b,h2/d1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1 Transformer As introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1, the transformer layer consists of an attention block followed by a feed-forward block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We will describe how to use row- and column-first tensor parallelism in these two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Feed-forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' There are two MLP layers and an acti- vation function in feed-forward (Y = GeLU(XA),Z = YB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Megatron-LM uses column parallelism for the first MLP layer and row parallelism for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Similarly, in our approach, we use column-first tensor parallelism for the first MLP layer and row-first for the second, as shown in Figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The sharded feed-forward block can be expressed as follows: Yshard = f3(XshardAshard) Zshard = f4(GeLU(Yshard)Bshard) The sharding specs of weight matrix Ashard and Bshard are [Shard(1),Shard(0)] and [Shard(0),Shard(1)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The input(Xshard) and output tensor(Zshard) have the same sharding spec: [Replicate,Shard(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' f3 and f4 are two con- jugate communication operations: f3 performs an all-reduce on the first dimension of the device mesh in the forward pass and on the second dimension in the backward pass, while f4 is the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Attention block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this block, we use a similar strategy to the feed-forward block, using column-first tensor parallelism in the QKV Linear and row-first in the Output Linear, as shown in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The main difference is that we want the computation of the Attention Core to be fully sharded, so the sharding spec of the input tensor should not include Replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To achieve this, we scatter the input tensor before the Attention Core, and gather the output tensor before the Output Linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' f1 and f2 are two conjugate communication operations, similar to f3 and f4 in the feed-forward block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 5 f2 f1 QKV Linear h/d2 3h/d1 Linear h/d2 Attention Core Column-First Tensor Parallelism Row-First Tensor Parallelism (a) Attention block GeLU f4 f3 MLP 1 h/d2 4h/d1 MLP 2 h/d2 (b) Feed-forward Figure 6: Parallel transformer layer with row- and column-first tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' f1,3 and f2,4 are conjugate operations: f1,3 performs an all-reduce on the first dimension of the device mesh in the forward pass and on the second dimension in the backward pass, while f2,4 is the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' f1 has a smaller amount of communication compared to f3 because the output hidden size of the QKV Linear is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Different device meshes correspond to different tensor sizes and communication groups for row- and column-first tensor parallelism, which can result in different optimal tensor par- allelism strategies for different interconnect topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, if we have 2n devices, then we will have n+1 kinds of 2D device meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This forms a search space in which we need to find the device mesh with the optimal performance for tensor parallelism in a specific topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Under the DeviceMesh(N,1) device mesh, our approach is similar to the tensor parallelism in Megatron-LM [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Google proposed a novel tensor parallelism algorithm [18] to reduce the inference latency of the giant transformer model on TPU clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On q2 TPUs with a 2D torus topology, Google’s algorithm is similar to our proposed tensor parallelism on DeviceMesh(q/2,2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These tensor parallelism methods are all contained in our search space, which demonstrates the potency of our search space and guarantees that the final performance will not be worse than these traditional tensor parallelism methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='3 Theoretical Analysis To analyze the communication cost theoretically, we assume that B1 is the all-reduce bandwidth on the first dimension of the device mesh and B2 is the bandwidth on the second dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For an L-layer transformer model with an input size of [b,s,h], where b, s, and h are the batch, sequence, and hidden size dimensions, the communication time of the all- reduce introduced by tensor parallelism is: Tcomm = 2Lbs×(Tf1 +Tf2 +Tf3 +Tf4) = 2Lbs×( 3h d1B2 + h d2B1 + 4h d1B2 + h d2B1 ) = 2Lbs×( 7h d1B2 + 2h d2B1 ) (2) It is worth noting that as the number of tensor-parallel workers (d1 ×d2) increases, the communication time (Tcomm) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This communication characteristic helps to extend tensor parallelism to a larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' From the perspective of the communication group, Megatron-LM requires the participation of all tensor-parallel workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On a 2D device mesh, communication is grouped, meaning that each communication requires only one row or column of workers on the device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This communica- tion pattern has the potential to better exploit complicated interconnect topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='4 Hierarchical Communication Matrix As previously mentioned, the goal is to find the optimal device mesh that provides the best performance in a specific topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Currently, many performance analysis and automatic paral- lelism frameworks for model training rely on communication volume to measure communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, to more effectively find the optimal device mesh, we utilize hierar- chical communication matrix to account for communication costs in complex topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' One of the main features of modern accelerator intercon- nects is their hierarchical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Within nodes, multiple GPUs are connected using PCIe or NVLink, and different nodes are connected using InfiniBand or Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The traditional method of describing the communication capabilities of a topology is through a P2P communication matrix, which is a N ×N matrix that represents the bandwidth of the interconnec- tions between the N ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, we have introduced the concepts of hierarchy and group bandwidth into the communi- cation matrix to more accurately represent the communication capabilities of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Figure 7 shows some examples of hierarchical communication matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The hierarchical communication matrix has multiple lay- ers, each containing a P2P communication matrix and group bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' At higher layers, each rank may represent a group of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, in Figure 7(a), there are two layers, with each rank representing a node (consisting of 4 devices) at the higher level and a single device at the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The P2P communication matrix is the aggregate bandwidth be- tween the two ranks, which at a higher level is the two groups of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The group bandwidth for each rank represents the aggregate bandwidth of that group of devices to the out- side world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In Figure 7(a), we can see that there are 4 GPUs 6 25 25 25 25 25 25 0 1 2 3 0 1 2 3 200 200 200 200 200 200 0 1 2 3 0 1 2 3 Group Bandwidth：600 GB/s Group Bandwidth：25 GB/s (a) Interconnect Topology 1 100 100 100 100 100 100 100 100 0 1 2 3 0 1 2 3 25 25 25 25 25 25 25 25 0 1 2 3 0 1 2 3 Group Bandwidth：50 GB/s Group Bandwidth：200 GB/s (b) Interconnect Topology 2 Figure 7: Examples of hierarchical communication matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' (a) is a cluster of four nodes connected via 200 Gbps HDR, with four GPUs per node interconnected using NVLinkv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' (b) is 4×4 devices with 2D Torus interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' interconnected by 4 NVLinks, each providing 50 GB/s of bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Therefore, the P2P communication bandwidth is 200 GB/s, while the group bandwidth for each GPU is 600 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Figure 7(b) shows a 4x4 device mesh with a 2D Torus interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Assuming that the bandwidth of each link is 25 GB/s, the second level of the matrix shows a torus ring in which devices i and i + 1, i − 1 are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For the first level, there are 4 links between rings (one for each device), resulting in a P2P bandwidth of 100 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The links in both directions on the ring make the group bandwidth twice as large as P2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5 Adaptive Tensor Parallelism By combining the search space (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2) and the hierarchical communication matrix (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='4), we can propose Adaptive Tensor Parallelism (ATP) for automatically selecting the optimal tensor parallelism strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In the search space, we can estimate the communication time for each device mesh and identify the optimal mesh with the minimum communication time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' First, we estimate the bandwidth of all-reduce communica- tion for each device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='3, on the device mesh, all-reduce is required for one column or one row of workers, whose bandwidths correspond to B1 and B2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We use the hierarchical communication matrix to estimate the bandwidth that can be provided by the intercon- nect link under all-reduce, resulting in B ′ 1 and B ′ 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Assuming that the hierarchical communication matrix has l layers with Ri ranks per layer, and denoting the group band- width of layer j as GroupBWj, we can compute B′ 1 and B′ 2 as follows: when considering DeviceMesh(d1,d2), the first dimension involves layers 1 to i, and the second dimension is i(+1) to l (i+1 when d1 ̸= ∏j:1→i Rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The performance of all-reduce depends on the bandwidth of the bottleneck, so B′ 1 and B′ 2 can be calculated using the following formula: B′ 1 = min(GroupBW1→i/d2) B′ 2 = min(GroupBWi+(1)→l) (3) When performing all-reduce in the first dimension of the device mesh, there are d2 groups of all-reduce, each with d1 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Since d2 groups of all-reduce share the interconnec- tion bandwidth, so we need to divide the bandwidth by d2 when calculating B ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The P2P communication matrix can be used to correct the group bandwidth when the all-reduce al- gorithm cannot utilize the full group bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, considering DeviceMesh(8,2) in Figure 7(a), B ′ 2 corresponds to the bandwidth between the two GPUs, which is equal to 200 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Note that B ′ 2 is lower than the group bandwidth (600 GB/s) due to the constraints of the P2P communication matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' B ′ 2 corresponds to 8-GPU inter-node communication, and since the two groups of all-reduce will share the inter- node bandwidth, B ′ 1 is equal to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We assume that the cost of all-reduce follows the cost model of Rabenseifner’s algorithm [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Ignoring the latency part, we can calculate the algorithm bandwidth B1 and B2 for DeviceMesh(d1,d2) using the following formula: B1 = d1 2(d1 −1)B ′ 1 B2 = d2 2(d2 −1)B ′ 2 (4) After obtaining B1, B2 from the hierarchical communica- tion matrix, we can calculate the tensor parallel communica- tion time based on Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP selects the device mesh with the smallest Tcomm out of all the possible device meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 4 Implementation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1 Chunk-based Overlapping Unlike data parallelism, which introduces all-reduce commu- nication that can overlap with backward computation, tensor parallelism introduces synchronous all-reduce communica- tion that can cause more severe bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To reduce over- head, we propose a chunk-based overlapping technique in which the tensor is computed sequentially in chunks, creating overlapping opportunities Since there is no dependency between the computation and communication of different samples in a batch, we can chunk on the batch dimension to reduce overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Figure 8 shows a chunk size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For each block, the data from two chunks can be processed independently, with communication 7 QKV Linear MLP1 MLP2 Attention Linear Attention Core f1 f2 f3 f4 Transformer Layer MLP 2 f3 f4 QKV Linear f1 f1 #1 f1 #2 f2 #1 f2 #2 f3 #1 f3 #2 f4 #1 f4 #2 QKV Linear #1 QKV Linear #2 AttCore #1 AttCore #2 AttLinear #1 AttLinear #2 MLP1 #1 MLP1 #2 MLP2 #1 MLP2 #2 Chunk = 2 Figure 8: Chunk-based overlapping for one transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This example shows chunk sizes of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The lighter color is for computation and communication of the first chunk, and the darker color is for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' triggered after the first chunk is processed and the second chunk is processed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' While a larger chunk size allows for more overlapping, it can also lead to inefficient computation and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Therefore, in practice, it is usually sufficient to set the chunk size to 2 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Unlike some previous work that fused all-reduce with GEMM [10], chunk-based overlapping combines the trans- former model structure with overlapping on a larger scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2 Other Details For linear layers, the chunk-based technique is used to overlap computation and communication in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' How- ever, in the backward pass, the gradient of the input and the weight are two matrix products that do not depend on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The back-propagation equations for the matrix-matrix product Y = XW are as follows: ∂L ∂X = ∂L ∂Y W T, ∂L ∂W = XT ∂L ∂Y After computing the gradient of X, an all-reduce com- munication is required, which can be overlapped with the computation of the gradient of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To prevent the commu- nication kernel from being delayed and reducing the effec- tiveness of overlapping, we set the environment variable CUDA_DEVICE_MAX_CONNECTIONS to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To reduce the development cost of distributed models, ATP offers easy-to-use APIs for higher-level frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Fig- ure 9 shows an example code of column-first tensor par- allelism on DeviceMesh(2,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To use ATP, users simply call the init_mesh function and use the ATPLinear mod- ule instead of the Linear module in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The com- munication groups in the device mesh and the placement strategy are based on the PyTorch SPMD implementation (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='com/pytorch/tau/tree/89700fd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 5 Evaluation The evaluation aim to answer the following questions: import atp atp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='init_mesh((2, 2)) shard_strategy = [atp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='spmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='Shard(1), atp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='spmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='Shard(0)] atp_linear = atp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='ATPLinear(256, 1024, shard_strategy) output_ = atp_linear(torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='randn(4, 512, 128)) Figure 9: An example code snippet for ATP’s API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this example, the user creates a DeviceMesh(2,2) using the init_mesh and specifies the column-first tensor parallelism ATPLinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Q1: How does ATP compare with the state-of-the-art tensor parallelism approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Q2: What is the impact of the overlapping optimization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Q3: Does hierarchical communication matrix help ATP to select to the optimal strategy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Q4: How effective is ATP on different interconnect topologies, and how should future interconnect topolo- gies be designed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In our experiments, we use three different hardware envi- ronments: two single-node machines (Machine A and Ma- chine B) and a multi-node cluster (Cluster C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Both Machine A and Machine B are equipped with 8 NVIDIA A100 80 GB GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Machine A is equipped with NVSwitch and Machine B is equipped with dual-GPU NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The interconnection topology of these machines can be seen in Figures 2(a) and 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Cluster C, which is a multi-node cluster comprised of NVIDIA A100 GPUs, has a inter-node bandwidth of 200 Gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In these three hardware environments, we conduct experi- ments using four different interconnects: (1) Machine A with- out NVLink (IC1), (2) Machine B with NVLink (IC2), (3) Ma- chine A with NVLink (IC3), and (4) Cluster C with InfiniBand (IC4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We evaluate the performance of PCIe on Machine A by setting the NCCL environment variable NCCL_P2P_DISABLE to 1, which disables NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We use appropriately sized GPT models with half-precision (FP16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' There are four sizes of GPT models (M1, M2, M3, M4) as listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We use the metric of achieved ter- aFlOP/s per GPU to evaluate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To calculate the number of floating-point operations (FLOPs) in a transformer layer, we use the formula from Megatron-LM [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We con- sider FLOPs in both the forward and backward passes, but do not consider activation checkpointing and therefore do not need to multiply the forward FLOPs by 2 as in Megatron-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' By default, we set the batch size to 4 and the sequence length to 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 8 M1 M2 M3 M4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='0 Achieved teraFLOP/s per GPU +64% +37% +53% +58% +70% +25% +4% 21% Megatron-LM 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D ATP (a) IC1 M1 M2 M3 M4 0 25 50 75 100 Achieved teraFLOP/s per GPU +0% +5% +9% +10% (b) IC2 M1 M2 M3 M4 0 50 100 150 Achieved teraFLOP/s per GPU +9% +10% +4% +4% (c) IC3 M2 M3 M4 0 5 10 15 Achieved teraFLOP/s per GPU 16% +2% +8% (d) IC4 Figure 10: Comparison with the SOTA Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The percentages on the bars represent the performance improvement of ATP over Megatron-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Table 2: Four sizes of GPT models for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' hidden size heads #billion params per layer #TFLOPs per layer M1 2048 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='048 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='625 M2 4096 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='192 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='75 M3 8192 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='768 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5 M4 12288 96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='728 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1 Comparison with the SOTA Methods To answer Q1, we compare ATP with Megatron-LM [15] and 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism [30,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In order to ensure fairness in the comparison, we do not use optimized kernels in any of the implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As baseline implementations, we use Megatron-LM public code v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='6 and used ColossalAI [1] v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='12 for 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For each of the four interconnects (IC1,2,3,4), we scale the training to the maximum number of GPUs: 8 GPUs for IC1,2,3 and 16 GPUs for IC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We treat 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism as a single baseline, using 2D for 16 GPUs and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D for 8 GPUs (2D tensor parallelism is only suitable for devices with square numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As shown in Figure 10, ATP consistently outperform Megatron-LM and 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism across vari- ous topologies and model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The optimization effect of ATP varies significantly depending on the interconnect topol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On PCIe-based IC1, ATP demonstrated the greatest per- formance improvement, ranging from 37% to 64%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This is mainly due to the use of DeviceMesh(2,4) in ATP, which significantly reduced communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On NVLink-based IC2,3, the performance improvement is smaller, with a max- imum improvement of about 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For these interconnects, ATP and Megatron-LM have similar communication costs because ATP select the optimal DeviceMesh(N,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The per- formance improvement in ATP is mainly due to better com- putation and communication overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On InfiniBand-based IC4 (16 devices), ATP shows a 4% improvement for larger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' A more detailed analysis can be found in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism performs significantly worse than Megatron-LM and ATP on IC2,3,4 due to the reasons discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On IC1, ATP performs better than 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D except for the smallest model size (M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As the model size increased, the performance advantage of 2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D relative to Megatron-LM decreases, and it performed poorly with M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2 Overlapping Optimization To investigate Q2, we compare ATP with different chunk sizes in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The table shows the training performance of M2,3,4 on four interconnect topologies scaled to 8 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The baseline represents the performance under the optimal ATP strategy, and then we add chunk optimization by setting the chunk size to 2 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Table 3: Overlapping Optimization for Different Interconnect Topologies (TeraFLOP/s per GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' IC1 IC2 IC3 IC4 M2 Baseline 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='45 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='82 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='45 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='31 Chunk=2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='46 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='59 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='38 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='20 Chunk=4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='51 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='73 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='20 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='62 M3 Baseline 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='68 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='61 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='98 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='47 Chunk=2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='79 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='27 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='64 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='09 Chunk=4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='72 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='95 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='79 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='18 M4 Baseline 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='71 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='83 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='19 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='52 Chunk=2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='70 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='25 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='17 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='95 Chunk=4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='70 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='23 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='78 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='81 9 We make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 1) The chunk-based overlapping optimization generally improves performance across different model sizes and interconnect topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2) the effect of chunk optimization is more significant on some interconnect, such as IC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' On IC4, larger chunk sizes also have greater performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For example, setting the chunk size to 4 results in a 16% to 21% improvement in end-to-end performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 3) For intra-node interconnects (IC1,2,3), chunk-based overlapping typically resultes in a 1% to 3% improvement in end-to-end performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 4) In some cases, performance may degrade as the chunk size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This can be due to reduced parallelism, leading to inefficient hardware utilization for smaller GEMMs and other operators, as well as increased overhead at the framework level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='3 Optimal Strategy for ATP To understand Q3 and better compare the performance of ATP under different device mesh configurations, we evaluated the performance of different device meshes without commu- nication optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The results for different interconnect topologies and model sizes are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP-i refers to ATP with DeviceMesh(N/i,i) (N devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For fully-connected topologies IC3,4, the hierarchical com- munication matrix has only one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Using equations 2 and 4, ATP selects ATP-1 when the number of devices is less than 8 and ATP-2 when it is greater than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The optimal ATP strategy is ATP-1 for IC3 with 8 GPUs and ATP-2 for IC4 with 16 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As shown in Figure 11(c), these results are consistent with the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For IC4, ATP- 2 demonstrated a slight improvement on M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, the performance degrades when the model is small due to the overhead introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For the dual-GPUs interconnect IC2, the hierarchical com- munication matrix has two layers, the first layer is 4 dual- GPUs interconnected by PCIe and the second layer con- sists of 2 GPUs connected by NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We mainly com- pare the second item in formula 2, 2h/d2B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Under ATP-1, this item is 2h/GroupBW2, while under ATP-2/4, we can get B1 = GroupBW2/d2 using formula 3, which is the same for all ATP strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, the first item in ATP-1 is 0, mak- ing ATP-1 the most optimal strategy in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This trend is also shown in Figure 11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For IC1, we find that the all-reduce bandwidth could not be accurately evaluated using the hierarchical communication matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In this case, we need to calibrate the bandwidths B1 and B2 using measured performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For DeviceMesh(2,4), B1 is calibrated to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='20 GB/s and B2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='95 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For DeviceMesh(8,1), B1 is calibrated to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='97 GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Using Equa- tion 2, the Tcomm of ATP-4 was theoretically 46% lower than that of ATP-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' As shown in Figure 11(a), the actual end-to-end training performance of ATP-4 was significantly improved, consistent with the calibrated theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='4 Interconnect Topologies In the above experiments, all interconnects except IC1 demon- strated relatively good performance under ATP-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' With the common GPU interconnect topologies currently available, ATP is optimal on DeviceMesh(N,1) and does not show the advantage of ATP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To investigate which interconnect topologies ATP can show superiority, we analyze the following two interconnect topolo- gies from a theoretical perspective: NVLink-Network Switch (IC5) [9] and 2D Torus (IC6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For IC5, because all GPUs are interconnected through NVLink-Network Switch, they have the same communication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Therefore, its hi- erarchical communication matrix has only one layer, and we can assume B ′ 1 = B ′ 2 = GroupBW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For IC6, the hierarchical communication matrix is shown in 7(b), in which case both B ′ 1 and B ′ 2 are also equal to GroupBW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Next, we combine equations 2 and 4 to obtain the tensor-parallel communication time for DeviceMesh(d1,d2): Tcomm = 2Lbsh GroupBW × 14d2 +4d1 −18 d1d2 Assuming that ∆ is equal to 2Lbsh/GroupBW, we visualize the ATP under different device mesh in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We observe that the communication cost increases with scaling in ATP- 1, while the cost decreases in ATP-2 and ATP-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP-OPT corresponds to the communication overhead in the optimal device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The communication cost of Megatron-LM has the same curve as ATP-1, with the communication cost rising with scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' The communication cost of ATP gradually de- creases as it scales up, and ATP will have a more significant advantage at larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In summary, in response to Q4, we suggest that future inter- connect topologies should consider using accelerator-specific systems to avoid the bottlenecks of current CPU-oriented communication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Additionally, fully-connected and torus topologies are expected to have good performance and scalability, with the latter being more cost-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 6 Related Work Foundation Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Foundation models have been widely adopted in many fields such as language [3, 5, 31], vision [20,35], and reasoning [17,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' To train foundation models on ultra-large datasets and model sizes, custom software systems such as Megatron-LM [15], DeepSpeed [23], ColossalAI [1] have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These systems are built on paralleliza- tion techniques such as tensor parallelism [15], pipeline paral- lelism [7,14] and ZeRO redundant optimizer [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Tensor parallelism is critical for the efficiency of training giant foun- dation models, but it has been rarely studied systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP provides an adaptive tensor parallelism framework to improve training efficiency on complicated interconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 10 M1 M2 M3 M4 0 5 10 Achieved teraFLOP/s per GPU (a) IC1 M1 M2 M3 M4 0 25 50 75 100 Achieved teraFLOP/s per GPU (b) IC2 M1 M2 M3 M4 0 50 100 150 Achieved teraFLOP/s per GPU (c) IC3 M2 M3 M4 0 5 10 Achieved teraFLOP/s per GPU ATP-1 ATP-2 ATP-4 ATP-8 (d) IC4 Figure 11: Performance comparison for ATP with different device meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP-i refers to ATP with DeviceMesh(N/i,i) (N devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For IC1,2,3 with 8 devices, we compare ATP-1/2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' For IC4 with 16 devices, we compare ATP-1/2/4/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2 4 8 16 32 64 128 The number of devices 2D 4D 7D Tcomm ATP-1 (Megatron-LM) ATP-2 ATP-4 ATP-OPT Figure 12: Communication time of IC5,6 with increasing num- ber of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP-i refers to DeviceMesh(N/i,i) and ATP- OPT refers to the optimal device mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Tensor Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Megatron-LM introduced tensor par- allelism for training GPT and BERT models, making it a standard technique for training foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Several recent works, such as 2D tensor parallelism [32] and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism [30], use the Scalable Universal Matrix Multiplication Algorithm (SUMMA) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2D tensor paral- lelism splits both activations and weights, and has a smaller memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='5D tensor parallelism extends the 2D approach to support an arbitrary number of GPUs, but still suffers from high communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Other works such as Sagemaker [13] and GSPMD [33] split activations along the hidden dimension, but also have inefficient communica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, none of these approaches consider complicated GPU interconnects, leaving room for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP ad- dresses this by proposing a search space of tensor parallelism strategies and using a hierarchical communication matrix to estimate the communication cost of each strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP is more adaptable and outperforms other tensor parallelism methods in some environments with complicated GPU inter- connects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Automatic Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Recent work has proposed meth- ods for optimizing distributed training through automatic parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' These methods, such as Alpa [37], Unity [28], GSPMD [33], and Whale [11], can find the optimal hybrid parallelization strategy within a certain search space rather than relying on manually-designed strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' However, these existing approaches only support regular tensor parallelism and simple performance model of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP offers a more general tensor Parallelism search space for automatic parallelism and can also better characterize the communica- tion ability of complicated interconnects through the use of hierarchical communication matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' This can improve the ac- curacy of these automatic parallelism methods in estimating communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 7 Conclusion In this work, we present Adaptive Tensor Parallelism (ATP) for foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' We propose column- and row-first tensor parallelism based on 2D device meshes, and use the hierarchical communication matrix to identify the optimal strategy with minimal communication cost on different inter- connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Additionally, we use chunk-based overlapping to reduce the communication overhead introduced by ATP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP outperforms state-of-the-art approaches for various intercon- nects and model sizes and shows theoretical improvement for some topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ATP can be implemented as a replacement for the current state-of-the-art tensor parallelism approach, providing improved performance, scalability, and adaptability to different (future) topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 11 References [1] Zhengda Bian, Hongxin Liu, Boxiang Wang, Haichen Huang, Yongbin Li, Chuan-Qing Wang, Fan Cui, and Yang You.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Colossal-ai: A unified deep learning system for large-scale parallel training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ArXiv, abs/2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='14883, 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Codella, Xiyang Dai, Jianfeng Gao, Houdong Hu, Xue- dong Huang, Boxin Li, Chunyuan Li, Ce Liu, Mengchen Liu, Zicheng Liu, Yumao Lu, Yu Shi, Lijuan Wang, Jian- feng Wang, Bin Xiao, Zhen Xiao, Jianwei Yang, Michael Zeng, Luowei Zhou, and Pengchuan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Florence: A new foundation model for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ArXiv, abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='11432, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' [36] Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher De- wan, Mona Diab, Xian Li, Xi Victoria Lin, Todor Mi- haylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel Simig, Punit Singh Koura, Anjali Sridhar, Tianlu Wang, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Opt: Open pre-trained trans- former language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' ArXiv, abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content='01068, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' [37] Lianmin Zheng, Zhuohan Li, Hao Zhang, Yonghao Zhuang, Zhifeng Chen, Yanping Huang, Yida Wang, Yuanzhong Xu, Danyang Zhuo, Joseph Gonzalez, and Ion Stoica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Alpa: Automating inter- and intra- operator parallelism for distributed deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' In USENIX Symposium on Operating Systems Design and Implementation, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' [38] Dimitrios Ziakas, Allen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Baum, Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Maddox, and Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Safranek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' Intel® quickpath intercon- nect architectural features supporting scalable system architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 2010 18th IEEE Symposium on High Performance Interconnects, pages 1–6, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf'} diff --git a/CNE4T4oBgHgl3EQfFgxh/content/tmp_files/2301.04886v1.pdf.txt b/CNE4T4oBgHgl3EQfFgxh/content/tmp_files/2301.04886v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b81ddd061527dbffef4c0d9ccb79315ad5d58ed5 --- /dev/null +++ b/CNE4T4oBgHgl3EQfFgxh/content/tmp_files/2301.04886v1.pdf.txt @@ -0,0 +1,1918 @@ +arXiv:2301.04886v1 [math.PR] 12 Jan 2023 +RANDOM VECTORS ON THE SPIN CONFIGURATION OF A +CURIE-WEISS MODEL ON ERDŐS-RÉNYI RANDOM GRAPHS +DOMINIK R. BACH +Max Planck UCL Centre for Computational Psychiatry and Ageing Research, University College London, UK +Abstract. This article is concerned with the asymptotic behaviour of random +vectors in a diluted ferromagnetic model. We consider a model introduced by +Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erdős- +Rényi random graph. +Here, directed connections between graph nodes are +uniformly drawn at random with a probability p that depends on the number +of nodes N and is allowed to go to zero in the limit. If Np −→ ∞ in this +model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, +and Kabluchko et al. +(2020) proved central limit theorems in probability. +Here, we generalise these results for β < 1 in the regime Np −→ ∞. We show +that all those random vectors on the spin configuration that have a limiting +distribution under the Curie-Weiss model converge weakly towards the same +distribution under the diluted model, in probability on graph realisations. This +generalises various results from the Curie-Weiss model to the diluted model. +As a special case, we derive a law of large numbers and central limit theorem +for two disjoint groups of spins. +Date: 14 March 2022. +Key words and phrases. Ising model, dilute Curie-Weiss model, Law of Large numbers, Central +Limit Theorem, random graphs. +The author would like to thank Prof. Werner Kirsch, FernUniversität in Hagen, Germany, for +critical support and guidance through this work. +1 + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +2 +1. Introduction +1.1. Background. Diluted ferromagnetic models might be taken to describe the +behaviour of a quenched alloy of ferromagnetic and a non-magnetic material [7, 8], +or the behaviour of voters who interact at random [14]. +The study of diluted +ferromagnetic models has historically focused on Ising type models (e.g. [4, 5, 8, 7]). +Bovier & Gayrard (1993) [2] introduced a diluted version of the Curie-Weiss model +on directed Erdős-Rényi random graphs. +They proved a law of large numbers +almost surely. For a very similar model, recent work by Kabluchko et al. (2020) +proved a central limit theorem in probability over graph configurations, for the case +where the external magnetic field h = 0 and β < 1, provided that N 2p3 −→ ∞ +[10]. In [12], they sharpened the approximations to the less tight regime Np −→ ∞ +for β < 1, and derived a central limit theorem for β = 1. In [11], they derived +(conditional) central limit theorems for the the cases β > 0 and external magnetic +field h > 0. Some physical properties of the model have been analysed in [9, 3], and +fluctuations of the partition function over graph configurations in [13]. The goal +of this note is to generalise the central limit theorem for Np −→ ∞ and β < 1 to +a wide set of random vectors, including a law of large numbers and central limit +theorem for the homogenous two-group case [14]. +1.2. Description of the model. We consider spin configurations on a sequence +of directed Erdős-Rényi random graphs with N = 1, 2, ... nodes. Each node i ∈ NN +can take spin xi ∈ {−1, +1} such that xN := (x1, x2, ..., xN) ∈ XN := {−1, +1}N. +The presence of a directed edge from node i to node j in graph N is denoted by +the indicator variable εN,i,j ∈ {0, 1}. Let p : N → ]0, 1] be an arbitrary function. +We define a product probability space over the entire graph sequence: +(1.1) +(Ωε, Aε, Pε) := + + × +(N,i,j)∈N3 +{0, 1} , +� +(N,i,j)∈N3 +P ({0, 1}) , +� +(N,i,j)∈N3 +PN,i,j + + +with PN,i,j (εN,i,j = 1) := p (N). For every fixed graph realisation, we define two +probability measures over (XN, P (XN)), corresponding to the well-known Curie- +Weiss model, and the Bovier-Gayrard model, with: +(1.2) +P(CW) +N +(x) := µ(CW) +N +(x) /Z(CW) +N +:= e +β +2N +N +� +i,j +xixj +/ +� +x∈XN +µ(CW) +N +(x) . +(1.3) +P(BG) +N +(x) := µ(BG) +N +(x) /Z(BG) +N +:= e +β +2Np +N +� +i,j +εN,i,jxixj +/ +� +x∈XN +µ(BG) +N +(x) . +We denote the total magnetisation (sum of all spins) of the graph with sN (xN); +recall that s2 +N (xN) = +N +� +i,j +xixj. We use x, x1, etc. when the number of elements N +is clear from the context. For the remainder of the paper, we assume β < 1 and +Np −→ ∞ as N −→ ∞. Furthermore, we define (fN)N∈N, fN : XN −→ R with +|fN| < M ∈ R for all N ∈ N. We denote the positive and negative parts of fN with +f + +N > 0 and f − +N > 0. + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +3 +2. Main results +Denote weak convergence with =⇒, and Pε-stochastic convergence with +Pε +−→. For +two probability measures P1 and P2, denote the Levy metric of weak convergence +with dL (P1, P2). Recall that (Pn =⇒ P) ⇔ (dL (Pn, P) −→ 0) (e.g. [6]). +Theorem 1. Let (YN)N∈N with YN : XN −→ Rd, d ∈ N, have limiting image +distribution P(CW) +Y +such that +� +P(CW) +N +◦ Y −1 +N +� +=⇒ P(CW) +Y +. Then +dL +� +P(BG) +N +◦ Y −1 +N , P(CW) +Y +� +Pε +−→ 0. +In shorthand notation, we write: +YN +Pε +=⇒ +BG P(CW) +Y +. +Theorem 2. Let s(1) +N (xN) and s(2) +N (xN) be the respective sums of two disjoint +subsets of spins with respective cardinality NN,1 and NN,2. +(1) Law of large numbers. +Let m (β) be the unique positive solution to x = +tanh (βx). Then +� +1 +NN,1 +s(1) +N (xN) , +1 +NN,2 +s(2) +N (xN) +� +Pε +=⇒ +BG +1 +2 +� +δ(−m(β),−m(β)) + δ(+m(β),+m(β)) +� +, +(2) Central limit theorem. Assume existence of +α1 := lim +N→∞ +NN,1 +N +, +α2 := lim +N→∞ +NN,2 +N +, +and define +C := +� +1 + α1 +β +1−β +√α1α2 +β +1−β +√α1α2 +β +1−β +1 + α2 +β +1−β +� +, +then: +� +1 +� +NN,1 +s(1) +N (xN) , +1 +� +NN,2 +s(2) +N (xN) +� +Pε +=⇒ +BG ξ, ξ ∼ N ((0, 0) , C) . +Proof. This theorem follows directly from theorem 1 and the results for the Curie- +Weiss model proven in [14]. +□ +3. Technical preparation +3.1. Results from previous work. The following definition and lemma are adapted +from [12]. In this reference, m is fixed at m = 1/5, but it is easy to show that the +results hold for any fixed 0 < m < 1. +Part (1) restates Lemmata 3.1 - 3.2 in +[12]. Part (2) is implicit in their proofs and made explicit here for clarity. For a +random variable G : Ωε → R and fixed N, we define its expectation over graph +configurations with Eε,N (G). +Definition 3. For fixed 0 < m < 1, define the following sets of “typical” spin +configurations, and pairs of spin configurations: +XT,N := +� +x ∈ XN : s2 +N (x) ≤ N (Np)m� +, + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +4 +X (2) +T,N := + + +(x1, x2) ∈ X 2 +N : s2 +N (x1) ≤ N (Np)m , s2 +N (x2) ≤ N (Np)m , +� N +� +i=1 +x1,ix2,i +�2 +≤ N (Np)m + + + . +Lemma 4. (1) For fixed xN ∈ XT,N and (xN,1, xN,2) ∈ X (2) +T,N: +Eε,N + + + + + + +µ(BG) +N +(xN) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + += exp +� +−β2 +8 + β +2N s2 +N (x) + cN,1 +� +, +Eε,N + + + + + + +µ(BG) +N +(xN,1) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� +µ(BG) +N +(xN,2) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + += exp +� +−β2 +4 + β +2N +� +s2 +N (xN,1) + s2 +N (xN,2) +� ++ cN,2 +� +, +where (cN,1)N∈N and (cN,2)N∈N are null sequences of real numbers that do not +depend on xN, xN,1, xN,2. +(2) For fixed xN, xN,1, xN,2 ∈ XN: +Eε,N + + + + + + +µ(BG) +N +(xN) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + += exp +�β2 +8 + β +2N s2 +N (xN) (1 + o (1)) + o (1) +� +Eε,N + + + + + + +µ(BG) +N +(xN,1) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� +µ(BG) +N +(xN,2) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + += exp + +−β2 +4 + o (1) + +� β +2N + o (1) +� � +s2 +N (xN,1) + s2 +N (xN,2) +� ++ o (1) +2N +� N +� +i +xi,1xi,2 +�2 + . +The following lemma is proven as corollary 4.15 in [6]: +Lemma 5. Let (Ω, A, P) be a probability space, let Yn, Y : (Ω, A) → (R, B) be +real-valued random variables for all n ∈ N. Then the following two statements are +equivalent: +1. Yn +P +−→ Y . +2. Every subsequence of (Yn)n∈N has a subsequence that converges to Y almost +surely. + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +5 +3.2. Technical preparation for the proofs of proposition 12. The following +lemmata and proofs generalise the approach taken in [10, 12]. +Definition 6. SN := {−N, −N + 2, .., N − 2, N}. +Lemma 7. Let δ ∈ R+. Then: +(3.1) +� +x∈(XT,N)c +f + +N (x) e +1−δ +2N s2 +N = 2No (1) . +Proof. For fixed s ∈ SN, let νN,s := |{x ∈ XN : sN = s}|, i.e. the number of spin +configurations with s+N +2 +positive spins. By the de Moivre-Laplace local limit the- +orem: +(3.2) +νN,s = +� +N +s+N +2 +� += 2N (1 + o (1)) +� +1 +2πN +exp +� +− s2 +2N +� +. +Recalling that f + +N (x) < M, we have: +(3.3) +� +x∈(XT,N)c +f + +N (x) e +1−δ +2N s2 +N +(3.4) +≤ 2N+1 (1 + o (1)) +� +1 +2πN +M +� +� +s∈SN:√ +N(Np)m≤s≤N +�e−δ s2 +2N +(by eq. (3.2)) +(3.5) +≤ 2N+1 M +√ +N +� +� +s∈SN:√ +N(Np)m≤s≤N +�e−δ s2 +2N , +for large enough N. Now for t ∈ +� +s/ +√ +N, (s + 2) / +√ +N +� +: +(3.6) +e−δ (s+2)2 +2N +≤ e−δ t2 +2 , +and so +(3.7) +2 +√ +N +e−δ (s+2)2 +2N +≤ +� (s+2)/ +√ +N +s/ +√ +N +e−δ t2 +2 dt. +Hence, continuing from eq. (3.5): +(3.8) +2N+1 M +√ +N +� +� +s∈SN:√ +N(Np)m≤s≤N +�e−δ s2 +2N +(3.9) +≤ 2NM +� (N+2)/ +√ +N +√ +(Np)m +e−δ t2 +2 dt + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +6 +(3.10) +≤ 2NM +� ∞ +√ +(Np)m e−δ t2 +2 dt +(3.11) += 2No (1) . +□ +Lemma 8. Let δ ∈ R+. Then: +(3.12) +� +(x1,x2)∈ +� +X (2) +T,N +�c +f + +N (x1) f + +N (x2) e +1−δ +2N +� +s2 +N (x1)+s2 +N (x2)+ +� N +� +i=1 +xi,1xi,2 +�2� +≤ +� +2No (1) +�2 . +Proof. For s, t, u ∈ ZN, define νN (s, t, u) := +���� +� +(x1, x2) ∈ X 2 +N : sN (x1) = s, sN (x2) = t, +N +� +i=1 +xi,1xi,2 = u +����� . +Then there exists a constant C ∈ R such that νN (s, t, u) < 22N +C +N 3/2 exp +� +− s2 +2N − t2 +2N − u2 +2N +� +(proof in [12]). Now let WN := +� +(s, t, u) ∈ S3 +N : s ≥ +� +N (Np)m ∨ t ≥ +� +N (Np)m ∨ u ≥ +� +N (Np)m� +. +Then: +(3.13) +� +(x1,x2)∈ +� +X (2) +T,N +�c +f + +N (x1) f + +N (x2) e +1−δ +2N +� +s2 +N (x1)+s2 +N(x2)+ +� N +� +i=1 +xi,1xi,2 +�2� +(3.14) +≤ 2M 2 +� +(s,t,u)∈WN +νN (s, t, u) e +1−δ +2N (s2+t2+u2) +(3.15) +< M 222N+1 +C +N 3/2 +� +(s,t,u)∈WN +e− +δ +2N (s2+t2+u2) +(3.16) += +� +2No (1) +�2 +(using eq. (3.7) for s, t, and u and expanding the range of integration as in eq. +(3.10)). +□ +Lemma 9. There exists some C ∈ R+ independent of N such that +Z(CW) +N +≥ 2NC. +Proof. We have: +(3.17) +Z(CW) +N += +� +x∈XN +e +β +2N s2 +N(x) +(3.18) +≥ 2N +√ +N +� +{s∈SN :0≤s≤N} +e(β−1) s2 +2N , +for large enough N, using eq. +(3.2). +Because β − 1 < 0, we have, for t ∈ +� +s/ +√ +N, (s + 2) / +√ +N +� +: + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +7 +(3.19) +e(β−1) s2 +2N ≥ e(β−1) t2 +2 , +and so +(3.20) +2 +√ +N +e(β−1) s2 +2N ≥ +� (s+2)/ +√ +N +s/ +√ +N +e(β−1) t2 +2 dt. +Hence, continuing from eq. (3.18): +(3.21) +2N +√ +N +� +{s∈SN:0≤s≤N} +e(β−1) s2 +2N +(3.22) +≥ 2N−1 +� (N+2)/ +√ +N +0 +e(β−1) t2 +2 dt +(3.23) +≥ 2N 1 +4 +� +π +2 (1 − β), +for large enough N, since erf (x) −→ 1 > 1/2 for x −→ ∞. +□ +4. Proof of the main results +4.1. Convergence of integral quotients. +4.1.1. Definitions and propositions. +Definition 10. For ease of notation, we define +(4.1) +E(µ,BG) +N +(YN) := +� +x∈XN +YN (x) µ(BG) +N +(x) , +(4.2) +E(P,BG) +N +(YN) := +� +x∈XN +YN (x) P(BG) +N +(x) = E(µ,BG) +N +(YN) /E(µ,BG) +N +(1) , +and similarly for E(µ,CW) +N +(YN) and E(P,CW) +N +(YN). Furthermore, we define +RN +� +f + +N +� := + + + + + + + +E(µ,BG) +N +(f + +N) +cosh( +β +2Np) exp +� +− β2 +8 + +N +� +i,j +εN,i,j +� +E(µ,CW ) +N +(f + +N) +� +E(µ,CW) +N +� +f + +N +� +̸= 0 +� +1 +� +E(µ,CW) +N +� +f + +N +� += 0 +� +TN +� +f + +N +� := +E(µ,BG) +N +� +f + +N +� +cosh +� +β +2Np +� +exp +� +− β2 +8 + +N +� +i,j +εN,i,j +� +Z(µ,CW) +N +Remark 11. E(µ,CW) +N +� +f + +N +� += 0 if and only if f + +N vanishes everywhere on XN, in +which case, E(µ,BG) +N +� +f + +N +� += E(µ,CW) +N +� +f + +N +� += 0. + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +8 +Proposition 12. +(1) If +lim +N→∞E(P,CW) +N +� +f + +N +� +> 0 exists, then RN +� +f + +N +� +Pε +−→ 1. +(2) If +lim +N→∞E(P,CW) +N +� +f + +N +� += 0, then TN +� +f + +N +� +Pε +−→ 0. +4.1.2. Preparation: expanding the expectations. +Lemma 13. If lim +N→∞E(P,CW) +N +� +f + +N +� +> 0 exists, then there exists a sequence (aN,1)N∈N +with aN,1 −→ 1 as Np −→ ∞, such that Eε,N +� +RN +� +f + +N +�� +≥ aN,1. +Proof. We split into typical and atypical spin configurations for some fixed 0 < +m < 1: +(4.3) +Eε,N + + + + + + +E(µ,BG) +N +� +f + +N +� +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + += Eε,N + + + + + + +� +x∈XN +f + +N (x) +µ(BG) +N +(x) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + +(4.4) +≥ +� +x∈XT,N +f + +N (x) Eε,N + + + + + + +µ(BG) +N +(x) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + +(because all terms are non-negative) +(4.5) += +� +x∈XT,N +f + +N (x) exp +� +−β2 +8 + β +2N s2 +N (x) + cN,1 +� +, +(by lemma 4) +(4.6) +≥ exp (− |cN,1|) e− β2 +8 +� +x∈XT,N +f + +N (x) e +β +2N s2 +N (x) +(4.7) += exp (− |cN,1|) e− β2 +8 E(µ,CW) +N +� +f + +N +� + + + +1 − +� +x∈X c +T,N +f + +N (x) e +β +2N s2 +N +E(P,CW) +N +� +f + +N +� +Z(CW) +N + + + + +(4.8) +≥ exp (− |cN,1|) e− β2 +8 E(µ,CW) +N +� +f + +N +� +(1 − o (1)) , +(for large enough N, using proof assumptions, lemma 7 with (1 − δ) = β < 1, +and lemma 9) +(4.9) += aN,1e− β2 +8 E(µ,CW) +N +� +f + +N +� +, +(4.10) +aN,1 := exp (− |cN,1|) (1 − o (1)) −→ 1, +Np −→ ∞. + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +9 +The lemma follows with definition 10. +□ +Lemma 14. If lim +N→∞E(P,CW) +N +� +f + +N +� +> 0 exists, then there exists a sequence (aN,2)N∈N +with aN,2 −→ 1 as Np −→ ∞, such that Eε,N +� +R2 +N +� +f + +N +�� +≤ aN,2. +Proof. +(4.11) +Eε,N + + + + + + + + + + + + +E(µ,BG) +N +� +f + +N +� +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + +2 + + + + + += Eε,N + + + + + + + + + + + + +� +x∈XN +f + +N (x) +µ(BG) (x) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + +2 + + + + + +(4.12) += +� +(x1,x2)∈X 2 +N +f + +N (x1) f + +N (x2) Eε,N + + + + + + +µ(BG) (x1) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� +µ(BG) (x2) +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + += +� +(x1,x2)∈X (2) +T,N +f + +N (x1) f + +N (x2) exp +� +−β2 +4 + β +2N +� +s2 +N (x1) + s2 +N (x2) +� ++ cN,2 +� ++ +� +(x1,x2)∈ +� +X (2) +T,N +�c +f + +N (x1) f + +N (x2) +(4.13) +·exp + +−β2 +4 + o (1) + +� β +2N + o (1) +� � +s2 +N (x1) + s2 +N (x2) +� ++ o (1) +2N +� N +� +i +xi,1xi,2 +�2 + , +by lemma 4. Because β < 1, there exists a δ ∈ R+ such that for large enough N: +� β +2N + o (1) +� � +s2 +N (x1) + s2 +N (x2) +� ++ o (1) +2N +� N +� +i +xi,1xi,2 +�2 +(4.14) +≤ (1 − δ) +2N + +s2 +N (x1) + s2 +N (x2) + +� N +� +i +xi,1xi,2 +�2 + . +Using this inequality, we have, for the atypical spin configurations and large +enough N: +� +(x1,x2)∈ +� +X (2) +T,N +�c +f + +N (x1) f + +N (x2) +(4.15) +· exp + +−β2 +4 + o (1) + +� β +2N + o (1) +� � +s2 +N (x1) + s2 +N (x2) +� ++ o (1) +2N +� N +� +i +xi,1xi,2 +�2 + + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +10 +(4.16) +≤ +� +2No (1) +�2 , +by lemma 8. +Inserting this into eq. (4.13) yields: +(4.17) +Eε,N + + + + + + + + + + + + +E(µ,BG) +N +� +f + +N +� +cosh +� +β +2Np +� +exp +� +N +� +i,j +εN,i,j +� + + + + + + +2 + + + + + +≤ exp (|cN,2|) e− β2 +4 +� +(x1,x2)∈X 2 +T,N +f + +N (x1) f + +N (x2) exp +�β +2 +�s2 +N (x1) + s2 +N (x2) +N +�� +(4.18) ++ +� +2No (1) +�2 +(4.19) +≤ exp (|cN,2|) e− β2 +4 +� � +x∈XN +f + +N (x) e +β +2N s2 +N(x) +�2 ++ +� +2No (1) +�2 +(4.20) += exp (|cN,2|) +� +e− β2 +8 E(µ,CW) +N +� +f + +N +��2 + +1 + +� +o (1) 2N +e− β2 +8 E(P,CW) +N +� +f + +N +� +Z(CW) +N +�2 + +(4.21) +≤ exp (|cN,2|) +� +e− β2 +8 E(µ,CW) +N +� +f + +N +��2 +(1 + o (1)) +(for large enough N, using proof assumptions and lemma 9) +(4.22) += aN,2 +� +e− β2 +8 E(µ,CW) +N +� +f + +N +��2 +, +with +(4.23) +aN,2 := exp (|cN,2|) (1 + o (1)) −→ 1, +Np −→ ∞. +□ +4.1.3. Proof of proposition 12, part 1. +Proof. We consider fixed δ > 0 and fixed N. By Markov’s inequality: +(4.24) +Pε +�� +ω ∈ Ωε : +��RN +� +f + +N +� +− 1 +�� > δ +�� +(4.25) +≤ δ−2Eε,N +�� +RN +� +f + +N +� +− 1 +�2� + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +11 +(4.26) +≤ δ−2 (aN,2 − 2aN,1 + 1) −→ 0, +as Np −→ ∞, using lemmata 13 and 14. +□ +4.1.4. Proof of proposition 12, part 2. +Proof. We consider fixed δ > 0 and fixed N. By Markov’s inequality: +(4.27) +Pε +�� +ω ∈ Ωε : +��TN +� +f + +N +��� > δ +�� +(4.28) +≤ δ−2Eε,N +� +T 2 +N +� +f + +N +�� +(4.29) += δ−2Eε,N + + +� +RN +� +f + +N +� E(µ,CW) +N +� +f + +N +� +Z(µ,CW) +N +�2 + +(4.30) += δ−2 � +E(P,CW) +N +� +f + +N +��2 +Eε,N +� +R2 +N +� +f + +N +�� +(4.31) +≤ δ−2 � +E(P,CW) +N +� +f + +N +��2 +aN,2 −→ 0, +as Np −→ ∞, using lemma 14 and proof assumptions. +□ +4.2. Convergence of bounded integrals. +Proposition 15. Assume that lim +N→∞E(P,CW) +N +� +f + +N +� +< ∞ and lim +N→∞E(P,CW) +N +� +f − +N +� +< +∞ exist. Then: +E(P,BG) +N +(fN) +Pε +−→ lim +N→∞E(P,CW) +N +(fN) . +Proof. First, we consider the case lim +N→∞E(P,CW) +N +� +f + +N +� +> 0 and lim +N→∞E(P,CW) +N +� +f − +N +� +> +0. +By lemmata 12 and 5, every subsequence of N has subsequences +� +n(1) +i +� +i∈N, +� +n(2) +i +� +i∈N, +� +n(3) +i +� +i∈N such that Rn(1) +i +(1) −→ 1, Rn(2) +i +� +f + +N +� +−→ 1 and Rn(3) +i +� +f − +N +� +−→ +1 almost surely. Now let (ni)i∈N := +� +n(1) +i +� +i∈N ∩ +� +n(2) +i +� +i∈N ∩ +� +n(3) +i +� +i∈N ̸= Ø (by +lemma 5). It is easy to see that almost surely, +(4.32) +lim +i→∞E(P,BG) +ni +� +f + +ni +� += lim +i→∞E(P,CW) +ni +� +f + +ni +� += lim +N→∞E(P,CW) +N +� +f + +N +� +, +similarly for f − +N , and by additivity of the expectation, also +lim +i→∞E(P,BG) +ni +(fni) = lim +N→∞E(P,CW) +N +(fN) . +By lemma 5, it follows that +E(P,BG) +N +(fN) +Pε +−→ lim +N→∞E(P,CW) +N +(fN) . +If lim +i→∞E(P,CW) +N +� +f + +N +� += 0, or lim +i→∞E(P,CW) +N +� +f + +N +� += 0, or both, the same result +holds. +□ + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +12 +4.3. Convergence in distribution (theorem 1). +Proof. We first consider d = 1. Let h ∈ Cb (R) and fN (x) := h ◦ YN (x). We have: +f + +N = h+ ◦YN and f − +N = h− ◦YN. Because f + and f − are bounded and by theorem +assumptions: +(4.33) +lim +N→∞E(P,CW) +N +� +f + +N +� +< ∞, +lim +N→∞E(P,CW) +N +� +f − +N +� +< ∞ +both exist. Using proposition 15, it follows that +� +R +h (x) d +� +P(BG) +N +◦ Y −1 +N +� += +� +x∈XN +h ◦ YN (x) P(BG) +N +(x) += E(P,BG) +N +(fN) +Pε +−→ lim +N→∞E(P,CW) +N +(fN) = +� +R +h (x) dP(CW). +This holds for any h ∈ Cb (R). We note that the mapping Ωε �→ +� +P(BG) +N +◦ Y −1 +N +� +is a random probability measure and Rd is a Polish space. With this and [1], it +follows that +dL +� +P(BG) +N +◦ Y −1 +N , P(CW)� +Pε +−→ 0. +For d > 1, we use h ∈ Cb +� +Rd� +and replace h◦YN (x) with h◦ +� +Y (1) +N , Y (2) +N , .., Y (d) +N +� +(x). +□ +References +[1] Patrizia Berti, Luca Pratelli, and Pietro Rigo. Almost sure weak convergence of random +probability measures. Stochastics, 78(2):91–97, apr 2006. +[2] Anton Bovier and Véronique Gayrard. The thermodynamics of the curie-weiss model with +random couplings. Journal of statistical physics, 72(3-4):643–664, 1993. +[3] Elena Agliari; Adriano Barra; Federico Camboni. Criticality in diluted ferromagnets. Journal +of Statistical Mechanics: Theory and Experiment, P10003, 2008. +[4] JT Chayes, L Chayes, and J Fröhlich. The low-temperature behavior of disordered magnets. +Communications in mathematical physics, 100(3):399–437, 1985. +[5] AG Dunn, JW Essam, and JM Loveluck. Scaling theory for the pair-connectedness in perco- +lation models. Journal of Physics C: Solid State Physics, 8(6):743, 1975. +[6] Jürgen Elstrodt. Maß- und Integrationstheorie. Springer Berlin Heidelberg, 2011. +[7] Hans-Otto Georgii. Spontaneous magnetization of randomly dilute ferromagnets. Journal of +Statistical Physics, 25(3):369–396, 1981. +[8] Robert B Griffiths and JL Lebowitz. Random spin systems: some rigorous results. Journal +of Mathematical Physics, 9(8):1284–1292, 1968. +[9] Luca De Sanctis; Francesco Guerra. Mean field dilute ferromagnet: High temperature and +zero temperature behavior. Journal of Statistical Physics, 132:759–785, 2008. +[10] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert. Fluctuations of the magnetiza- +tion for ising models on dense erdős–rényi random graphs. Journal of Statistical Physics, +177(1):78–94, 2019. +[11] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert. Fluctuations of the magnetization +for ising models on erdős-rényi random graphs – the regimes of low temperature and external +magnetic field. pre-print on https://arxiv.org/abs/2012.08204v2, 2020. +[12] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert. Fluctuations of the magnetiza- +tion for ising models on erdős–rényi random graphs—the regimes of small p and the critical +temperature. Journal of Physics A: Mathematical and Theoretical, 53(35):355004, aug 2020. + +RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS +13 +[13] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert. Fluctuations for the partition +function of ising models on erdös-rényi random graphs. Ann. Inst. H. Poincaré Probab. +Statist., 57(4):2017–2042, 2021. +[14] Werner Kirsch and Gabor Toth. Two groups in a curie-weiss model. Mathematical Physics, +Analysis and Geometry, 23(2), may 2020. +Max Planck UCL Centre for Computational Psychiatry and Ageing Research, +University College London, United Kingdom + diff --git a/CNE4T4oBgHgl3EQfFgxh/content/tmp_files/load_file.txt b/CNE4T4oBgHgl3EQfFgxh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30f79f7ada2936ed7144408be162cb85e1a6e481 --- /dev/null +++ b/CNE4T4oBgHgl3EQfFgxh/content/tmp_files/load_file.txt @@ -0,0 +1,378 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf,len=377 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='04886v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='PR] 12 Jan 2023 RANDOM VECTORS ON THE SPIN CONFIGURATION OF A CURIE-WEISS MODEL ON ERDŐS-RÉNYI RANDOM GRAPHS DOMINIK R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' BACH Max Planck UCL Centre for Computational Psychiatry and Ageing Research, University College London, UK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erdős- Rényi random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' If Np −→ ∞ in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (2020) proved central limit theorems in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Here, we generalise these results for β < 1 in the regime Np −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' This generalises various results from the Curie-Weiss model to the diluted model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Date: 14 March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Ising model, dilute Curie-Weiss model, Law of Large numbers, Central Limit Theorem, random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The author would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Werner Kirsch, FernUniversität in Hagen, Germany, for critical support and guidance through this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' 1 RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Diluted ferromagnetic models might be taken to describe the behaviour of a quenched alloy of ferromagnetic and a non-magnetic material [7, 8], or the behaviour of voters who interact at random [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The study of diluted ferromagnetic models has historically focused on Ising type models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' [4, 5, 8, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Bovier & Gayrard (1993) [2] introduced a diluted version of the Curie-Weiss model on directed Erdős-Rényi random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' They proved a law of large numbers almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For a very similar model, recent work by Kabluchko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (2020) proved a central limit theorem in probability over graph configurations, for the case where the external magnetic field h = 0 and β < 1, provided that N 2p3 −→ ∞ [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' In [12], they sharpened the approximations to the less tight regime Np −→ ∞ for β < 1, and derived a central limit theorem for β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' In [11], they derived (conditional) central limit theorems for the the cases β > 0 and external magnetic field h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Some physical properties of the model have been analysed in [9, 3], and fluctuations of the partition function over graph configurations in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The goal of this note is to generalise the central limit theorem for Np −→ ∞ and β < 1 to a wide set of random vectors, including a law of large numbers and central limit theorem for the homogenous two-group case [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Description of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We consider spin configurations on a sequence of directed Erdős-Rényi random graphs with N = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Each node i ∈ NN can take spin xi ∈ {−1, +1} such that xN := (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=', xN) ∈ XN := {−1, +1}N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The presence of a directed edge from node i to node j in graph N is denoted by the indicator variable εN,i,j ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let p : N → ]0, 1] be an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We define a product probability space over the entire graph sequence: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) (Ωε, Aε, Pε) := \uf8eb \uf8ed × (N,i,j)∈N3 {0, 1} , � (N,i,j)∈N3 P ({0, 1}) , � (N,i,j)∈N3 PN,i,j \uf8f6 \uf8f8 with PN,i,j (εN,i,j = 1) := p (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For every fixed graph realisation, we define two probability measures over (XN, P (XN)), corresponding to the well-known Curie- Weiss model, and the Bovier-Gayrard model, with: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) P(CW) N (x) := µ(CW) N (x) /Z(CW) N := e β 2N N � i,j xixj / � x∈XN µ(CW) N (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='3) P(BG) N (x) := µ(BG) N (x) /Z(BG) N := e β 2Np N � i,j εN,i,jxixj / � x∈XN µ(BG) N (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We denote the total magnetisation (sum of all spins) of the graph with sN (xN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' recall that s2 N (xN) = N � i,j xixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We use x, x1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' when the number of elements N is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For the remainder of the paper, we assume β < 1 and Np −→ ∞ as N −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Furthermore, we define (fN)N∈N, fN : XN −→ R with |fN| < M ∈ R for all N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We denote the positive and negative parts of fN with f + N > 0 and f − N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Main results Denote weak convergence with =⇒, and Pε-stochastic convergence with Pε −→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For two probability measures P1 and P2, denote the Levy metric of weak convergence with dL (P1, P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Recall that (Pn =⇒ P) ⇔ (dL (Pn, P) −→ 0) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let (YN)N∈N with YN : XN −→ Rd, d ∈ N, have limiting image distribution P(CW) Y such that � P(CW) N Y −1 N � =⇒ P(CW) Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then dL � P(BG) N Y −1 N , P(CW) Y � Pε −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' In shorthand notation, we write: YN Pε =⇒ BG P(CW) Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let s(1) N (xN) and s(2) N (xN) be the respective sums of two disjoint subsets of spins with respective cardinality NN,1 and NN,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (1) Law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let m (β) be the unique positive solution to x = tanh (βx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then � 1 NN,1 s(1) N (xN) , 1 NN,2 s(2) N (xN) � Pε =⇒ BG 1 2 � δ(−m(β),−m(β)) + δ(+m(β),+m(β)) � , (2) Central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Assume existence of α1 := lim N→∞ NN,1 N , α2 := lim N→∞ NN,2 N , and define C := � 1 + α1 β 1−β √α1α2 β 1−β √α1α2 β 1−β 1 + α2 β 1−β � , then: � 1 � NN,1 s(1) N (xN) , 1 � NN,2 s(2) N (xN) � Pε =⇒ BG ξ, ξ ∼ N ((0, 0) , C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' This theorem follows directly from theorem 1 and the results for the Curie- Weiss model proven in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Technical preparation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Results from previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The following definition and lemma are adapted from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' In this reference, m is fixed at m = 1/5, but it is easy to show that the results hold for any fixed 0 < m < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Part (1) restates Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Part (2) is implicit in their proofs and made explicit here for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For a random variable G : Ωε → R and fixed N, we define its expectation over graph configurations with Eε,N (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For fixed 0 < m < 1, define the following sets of “typical” spin configurations, and pairs of spin configurations: XT,N := � x ∈ XN : s2 N (x) ≤ N (Np)m� , RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 4 X (2) T,N := \uf8f1 \uf8f2 \uf8f3(x1, x2) ∈ X 2 N : s2 N (x1) ≤ N (Np)m , s2 N (x2) ≤ N (Np)m , � N � i=1 x1,ix2,i �2 ≤ N (Np)m \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (1) For fixed xN ∈ XT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='N and (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) ∈ X (2) T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='N: Eε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed µ(BG) N (xN) cosh � β 2Np � exp � N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j εN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = exp � −β2 8 + β 2N s2 N (x) + cN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Eε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed µ(BG) N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) cosh � β 2Np � exp � N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j εN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j � µ(BG) N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) cosh � β 2Np � exp � N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j εN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = exp � −β2 4 + β 2N � s2 N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) + s2 N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) � + cN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' where (cN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1)N∈N and (cN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2)N∈N are null sequences of real numbers that do not depend on xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (2) For fixed xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2 ∈ XN: Eε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed µ(BG) N (xN) cosh � β 2Np � exp � N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j εN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = exp �β2 8 + β 2N s2 N (xN) (1 + o (1)) + o (1) � Eε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed µ(BG) N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) cosh � β 2Np � exp � N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j εN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j � µ(BG) N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) cosh � β 2Np � exp � N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j εN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = exp \uf8eb \uf8ed−β2 4 + o (1) + � β 2N + o (1) � � s2 N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) + s2 N (xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) � + o (1) 2N � N � i xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2 �2\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The following lemma is proven as corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='15 in [6]: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let (Ω, A, P) be a probability space, let Yn, Y : (Ω, A) → (R, B) be real-valued random variables for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then the following two statements are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Yn P −→ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Every subsequence of (Yn)n∈N has a subsequence that converges to Y almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Technical preparation for the proofs of proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' The following lemmata and proofs generalise the approach taken in [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' SN := {−N, −N + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='., N − 2, N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let δ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) � x∈(XT,N)c f + N (x) e 1−δ 2N s2 N = 2No (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For fixed s ∈ SN, let νN,s := |{x ∈ XN : sN = s}|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' the number of spin configurations with s+N 2 positive spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' By the de Moivre-Laplace local limit the- orem: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) νN,s = � N s+N 2 � = 2N (1 + o (1)) � 1 2πN exp � − s2 2N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Recalling that f + N (x) < M, we have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='3) � x∈(XT,N)c f + N (x) e 1−δ 2N s2 N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='4) ≤ 2N+1 (1 + o (1)) � 1 2πN M � � s∈SN:√ N(Np)m≤s≤N �e−δ s2 2N (by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='5) ≤ 2N+1 M √ N � � s∈SN:√ N(Np)m≤s≤N �e−δ s2 2N , for large enough N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Now for t ∈ � s/ √ N, (s + 2) / √ N � : (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='6) e−δ (s+2)2 2N ≤ e−δ t2 2 , and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='7) 2 √ N e−δ (s+2)2 2N ≤ � (s+2)/ √ N s/ √ N e−δ t2 2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Hence, continuing from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='5): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='8) 2N+1 M √ N � � s∈SN:√ N(Np)m≤s≤N �e−δ s2 2N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='9) ≤ 2NM � (N+2)/ √ N √ (Np)m e−δ t2 2 dt RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 6 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='10) ≤ 2NM � ∞ √ (Np)m e−δ t2 2 dt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='11) = 2No (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let δ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='12) � (x1,x2)∈ � X (2) T,N �c f + N (x1) f + N (x2) e 1−δ 2N � s2 N (x1)+s2 N (x2)+ � N � i=1 xi,1xi,2 �2� ≤ � 2No (1) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For s, t, u ∈ ZN, define νN (s, t, u) := ���� � (x1, x2) ∈ X 2 N : sN (x1) = s, sN (x2) = t, N � i=1 xi,1xi,2 = u ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then there exists a constant C ∈ R such that νN (s, t, u) < 22N C N 3/2 exp � − s2 2N − t2 2N − u2 2N � (proof in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Now let WN := � (s, t, u) ∈ S3 N : s ≥ � N (Np)m ∨ t ≥ � N (Np)m ∨ u ≥ � N (Np)m� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='13) � (x1,x2)∈ � X (2) T,N �c f + N (x1) f + N (x2) e 1−δ 2N � s2 N (x1)+s2 N(x2)+ � N � i=1 xi,1xi,2 �2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='14) ≤ 2M 2 � (s,t,u)∈WN νN (s, t, u) e 1−δ 2N (s2+t2+u2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='15) < M 222N+1 C N 3/2 � (s,t,u)∈WN e− δ 2N (s2+t2+u2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='16) = � 2No (1) �2 (using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='7) for s, t, and u and expanding the range of integration as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' There exists some C ∈ R+ independent of N such that Z(CW) N ≥ 2NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We have: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='17) Z(CW) N = � x∈XN e β 2N s2 N(x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='18) ≥ 2N √ N � {s∈SN :0≤s≤N} e(β−1) s2 2N , for large enough N, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Because β − 1 < 0, we have, for t ∈ � s/ √ N, (s + 2) / √ N � : RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 7 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='19) e(β−1) s2 2N ≥ e(β−1) t2 2 , and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='20) 2 √ N e(β−1) s2 2N ≥ � (s+2)/ √ N s/ √ N e(β−1) t2 2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Hence, continuing from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='18): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='21) 2N √ N � {s∈SN:0≤s≤N} e(β−1) s2 2N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='22) ≥ 2N−1 � (N+2)/ √ N 0 e(β−1) t2 2 dt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='23) ≥ 2N 1 4 � π 2 (1 − β), for large enough N, since erf (x) −→ 1 > 1/2 for x −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof of the main results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Convergence of integral quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Definitions and propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For ease of notation, we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1) E(µ,BG) N (YN) := � x∈XN YN (x) µ(BG) N (x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2) E(P,BG) N (YN) := � x∈XN YN (x) P(BG) N (x) = E(µ,BG) N (YN) /E(µ,BG) N (1) , and similarly for E(µ,CW) N (YN) and E(P,CW) N (YN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Furthermore, we define RN � f + N � := \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 E(µ,BG) N (f + N) cosh( β 2Np) exp � − β2 8 + N � i,j εN,i,j � E(µ,CW ) N (f + N) � E(µ,CW) N � f + N � ̸= 0 � 1 � E(µ,CW) N � f + N � = 0 � TN � f + N � := E(µ,BG) N � f + N � cosh � β 2Np � exp � − β2 8 + N � i,j εN,i,j � Z(µ,CW) N Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' E(µ,CW) N � f + N � = 0 if and only if f + N vanishes everywhere on XN, in which case, E(µ,BG) N � f + N � = E(µ,CW) N � f + N � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 8 Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (1) If lim N→∞E(P,CW) N � f + N � > 0 exists, then RN � f + N � Pε −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (2) If lim N→∞E(P,CW) N � f + N � = 0, then TN � f + N � Pε −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Preparation: expanding the expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' If lim N→∞E(P,CW) N � f + N � > 0 exists, then there exists a sequence (aN,1)N∈N with aN,1 −→ 1 as Np −→ ∞, such that Eε,N � RN � f + N �� ≥ aN,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We split into typical and atypical spin configurations for some fixed 0 < m < 1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='3) Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed E(µ,BG) N � f + N � cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed � x∈XN f + N (x) µ(BG) N (x) cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='4) ≥ � x∈XT,N f + N (x) Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed µ(BG) N (x) cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (because all terms are non-negative) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='5) = � x∈XT,N f + N (x) exp � −β2 8 + β 2N s2 N (x) + cN,1 � , (by lemma 4) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='6) ≥ exp (− |cN,1|) e− β2 8 � x∈XT,N f + N (x) e β 2N s2 N (x) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='7) = exp (− |cN,1|) e− β2 8 E(µ,CW) N � f + N � \uf8eb \uf8ec \uf8ec \uf8ed1 − � x∈X c T,N f + N (x) e β 2N s2 N E(P,CW) N � f + N � Z(CW) N \uf8f6 \uf8f7 \uf8f7 \uf8f8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='8) ≥ exp (− |cN,1|) e− β2 8 E(µ,CW) N � f + N � (1 − o (1)) , (for large enough N, using proof assumptions, lemma 7 with (1 − δ) = β < 1, and lemma 9) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='9) = aN,1e− β2 8 E(µ,CW) N � f + N � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='10) aN,1 := exp (− |cN,1|) (1 − o (1)) −→ 1, Np −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 9 The lemma follows with definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' If lim N→∞E(P,CW) N � f + N � > 0 exists, then there exists a sequence (aN,2)N∈N with aN,2 −→ 1 as Np −→ ∞, such that Eε,N � R2 N � f + N �� ≤ aN,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='11) Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed E(µ,BG) N � f + N � cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 2\uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed � x∈XN f + N (x) µ(BG) (x) cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 2\uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='12) = � (x1,x2)∈X 2 N f + N (x1) f + N (x2) Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed µ(BG) (x1) cosh � β 2Np � exp � N � i,j εN,i,j � µ(BG) (x2) cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = � (x1,x2)∈X (2) T,N f + N (x1) f + N (x2) exp � −β2 4 + β 2N � s2 N (x1) + s2 N (x2) � + cN,2 � + � (x1,x2)∈ � X (2) T,N �c f + N (x1) f + N (x2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='13) exp \uf8eb \uf8ed−β2 4 + o (1) + � β 2N + o (1) � � s2 N (x1) + s2 N (x2) � + o (1) 2N � N � i xi,1xi,2 �2\uf8f6 \uf8f8 , by lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Because β < 1, there exists a δ ∈ R+ such that for large enough N: � β 2N + o (1) � � s2 N (x1) + s2 N (x2) � + o (1) 2N � N � i xi,1xi,2 �2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='14) ≤ (1 − δ) 2N \uf8eb \uf8eds2 N (x1) + s2 N (x2) + � N � i xi,1xi,2 �2\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Using this inequality, we have, for the atypical spin configurations and large enough N: � (x1,x2)∈ � X (2) T,N �c f + N (x1) f + N (x2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='15) exp \uf8eb \uf8ed−β2 4 + o (1) + � β 2N + o (1) � � s2 N (x1) + s2 N (x2) � + o (1) 2N � N � i xi,1xi,2 �2\uf8f6 \uf8f8 RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 10 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='16) ≤ � 2No (1) �2 , by lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Inserting this into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='13) yields: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='17) Eε,N \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed E(µ,BG) N � f + N � cosh � β 2Np � exp � N � i,j εN,i,j � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 2\uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ≤ exp (|cN,2|) e− β2 4 � (x1,x2)∈X 2 T,N f + N (x1) f + N (x2) exp �β 2 �s2 N (x1) + s2 N (x2) N �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='18) + � 2No (1) �2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='19) ≤ exp (|cN,2|) e− β2 4 � � x∈XN f + N (x) e β 2N s2 N(x) �2 + � 2No (1) �2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='20) = exp (|cN,2|) � e− β2 8 E(µ,CW) N � f + N ��2 \uf8eb \uf8ed1 + � o (1) 2N e− β2 8 E(P,CW) N � f + N � Z(CW) N �2\uf8f6 \uf8f8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='21) ≤ exp (|cN,2|) � e− β2 8 E(µ,CW) N � f + N ��2 (1 + o (1)) (for large enough N, using proof assumptions and lemma 9) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='22) = aN,2 � e− β2 8 E(µ,CW) N � f + N ��2 , with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='23) aN,2 := exp (|cN,2|) (1 + o (1)) −→ 1, Np −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof of proposition 12, part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We consider fixed δ > 0 and fixed N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' By Markov’s inequality: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='24) Pε �� ω ∈ Ωε : ��RN � f + N � − 1 �� > δ �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='25) ≤ δ−2Eε,N �� RN � f + N � − 1 �2� RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 11 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='26) ≤ δ−2 (aN,2 − 2aN,1 + 1) −→ 0, as Np −→ ∞, using lemmata 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof of proposition 12, part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We consider fixed δ > 0 and fixed N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' By Markov’s inequality: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='27) Pε �� ω ∈ Ωε : ��TN � f + N ��� > δ �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='28) ≤ δ−2Eε,N � T 2 N � f + N �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='29) = δ−2Eε,N \uf8eb \uf8ed � RN � f + N � E(µ,CW) N � f + N � Z(µ,CW) N �2\uf8f6 \uf8f8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='30) = δ−2 � E(P,CW) N � f + N ��2 Eε,N � R2 N � f + N �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='31) ≤ δ−2 � E(P,CW) N � f + N ��2 aN,2 −→ 0, as Np −→ ∞, using lemma 14 and proof assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Convergence of bounded integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Assume that lim N→∞E(P,CW) N � f + N � < ∞ and lim N→∞E(P,CW) N � f − N � < ∞ exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Then: E(P,BG) N (fN) Pε −→ lim N→∞E(P,CW) N (fN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' First, we consider the case lim N→∞E(P,CW) N � f + N � > 0 and lim N→∞E(P,CW) N � f − N � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' By lemmata 12 and 5, every subsequence of N has subsequences � n(1) i � i∈N, � n(2) i � i∈N, � n(3) i � i∈N such that Rn(1) i (1) −→ 1, Rn(2) i � f + N � −→ 1 and Rn(3) i � f − N � −→ 1 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Now let (ni)i∈N := � n(1) i � i∈N ∩ � n(2) i � i∈N ∩ � n(3) i � i∈N ̸= Ø (by lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' It is easy to see that almost surely, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='32) lim i→∞E(P,BG) ni � f + ni � = lim i→∞E(P,CW) ni � f + ni � = lim N→∞E(P,CW) N � f + N � , similarly for f − N , and by additivity of the expectation, also lim i→∞E(P,BG) ni (fni) = lim N→∞E(P,CW) N (fN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' By lemma 5, it follows that E(P,BG) N (fN) Pε −→ lim N→∞E(P,CW) N (fN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' If lim i→∞E(P,CW) N � f + N � = 0, or lim i→∞E(P,CW) N � f + N � = 0, or both, the same result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Convergence in distribution (theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We first consider d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Let h ∈ Cb (R) and fN (x) := h ◦ YN (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We have: f + N = h+ ◦YN and f − N = h− ◦YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Because f + and f − are bounded and by theorem assumptions: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='33) lim N→∞E(P,CW) N � f + N � < ∞, lim N→∞E(P,CW) N � f − N � < ∞ both exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Using proposition 15, it follows that � R h (x) d � P(BG) N Y −1 N � = � x∈XN h ◦ YN (x) P(BG) N (x) = E(P,BG) N (fN) Pε −→ lim N→∞E(P,CW) N (fN) = � R h (x) dP(CW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' This holds for any h ∈ Cb (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' We note that the mapping Ωε �→ � P(BG) N Y −1 N � is a random probability measure and Rd is a Polish space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' With this and [1], it follows that dL � P(BG) N Y −1 N , P(CW)� Pε −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' For d > 1, we use h ∈ Cb � Rd� and replace h◦YN (x) with h◦ � Y (1) N , Y (2) N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='., Y (d) N � (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' □ References [1] 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dilute ferromagnet: High temperature and zero temperature behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Journal of Statistical Physics, 132:759–785, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' [10] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Fluctuations of the magnetiza- tion for ising models on dense erdős–rényi random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Journal of Statistical Physics, 177(1):78–94, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' [11] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Fluctuations of the magnetization for ising models on erdős-rényi random graphs – the regimes of low temperature and external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' pre-print on https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='org/abs/2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content='08204v2, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' [12] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Fluctuations of the magnetiza- tion for ising models on erdős–rényi random graphs—the regimes of small p and the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Journal of Physics A: Mathematical and Theoretical, 53(35):355004, aug 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' RANDOM VECTORS ON A CURIE-WEISS MODEL WITH RANDOM COUPLINGS 13 [13] Zakhar Kabluchko, Matthias Löwe, and Kristina Schubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Fluctuations for the partition function of ising models on erdös-rényi random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Poincaré Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=', 57(4):2017–2042, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' [14] Werner Kirsch and Gabor Toth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Two groups in a curie-weiss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Mathematical Physics, Analysis and Geometry, 23(2), may 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} +page_content=' Max Planck UCL Centre for Computational Psychiatry and Ageing Research, University College London, United Kingdom' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf'} diff --git a/CNFRT4oBgHgl3EQfvzii/content/tmp_files/2301.13636v1.pdf.txt b/CNFRT4oBgHgl3EQfvzii/content/tmp_files/2301.13636v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc59a6d7b655919bc3cb0dc116743fcf2e9db175 --- /dev/null +++ b/CNFRT4oBgHgl3EQfvzii/content/tmp_files/2301.13636v1.pdf.txt @@ -0,0 +1,2117 @@ +Transport with Support: Data-Conditional Diffusion Bridges +Ella Tamir 1 Martin Trapp 1 Arno Solin 1 +Abstract +The dynamic Schrödinger bridge problem pro- +vides an appealing setting for solving optimal +transport problems by learning non-linear diffu- +sion processes using efficient iterative solvers. Re- +cent works have demonstrated state-of-the-art re- +sults (e.g., in modelling single-cell embryo RNA +sequences or sampling from complex posteriors) +but are limited to learning bridges with only initial +and terminal constraints. Our work extends this +paradigm by proposing the Iterative Smoothing +Bridge (ISB). We integrate Bayesian filtering and +optimal control into learning the diffusion pro- +cess, enabling constrained stochastic processes +governed by sparse observations at intermediate +stages and terminal constraints. We assess the +effectiveness of our method on synthetic and real- +world data and show that the ISB generalises well +to high-dimensional data, is computationally ef- +ficient, and provides accurate estimates of the +marginals at intermediate and terminal times. +1. Introduction +Generative diffusion models have gained increasing popular- +ity and achieved impressive results in a variety of challeng- +ing application domains, such as computer vision (e.g., Ho +et al., 2020; Song et al., 2021a; Dhariwal & Nichol, 2021), +reinforcement learning (e.g., Janner et al., 2022), and time +series modelling (e.g., Rasul et al., 2021; Vargas et al., 2021; +Tashiro et al., 2021; Park et al., 2022). Recent works have +explored connections between denoising diffusion models +and the dynamic Schrödinger bridge problem (SBP, e.g., +Vargas et al., 2021; De Bortoli et al., 2021; Shi et al., 2022) +to adopt iterative schemes for solving the dynamic optimal +transport problem more efficiently. The solution of the SBP +then acts as a denoising diffusion model in finite time and is +the closest in Kullback–Leibler (KL) divergence to the for- +ward noising process of the noising model under marginal +constraints. Data may then be generated by time reversal of +1Department of Computer Science, Aalto University. Espoo, +Finland. Correspondence to: Ella Tamir . +Sparse observations +π0 ∼ N(0, I) +πT ∼ N((10, 0)⊤, I) +Figure 1. Illustrative example transport between a unit Gaussian +and a shifted unit Gaussian. The Iterative Smoothing Bridge can +constrain the transport problem onto sparse observations ( ) forc- +ing the process to separate into two modes at intermediate times. +the process, i.e., through the denoising process. +In many applications, the interest is not purely in modelling +transport between an initial and terminal state distribution. +For example, in naturally occurring generative processes, we +typically observe snapshots of realizations along intermedi- +ate stages of individual sample trajectories (see Fig. 1). Such +problems arise in medical diagnosis (e.g., tissue changes +and cell growth), demographic modelling, environmental +dynamics, and animal movement modelling—see Fig. 4 for +modelling bird migration and wintering patterns. Recently, +constrained optimal control problems have been explored +by adding additional fixed path constraints (Maoutsa et al., +2020; Maoutsa & Opper, 2021) or modifying the prior pro- +cesses (Fernandes et al., 2021). However, defining meaning- +ful fixed path constraints or prior processes for the optimal +control problems can be challenging, while sparse observa- +tional data are accessible in many real-world applications. +In this work, we propose the Iterative Smoothing Bridge +(ISB), an iterative method for solving control problems +under constraints on both the initial and terminal distribu- +tion and sparse observational data constraints. The sparse +observational constraints act as a way to encourage the +paths sampled from the transport process to lie close to +the observed data points. We perform the conditioning by +arXiv:2301.13636v1 [cs.LG] 31 Jan 2023 + +- +口 +口 +1Transport with Support: Data-Conditional Diffusion Bridges +leveraging the iterative pass idea from the Iterative Propor- +tional Fitting procedure (IPFP) (Kullback, 1968; De Bortoli +et al., 2021) and applying differentiable particle filtering +(Reich, 2013; Corenflos et al., 2021) within the outer loop. +Integrating sequential Monte Carlo methods (e.g., Doucet +et al., 2001; Chopin & Papaspiliopoulos, 2020) into the +IPFP framework in such a way is non-trivial and can be +understood as a novel iterative version of the algorithm by +Maoutsa & Opper (2021) but with more general marginal +constraints and additional path constraints defined by data. +We summarize the contributions as follows. (i) We propose +a novel method for solving constrained optimal transport +as a bridge problem under sparse observational data +constraints. (ii) Thereof, we utilize the strong connections +between the constrained bridging problem and particle +filtering in sequential Monte Carlo, extending those links +from pure inference to learning. +Additionally, (iii) we +demonstrate practical efficiency and show that the iterative +smoothing bridge approach scales to high-dimensional data. +1.1. Related Work +Schrödinger bridges The problem of learning a stochastic +process moving samples from one distribution to another +can be posed as a type of transport problem known +as a dynamic Schrödinger bridge problem (SBP, e.g., +Schrödinger, 1932; Léonard, 2014), where the marginal +densities of the stochastic process are desired to resemble +a given reference measure. In machine learning literature, +the problem has been studied through learning the drift +function of the dynamical system (De Bortoli et al., 2021; +Wang et al., 2021; Vargas et al., 2021; Bunne et al., 2022). +When an SDE system also defines the reference measure, +the bridge problem becomes a constrained optimal control +problem (e.g., Caluya & Halder, 2022; 2021; Chen et al., +2021; Liu et al., 2022), which has been leveraged in learning +Schrödinger bridges by Tianrong Chen (2022) through +forward–backward SDEs. +Moreover, neural stochastic +control has been studied in Zhang et al. (2022). An optimal +control problem with both initial and terminal distribution +constraints and a fixed path constraint has been studied in +Maoutsa et al. (2020) and Maoutsa & Opper (2021), where +particle filtering is applied to continuous path constraints +but the boundary constraints are defined by a single point. +Furthermore, the combination of Schrödinger bridges and +state-space models has been studied by Reich (2019), in +a setting where Schrödinger bridges are applied to the +transport problem between filtering distributions. +Diffusion models in machine learning +The recent +advances in diffusion models in machine learning literature +have been focused on generating samples from complex +distributions defined by data through transforming samples +from an easy-to-sample distribution by a dynamical system +(e.g., Ho et al., 2020; Song et al., 2021b;a; Nichol & +Dhariwal, 2021). The concept of reversing SDE trajectories +via score-based learning (Hyvärinen & Dayan, 2005; +Vincent, 2011) has allowed for models scalable enough to +be applied to high-dimensional data sets directly in the data +space. In earlier work, score-based diffusion models have +been applied to problems where the dynamical system itself +is of interest, for example, for the problem of time series +amputation in Tashiro et al. (2021), inverse problems in +imaging in Song et al. (2022) and for importance sampling +Doucet et al. (2022). Interpreting the diffusion modelling +problem as optimal control has recently been studied in +Berner et al. (2022). Other dynamical models parametrized +by neural networks have been applied to modelling latent +time-series based on observed snapshots of dynamics +(Rubanova et al., 2019; Li et al., 2020), but without further +constraints on the initial or terminal distributions. +State-space models +In their general form, state-space +models combine a latent space dynamical system with an +observation (likelihood) model. Evaluating the latent state +distribution based on observational data can be performed +by applying particle filtering and smoothing (Doucet et al., +2000) or by approximations of the underlying state distribu- +tion of a non-linear state-space model by a specific model +family, for instance, a Gaussian (see Särkkä, 2013, for an +overview). Speeding up parameter inference and learning +in state-space models has been widely studied (e.g., Schön +et al., 2011; Svensson & Schön, 2017; Kokkala et al., 2014). +Particle smoothing can be connected to Schrödinger bridges +via the two-filter smoother (e.g., Bresler, 1986; Briers et al., +2009; Hostettler, 2015), where the smoothing distribution is +estimated by performing filtering both forward from the ini- +tial constraint and backwards from the terminal constraint. +We refer to Mitter (1996) and Todorov (2008) for a more +detailed discussion on the connection of stochastic control +and filtering and to Chopin & Papaspiliopoulos (2020) for +an introduction to particle filters. +2. Background +Let C = C([0, T], Rd) denote the space of continuous func- +tions from [0, T] to Rd and let B(C) denote the Borel σ- +algebra on C. Let P(π0, πT ) denote the space of probability +measures on (C, B(C)) such that the marginals at 0, T coin- +cide with probability densities π0 and πT , respectively. The +KL divergence from measure Q to measure P is written as +DKL [Q ∥ P], where we assume that Q ≪ P. For modelling +the time dynamics, we assume a (continuous-time) state- +space model consisting of a non-linear latent Itô SDE (see, +e.g., Øksendal, 2003; Särkkä & Solin, 2019) in [0, T] × Rd +with drift function fθ(·) and diffusion function g(·), and a +Gaussian observation model, i.e., +x0 ∼ π0, +dxt = fθ(xt, t) dt + g(t) dβt, +(1) + +Transport with Support: Data-Conditional Diffusion Bridges +and yk ∼ N(yk | xt, σ2 Id) +�� +t=tk where the drift function +fθ : Rd × [0, T] → Rd is a mapping modelled by a +neural network (NN) parameterized by θ ∈ Θ, diffusion +g : [0, T] → R and βt denotes standard d-dimensional +Brownian motion. xt denotes the latent stochastic process +and yt denotes the observation-space process. In practice, +we consider the continuous-discrete time setting, where the +process is observed at discrete time instances tk such that +observational data can be given in terms of a collection of +input–output pairs {(tj, yj)}M +j=1. +2.1. Schrödinger Bridges and Optimal Control +The Schrödinger bridge problem (SBP, Schrödinger, 1932; +Léonard, 2014) is an entropy-regularized optimal transport +problem where the optimality is measured through the KL +divergence from a reference measure P to the posterior Q, +with fixed initial and final densities π0 and πT , i.e., +min +Q∈P(π0,πT ) DKL [Q ∥ P] . +(2) +In this work, we consider only the case where the mea- +sures P and Q are constructed as the marginals of an SDE, +i.e., Qt is the probability measure of the marginal of the +SDE in Eq. (1) at time t, whereas Pt corresponds to the +probability measure of the marginal of a reference SDE +dxt = f(xt, t) dt + g(t) dβt, at time t, where we call f +the reference drift. Under the optimal control formulation +of the SBP (Caluya & Halder, 2021) the KL divergence in +Eq. (2) reduces to +E +� � T +0 +1 +2g(t)2 ∥fθ(xt, t) − f(xt, t)∥2 dt +� +, +(3) +where the expectation is over paths from Eq. (1). Rüschen- +dorf & Thomsen (1993) and Ruschendorf (1995) showed +that a solution to the SBP can be obtained by iteratively +solving two half-bridge problems using the Iterative Propor- +tional Fitting procedure (IPFP) for l = 0, 1, . . . , L steps, +Q2l+1 = arg min +Q∈P(·,πT ) +DKL [Q ∥ Q2l] +and +(4) +Q2l+2 = arg min +Q∈P(π0,·) +DKL [Q ∥ Q2l+1] , +(5) +where Q0 is set as the reference measure, and P(π0, ·) and +P(·, πT ) denote the sets of probability measures with only +either the marginal at time 0 or time T coinciding with π0 or +πT , respectively. Recently, the IPFP to solving Schrödinger +bridges has been adapted as a machine learning problem +(Bernton et al., 2019; Vargas et al., 2021; De Bortoli et al., +2021). In practice, the interval [0, T] is discretized and +the forward drift fθ and the backward drift bφ of the cor- +responding reverse-time process (Haussmann & Pardoux, +1986; Föllmer, 1988) are modelled by NNs. Under the Gaus- +sian transition approximations, each step in the discrete-time +diffusion model can be reversed by applying an objective +based on mean-matching. +3. Methods +Given an initial and terminal distribution π0 and πT , we +are interested in learning a data-conditional bridge between +π0 and πT . Let D = {(tj, yj)}M +j=1 be a set of M sparsely +observed values, i.e., only a few or no observations are +made at each point in time and let the state-space model +of interest be given by Eq. (1). Note that we deliberately +use (tj, yj) (instead of (tk, yk)) to highlight that we allow +for multiple observations at the same time point tk. Our +aim is to find a parameterization of the drift function fθ +such that evolving N particles xi +t, with xi +0 ∼ π0 (with +i = 1, 2, . . . , N), according to Eq. (1) will result in samples +xi +T from the terminal distribution πT . Inspired by the IPFP +by De Bortoli et al. (2021), which decomposes the SBP +into finding two half-bridges, we propose to iteratively +solve the two half-bridge problems while accounting for the +additional sparse observations simultaneously. For this, let +dxt = fl,θ(xt, t) dt + g(t) dβt, +x0 ∼ π0, +(6) +dzt = bl,φ(zt, t) dt + g(t) d ˆβt, +z0 ∼ πT , +(7) +denote the forward and backward SDE at iteration +l = 1, 2, . . . , L, where ˆβt is the reverse-time Brownian +motion. For simplicity, we denote βt = ˆβt when the +direction of the SDE is clear. +To learn the data-conditioned bridge, we iteratively employ +the following steps: 1 evolve forward particle trajectories +according to Eq. (6) with drift fl−1,θ and filter wrt the ob- +servations {(tk, yk)}M +k=1, 2 learn the drift function bl,φ +for the reverse-time SDE, 3 evolve backward particle tra- +jectories according to Eq. (7) with the drift bl,φ learned in +step 2 and filter wrt the observations {(tk, yk)}M +k=1, and +4 learn the drift function fl,θ for the forward SDE based +on the backward particles. Fig. 2 illustrates the forward +and backward process of our iterative scheme for a data- +conditioned denoising diffusion bridge. Next, we will go +through steps 1 – 4 in detail and introduce the Iterative +Smoothing Bridge method for solving data-conditional dif- +fusion bridges. +3.1. The Iterative Smoothing Bridge +The Iterative Smoothing Bridge (ISB) method iteratively +generates particle filtering trajectories (steps 1 and 3 in +Fig. 2) and learns the parameterizations of the forward and +backward drift functions fl,θ and bl,φ (steps 2 and 4 ) +by minimizing a modified version of the mean-matching +objective presented by De Bortoli et al. (2021). Note that +steps 2 and 4 are dependent on applying differential +resampling in the particle filtering steps 1 and 3 for re- + +Transport with Support: Data-Conditional Diffusion Bridges +⇋ +· · · +⇋ +2 +2 +⇋ +⇋ +4 +Forward 1 +Backward 3 +... +... +... +Forward 1 +Backward 3 +ISB 1 +ISB 6 +· · · +time t +t = 0 +t = T +Sparse observations +π0 +πT +Figure 2. Sketch of a diffusion bridge between a 2D data distribution (π0) and an isotropic Gaussian (πT ) constrained by sparse +observations ( +). The forward diffusion at the first iteration (ISB 1) learns to account for the sparse observations but does not converge +to the correct terminal distribution (t = T), and the backward diffusion vice versa. After iterating (ISB 6), the forward and backward +diffusions converge to the correct targets and are able to account for the sparse observational data. +versing the generated trajectories. We will now describe the +forward trajectory generating step 1 and the backward drift +learning step 2 in detail. Steps 3 and 4 are given by +application of 1 and 2 on their reverse-time counterparts. +Step 1 (and 3 ): +Given a fixed discretization of the +time interval [0, T] denoted as {tk}K +k=1 with t1 = 0 and +tK = T, denote the time step lengths as ∆k = tk+1 − tk. +By truncating the Itô–Taylor series of the SDE, we can +consider an Euler–Maruyama (e.g., Ch. 8 in Särkkä & Solin, +2019) type of discretization. We give the time-update of the +ith particle at time tk evolved according to Eq. (6) +˜xi +tk = xtk−1 + fl−1,θ(xtk−1,tk−1)∆k + g(tk−1) +� +∆k ξi +k, +(8) +where ξi +k ∼ N(0, I). Notice that we have not yet condi- +tioned on the observational data. In step 3 , the particles +˜zi +tk of the backward SDE Eq. (7) are similarly obtained. +The SDE dynamics sampled in steps 1 and 3 apply the +learned drift functions fl−1,θ and bl,φ from the previous +step and do not require sampling from the underlying SDE +model. For times tk at which no observations are available, +we set xi +t = ˜xi +t (and zi +tk = ˜zi +tk respectively) and otherwise +compute the particle filtering weights wi +tk based on the ob- +servations {(tj, yj) ∈ D | tj = tk} for resampling. See +Sec. 3.2 for details on the particle filtering. +For resampling, we employ a differentiable resampling pro- +cedure, where the particles and weights (˜xi +tk, wi +tk) are trans- +ported to uniformly weighted particles (xi +tk, 1 +N ) by solving +an entropy-regularized optimal transport problem (Cuturi, +2013; Peyré & Cuturi, 2019; Corenflos et al., 2021) (see +App. D). Through application of the ε-regularized optimal +transport map T(ε) ∈ RN×N (see Corenflos et al., 2021) +the particles are resampled via the map to xi +tk = ˜X⊤ +tk T(ε),i, +where ˜Xtk ∈ RN×d denotes the stacked particles {˜xi +tk}N +i=1 +at time tk before resampling. +Step 2 (and 4 ): +Given the particles {xi +tk}K,N +k=1,i=1, we +now aim to learn the drift function for the respective reverse- +time process. In case no observation is available at time tk, +we apply the mean-matching loss based on a Gaussian tran- +sition approximation proposed in De Bortoli et al. (2021): +ℓi +k+1,nobs = ∥bl,φ(xi +tk+1, tk+1)∆k − xi +tk+1 +(9) +− fl−1,θ(xi +tk+1, tk)∆k + xi +tk + fl−1,θ(xi +tk, tk)∆k∥2. +In case an observation is available at time tk the particle +values ˜Xtk will be coupled through the optimal transport +map. Therefore, the transition density is a sum of Gaussian +variables (see App. A for details and a derivation), and the + +SSSSTransport with Support: Data-Conditional Diffusion Bridges +mean-matching loss is therefore given by ℓi +k+1,obs = +∥bl,φ(xi +tk+1, tk+1)∆k − xi +tk+1 − fl−1,θ(xi +tk+1, tk)∆k ++ �N +n=1 T(ε),i,n +� +xn +tk + fl−1,θ(xn +tk, tk)∆k +� +∥2. +(10) +The overall objective function is a combination of both +loss functions, with the respective mean-matching loss +depending on whether tk is an observation time. The final +loss function is written as: +ℓ(φ) = +N +� +i=1 +� K +� +k=1 +ℓi +k,obs(φ)Iytk ̸=∅ + ℓi +k,nobs(φ)Iytk =∅ +� +, +(11) +where Icond. denotes an indicator function that returns ‘1’ +iff the condition is true, and ‘0’ otherwise. Consequently, +the parameters φ of bl,φ are learned by minimizing +Eq. (11) through gradient descent. In practice, a cache of +trajectories {xi +tk}K,N +k=1,i=1 is maintained through training +of the drift functions, and refreshed at a fixed number of +inner loop iterations, as in De Bortoli et al. (2021), avoiding +differentiation over the SDE generation computational +graph. The calculations for step 4 follow similarly. We +present a high-level description of the ISB steps in Alg. 1. +The learned backward drift bl,φ can be interpreted as +an analogy of the backward drift in Maoutsa & Opper +(2021), connecting our approach to solving optimal control +problems through Hamilton–Jacobi equations, see App. A.2 +for an analysis of the backwards SDE and the control +objective. While we are generally considering problem +settings where the number of observations is low, we +propose that letting M → ∞ yields the underlying marginal +distribution, see Prop. 2 in App. A.3. +3.2. Computational Considerations +The ISB algorithm is a generic approach to learn data- +conditional diffusion bridges under various choices of, +e.g., the particle filter proposal density or the reference +drift. +Next, we cover practical considerations for the +implementation of the method and highlight the model +choices in the experiments. +Multiple observations per time step +Naturally, we can +make more than one observation at a single point in time +tk, denoted as Dtk = {(tj, yj) ∈ D | tj = tk}. To compute +particle weights wi +tk for the ith particle we consider only the +H-nearest neighbours of xi +tk in Dtk instead of all observa- +tions in Dtk. By restricting to the H-nearest neighbours, de- +noted as DH +tk, we introduce an additional locality to the pro- +posal density computation, which can be helpful in the case +of multimodality. On the other hand, letting H > 1 results +in weights which take into account the local density of the +observations, not only the distance to the nearest neighbour. +In experiments with few observations, we set H = 1, the +choice of H is discussed when we have set the value higher. +Algorithm 1 The Iterative Smoothing Bridge +Input: Marginal constraints (π0, πT ), observations D = +{(tj, yj)}M +j=1, initial drift function f0,θ, iterations L, dis- +cretization steps K, number of particles N, observation +noise schedule κ(l) +Output: Learned forward and backward drift (fθ, bφ) +for l = 1 to L do +Forward process +Initialize forward particles {xi +0}N +i=1 ∼ π0 +for k = 1 to K do +Generate {xi +k}N +i=1 using {xi +k−1}N +i=1 +▷ Eq. (6) +if Observations at tk then +{xi +k}N +i=1 ← DiffResample({xi +k}N +i=1, κ(l)) +end if +end for +Optimize the forward loss function wrt φ ▷ Eq. (11) +Backward process +Initialize backward particles {zi +K}N +i=1 ∼ πT +for k = K to 1 do +Generate {zi +k−1}N +i=1 using {zi +k}N +i=1 +▷ Eq. (7) +if Observations at tk then +{zi +k−1}N +i=1 ← DiffResample({zi +k−1}N +i=1, κ(l)) +end if +end for +Optimize the backwards loss function wrt θ ▷ Eq. (14) +end for +Particle filtering proposal +The proposal density cho- +sen for the ISB is the bootstrap filter, where the proposal +matches the Gaussian transition density p(xtk | xtk−1). As- +suming a Gaussian noise model N(0, σ2I), the unnormal- +ized log-weights for the ith particle at time tk are given by +log wi +tk = −1/2σ2 � +yj∈DH +tk ∥xi +tk − yj∥2. +Observational noise schedule +In practice, using a con- +stant observation noise variance σ2 can result in an iterative +scheme which does not have a stationary point as L → ∞. +Even if the learned drift function fl,θ was optimal, the filter- +ing steps 1 and 3 would alter the trajectories unless all +particles would have uniform weights. Thus, we introduce +a noise schedule κ(l) which ensures that the observation +noise increases in the number of ISB iterations, causing ISB +to converge to the IPFP (De Bortoli et al., 2021) as L → ∞. +We found that letting the observation noise first decrease +and then increase (in the spirit of simulated annealing) +often outperformed a strictly increasing observation noise +schedule. The noise schedule is studied in App. C, where +we derive the property that letting L → ∞ yields IPFP. +Drift initialization Depending on the application, one may +choose to incorporate additional information by selecting +an appropriate initial drift. A possible choice includes a pre- +trained neural network drift learned to transport π0 to πT +without accounting for observations. However, starting from +a drift for the unconstrained SBP can be problematic in cases + +Transport with Support: Data-Conditional Diffusion Bridges +where the observations are far away from the unconstrained +bridge. To encourage exploration, one may choose f0 = 0 +for the initial drift. In various problem settings, we found a +zero drift and starting from the SBP to be successful in the +experiments. See App. C for discussion. +4. Experiments +To assess the properties and performance of the ISB, we +present a range of experiments that demonstrate how the +iterative learning procedure can incorporate both observa- +tional data and terminal constraints. We start with simple +examples that build intuition (cf. Fig. 1 and Fig. 2) and +show standard ML benchmark tasks. For quantitative as- +sessment, we design an experiment with a non-linear SDE +for which the marginal distributions are available in closed- +form. Finally, we demonstrate our model both in a highly +multimodal bird migration task, conditioned image genera- +tion, and in a single-cell embryo RNA modelling problem. +Ablation studies are found in App. C. +Experiment setup +In all experiments, the forward and +backward drift functions fθ and bφ are parametrized as +neural networks. For low-dimensional experiments, we +apply the MLP block design as in De Bortoli et al. (2021), +and for the image experiment an U-Net as in Nichol & +Dhariwal (2021). The latent state SDE was simulated by +Euler–Maruyama with a fixed time-step of 0.01 over 100 +steps and 1000 particles if not otherwise stated. All low- +dimensional (at most d = 5) experiments were run on a +MacBook Pro laptop CPU, whereas the image experiments +used a single Nvidia A100 GPU and ran for 5 h 10 min. +Notice that since ISB only performs particle filtering outside +the stochastic gradient training loop, the training runtime +is in the same order as in the earlier Schrödinger bridge +image generation experiments of De Bortoli et al. (2021). +Thus we omit any wall-clock timings. Full details for all the +experiments are included in App. B. +All experiment settings include a number of hyperparameter +choices, some typical to all diffusion problems and some +specific to particle filtering and smoothing. The diffusion +g(t) is a pre-determined function not optimized during +training. We divide the experiments into two main subsets: +problems of ‘sharpening to achieve a data distribution’ +and ‘optimal transport problems’. In the former, the initial +distribution has a support overlapping with the terminal dis- +tribution and the process noise level g(t) goes from high to +low as time progresses. Conversely, in the latter setting, the +particles sampled from the initial distribution must travel to +reach the support of the terminal distribution, and we chose +to use a constant process noise level. Perhaps the most sig- +nificant choice of hyperparameter is the observational noise +level, as it imposes a preference on how closely should the +observational points be followed, see App. C.1 for details. +t = 0 +t = 1/4 +t = 1/2 +t = 3/4 +t = T +Figure 3. 2D toy experiments from scikit-learn with both cases +starting from a Gaussian: The TWO CIRCLES (top) and TWO +MOONS (bottom) data sets, with observations (red markers) con- +straining the problem. For the circles, the 10 circular observations +at t = 0.5 first force the method to create a circle that then splits +into two; in the lower plot the observations at t ∈ [0.25, 0.5, 0.75] +split the data into clusters before joining them into two moons. See +Fig. 7 in the Appendix for the IPFP result. +2D toy examples We show illustrative results for the TWO +MOONS and CIRCLES from scikit-learn. We add artificial +observation data to bias the processes. For the circles, the +observational data consists of 10 points, spaced evenly on +the circle. The points are all observed simultaneously, at +halfway through the process, forcing the marginal density +of the generating SDE to collapse to the small circle, and +then to expand. +For the two moons, the observational +data is collected from 10 trajectories of a diffusion model, +which generates the two moons from noise, and these +10 trajectories are then observed at three points in time. +Results are visualized in Fig. 3 (see videos in supplement). +For reference, we have included plots of the IPFP dynamics +in the supplement, see Fig. 7. +Quantitative comparison on the Beneš SDE In order to +quantify how observing a process in between its initial and +terminal states steers the ISB model to areas with higher +likelihood, we test its performance on a Beneš SDE model +(see, e.g. Särkkä & Solin, 2019). The Beneš SDE is a non- +linear one-dimensional SDE of form dxt = tanh(xt) dt + +dβt with x0 = 0, but its marginal density is available in +closed-form, allowing for negative log-likelihood evaluation. +We simulate trajectories from the Beneš SDE and from the +reverse drift and stack the reversed trajectories. The terminal +distribution is shifted and scaled so that the Beneš SDE +itself does not solve the transport problem from π0 to πT , +see App. B.2 for details and visualizations of the processes. +We fit a Schrödinger bridge model with no observational +data as a baseline, using the Beneš SDE drift as the reference +model. The ISB model is initialized with a zero-drift model +(not with the Beneš as reference), thus making learning more +challenging. We compare the models in terms of negative +log predictive density in Table 1, where we see that the ISB +model captures the process well on average (over the entire + +Transport with Support: Data-Conditional Diffusion Bridges +Summer +Winter +Example sightings during migration +Bird observations +ISB result +Figure 4. Bird migration example. The top row describes nesting and wintering areas and example sightings during migration. The bottom +shows the marginal densities of the ISB model from the initial to terminal distribution, matching bird sightings along the migration. +time-horizon) and at selected marginal times. +Table 1. Results for the Beneš experiment. +Negative log predictive density +METHOD +AVERAGE +MIDDLE +END +Schrödinger B +4.787 +3.565 +0.1919 +Iterative smoothing B +3.557 +2.985 +0.1567 +Bird migration +Bird migration can be seen as a regular +seasonal transport problem, where birds move (typically +North–South) along a flyway, between breeding and win- +tering grounds. We take this as a motivating example of +constrained optimal transport, where the geographical and +constraints and preferred routes are accounted for by bird +sighting data (see Fig. 4 top). By adapting data from Am- +brosini et al. (2014) and Pellegrino et al. (2015), we propose +a simplified data set for geese migration in Europe (OIBMD: +ornithologically implausible bird migration data; available +in the supplement). We applied the ISB for 12 iterations, +with a linear observation noise schedule from 1 to 0.2, and +constant diffusion noise 0.05. The drift function was initial- +ized as a zero-function, and thus the method did not rely on +a separately fit model optimized for generating the wintering +distribution based on the breeding distribution. For compar- +ison, we include the Schrödinger bridge results in App. B.3. +Constraining an image generation process +We demon- +strate that the ISB approach scales well to high-dimensional +inputs by studying a proof-of-concept image generation +task. We modify the diffusion generative process of the +MNIST (LeCun et al., 1998) digit 8 by artificial observa- +tions steering the dynamical system in the middle of the +generation process. While the concept of observations in +case of image generation is somewhat unnatural, it show- +cases the scalability of the method to high-dimensional data +spaces. Here, the drift is initialized using a pre-trained neu- +ral network obtained by first running a Schrödinger bridge +model for image generation. The process is then given an +observation in the form of a bottom-half of a MNIST digit +8 in the middle of the dynamical process. As the learned +model uses information from the observation both before +and after the observation time, the lower half of the image +is sharper than the upper half. We provide further details on +this experiment and sampled trajectories in App. B.4. +Single-cell embryo RNA-seq Lastly, we evaluated our ap- +proach on an Embryoid body scRNA-seq time course (Tong +et al., 2020). The data consists of RNA measurements col- +lected over five time ranges from a developing human em- +bryo system. No trajectory information is available, instead +we only have access to snapshots of RNA data. This leads +to a data set over 5 time ranges, the first from days 0–3 and +the last from days 15–18. In the experiment, we followed +the protocol by Tong et al. (2020), reduced the data dimen- +sionality to d = 5 using PCA, and used the first and last +time ranges as the initial and terminal constraints. All other +time ranges are considered observational data. Contrary to +the other experiments, intermediate data are imprecise (only +Table 2. Results for single-cell embryo RNA experiment. +Earth mover’s distance +METHOD +t=0 +t=1 +t=2 +t=3 +t=T +TrajectoryNet +0.62 +1.15 +1.49 +1.26 +0.99 +IPML +0.34 +1.13 +1.35 +1.01 +0.49 +IPFP (no obs) +0.57 +1.53 +1.86 +1.32 +0.85 +ISB (single-cell obs) +0.57 +1.04 +1.24 +0.94 +0.83 + +Transport with Support: Data-Conditional Diffusion Bridges +Principal axis #1 +PA #2 +t=0 +t=1 +t=2 +t=3 +t=T +(a) Schrödinger bridge (via IPFP) +Principal axis #1 +PA #2 +t=0 +t=1 +t=2 +t=3 +t=T +(b) Iterative Smoothing Bridge +Figure 5. Illustration of the trajectories of the single-cell experiment for the Schrödinger bridge (a) and the ISB (b), projected onto the first +two principal components. The first five trajectories are highlighted in colour, and intermediate observation densities visualized as slices. +a time range of multiple days is known) but abundant. +We learned the ISB using a zero drift and compared it against +an unconditional bridge obtained through the IPFP (De Bor- +toli et al., 2021)—see Fig. 5. The ISB learns to generate +trajectories with marginals closer to the observed data while +performing comparably to the IPFP at the initial and termi- +nal stages. This improvement is also verified numerically +in Table 2, showing that the ISB obtains a lower Earth +mover’s distance between the generated marginals and the +observational data than IPFP. Additionally, Table 2 lists the +performance of previous works that do not use the interme- +diate data during training (Tong et al., 2020) or only use +it to construct an informative reference drift (Vargas et al., +2021), see App. B.5 for details. In both cases, ISB outper- +forms the other approaches w.r.t. the intermediate marginal +distributions (t = 1, 2, 3), while IPML (Vargas et al., 2021) +outperforms ISB at the initial and terminal stages due to its +data-driven reference drift. Notice that while we reduced +the dimensionality via PCA to 5 for fair comparisons to +Vargas et al. (2021), the ISB model would also allow mod- +elling the full state-space model, with observations in the +high-dimensional gene space and a latent SDE. +5. Discussion and Conclusion +The dynamic Schrödinger bridge problem provides an ap- +pealing setting for posing optimal transport problems as +learning non-linear diffusion processes and enables efficient +iterative solvers. However, while recent works have state-of- +the-art performance in many complex application domains, +they are typically limited to learning bridges with only initial +and terminal constraints dependent on observed data. In this +work, we have extended this paradigm and introduced the +Iterative Smoothing Bridge (ISB), an iterative algorithm for +learning data-conditional smoothing bridges. For this, we +leveraged the strong connections between the constrained +bridging problem and particle filtering in sequential Monte +Carlo, extending them from pure inference to learning. We +thoroughly assessed the applicability and flexibility of our +approach in various experimental settings, including syn- +thetic data sets and complex real-world scenarios (e.g., bird +migration, conditional image generation, and modelling +single-cell RNA-sequencing time-series). Our experiments +showed that ISB generalizes well to high-dimensional data, +is computationally efficient, and provides accurate estimates +of the marginals at initial, terminal, and intermediate times. +Accurately modelling the dynamics of complex systems un- +der both path constraints induced by sparse observations and +initial and terminal constraints is a key challenge in many +application domains. These include biomedical applications, +demographic modelling, and environmental dynamics, but +also machine learning specific applications such as rein- +forcement learning, planning, and time-series modelling. +All these applications have in common that the dynamic +nature of the problem is driven by the progression of time, +and not only the progression of a generative process as often +is the case in, e.g., generative image models. Thus, con- +straints over intermediate stages have a natural role and in- +terpretation in this wider set of dynamic diffusion modelling +applications. 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Neural stochastic control. +In Advances in Neural Information Processing Systems +35, 2022. + +Transport with Support: Data-Conditional Diffusion Bridges +A. Method Details +We present the details of the objective function derivation in App. A.1 and explain the connection of the backward drift +function to Hamilton–Jacobi equations in App. A.2. In App. A.3, we discuss the behaviour of our model at the limit +M → ∞, that is, when the observations fully represent the marginal densities of the stochastic process. +A.1. Deriving the Mean-matching Loss at Observation Times +Recall that the forward loss is written as +ℓ(φ) = +N +� +i=1 +� K +� +k=1 +ℓi +k,obs(φ)Iytk ̸=∅ + ℓi +k,nobs(φ)Iytk =∅ +� +, +(12) +where the loss at observations ℓi +k,obs(φ) and loss elsewhere ℓi +k,nobs(φ) are +ℓi +k+1,nobs = ∥bl,φ(xi +tk+1, tk+1)∆k − xi +tk+1 +− fl−1,θ(xi +tk+1, tk)∆k + xi +tk + fl−1,θ(xi +tk, tk)∆k∥2, +ℓi +k+1,obs = ∥bl,φ(xi +tk+1, tk+1)∆k − xi +tk+1 − fl−1,θ(xi +tk+1, tk)∆k ++ +1 +Cε,i +�N +n=1 T(ε),i,n +� +xn +tk + fl−1,θ(xn +tk, tk)∆k +� +∥2, +(13) +For convenience, we state the backward loss functions which follow similarly to their forward versions. The backward loss +is defined as +←−− +ℓ(θ) = +N +� +i=1 +� K +� +k=1 +←−ℓ i +k,obs(θ)Iytk ̸=∅ + ←−ℓ i +k,nobs(θ)Iytk =∅ +� +, +(14) +where the loss at observations ←−ℓ i +k,obs(θ) and loss elsewhere ←−ℓ i +k,nobs(θ) are +←−ℓ i +k+1,nobs = ∥fl,θ(xi +tk+1, tk+1)∆k − xi +tk+1 +− bl,θ(xi +tk+1, tk)∆k + xi +tk + bl,θ(xi +tk, tk)∆k∥2, +←−ℓ i +k+1,obs = ∥fl,θ(xi +tk+1, tk+1)∆k − xi +tk+1 − bl,φ(xi +tk+1, tk)∆k ++ +1 +Cε,i +�N +n=1 T(ε),i,n +� +xn +tk + bl,φ(xn +tk, tk)∆k +� +∥2, +(15) +Proposition 1. Define the forward SDE as +dxt = fl,θ(xt, t) dt + g(t) dβt, +x0 ∼ π0, +(16) +and a backward SDE drift as +bl,φ(xtk+1, tk+1) = fl−1,θ(xtk+1, tk) − g(tk+1)2∇ ln ptk+1, +(17) +where ptk+1 is the particle filtering density after differential resampling at time tk+1. Then bl,φ(xtk+1, tk+1) minimizes the +loss function +ℓi +k+1,obs = ∥bl,φ(xi +tk+1, tk+1)∆k − xi +tk+1 − fl−1,θ(xi +tk+1, tk)∆k ++ +1 +Cε,i +�N +n=1 T(ε),i,n +� +xn +tk + fl−1,θ(xn +tk, tk)∆k +� +∥2, +(18) +where we denote Cε,i = +1 +g(tk+1)2∆k Var +��N +n=1 T(ε),i,n˜xn +tk+1 +� +, and {˜xi +tk+1}N +i=1 are the particles before resampling. + +Transport with Support: Data-Conditional Diffusion Bridges +Proof sketch. Our objective is to find a backward drift function bl,φ(xtk+1, tk+1) as in Eq. (17). Notice that at observation +times tk, this is not equivalent to finding the reverse drift of the SDE forward transition and differential resampling combined, +since the drift function fl−1,θ alone does not map the particles {xi +tk}N +i=1 to the particles {xi +tk+1}N +i=1. We will derive a loss +function for learning the backward drift as in Eq. (17) below, leaving the discussion on why it is a meaningful choice of a +backward drift to App. A.2. Our derivation closely follows the proof of Proposition 3 in De Bortoli et al. (2021), but we +provide the details here for the sake of completeness. +First, we give the transition density pxtk | xi +tk−1 (xk) and apply it to derive the observation time loss ℓi +k,obs. The derivation for +the loss ℓi +k,no obs is skipped since it is as in the proof of Proposition 3 in De Bortoli et al. (2021). Suppose that at tk, there are +observations. By definition, the particles before resampling {˜xi +tk+1}N +i=1 are generated by the Gaussian transition density +p(˜xtk+1 | xi +tk) = N(˜xtk+1 | xi +tk + δkfl(xi +tk, tk), g(tk+1)2∆kI). +(19) +Recall that the resampled particles are defined as a weighted average of all the particles, xi +tk = �N +n=1 ˜xn +tk T(ε),i,n. Thus, the +transition density from {xi +tk}N +i=1 to the particles {xi +tk+1}N +i=1 is also a Gaussian, +p(xti +k+1 | xi +tk) = N(˜xtk+1 | +N +� +n=1 +T(ε),i,n(xn +tk−1 + ∆kfl−1,θ(xn +tk, tk)), g(tk+1)2∆kCε,iId). +(20) +We will derive the loss function Eq. (10) by modifying the mean matching proof in De Bortoli et al. (2021) by the transition +mean Eq. (20) and the backward drift definition Eq. (17). Using the particle filtering approximation, the marginal density +can be decomposed as ptk+1(xk+1) = �N +i=1 ptk(xi +k)pxk+1 | xi +k(xk+1). By substituting the transition density Eq. (20) it +follows that +ptk+1(xtk+1) += 1 +Z +N +� +i=1 +ptk(xi +tk) exp +� +�− +∥ +��N +n=1 T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) +� +− xtk+1∥2 +2g(tk+1)2Cε,i∆k +� +� , +(21) +where Z is the normalization constant of Eq. (20). As in the proof of Proposition 3 of De Bortoli et al. (2021), we derive an +expression for the score function. Since ∇ ln ptk+1(xtk+1) = +∇xtk+1 ptk+1(xtk+1) +ptk+1(xtk+1 ) +, we first manipulate ∇xtk+1 ptk+1(xtk+1), +∇xtk+1 ptk+1(xtk+1) +(22) += 1 +Z +N +� +i=1 +∇xtk+1 p(xi +tk) exp +� +�− +∥ +��N +n=1 T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) +� +− xtk+1∥2 +2g(tk+1)2Cε,i∆k +� +� +(23) += 1 +Z +� N +� +i=1 +p(xi +tk) +� N +� +n=1 +1 +g(tk+1)2∆kCε,i +� +T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) − xtk+1 +� +� +(24) +exp +� +�− +∥ +��N +n=1 T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) +� +− xtk+1∥2 +2g(tk+1)2Cε,i∆k +� +� +� +. +(25) +Substituting ptk(xi +k) = +ptk+1(xtk+1)pxk+1 | xi +k (xk+1) +pxi +k | xk+1(xi +k) +to the equation above gives +∇xtk+1 ptk+1(xtk+1) += ptk+1(xtk+1) +N +� +i=1 +pxk+1 | xi +k(xi +k) +� N +� +n=1 +� +T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) − xtk+1 +� +g(tk+1)2∆kCε,i +� +, +(26) + +Transport with Support: Data-Conditional Diffusion Bridges +and dividing by ptk+1(xtk+1) yields +∇ ln ptk+1(xtk+1) += +N +� +i=1 +pxti +k | xtk+1 (xti +k) +� N +� +n=1 +� +T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) − xtk+1 +� +g(tk+1)2∆kCε,i +� +. +(27) +Substituting Eq. (27) to the definition of the optimal backward drift Eq. (17) gives +bl,φ(xtk+1, tk+1) += fl−1,θ(xtk+1, tk) − g(tk+1)2∇ ln ptk+1(xk+1) += fl−1,θ(xtk+1, tk)− +g(tk+1)2 +N +� +i=1 +pxti +k | xtk+1 (xtk+1) +� N +� +n=1 +� +T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) − xtk+1 +� +g(tk+1)2∆kCε,i +� +, +(28) +where taking fl−1,θ(xtk+1, tk) inside the sum yields +bl,φ(xtk+1, tk+1) = +� N +� +i=1 +pxti +k | xtk+1 (xtk+1) +� +1 +Cε,i +� N +� +n=1 +T(ε),i,n(xi +tk + fl−1,θ(xtk, tk)) +� +− xtk+1 +Cε,i +− ∆kfl−1,θ(xtk+1, tk) +� +/∆k) +� +. +(29) +Multiplying the equation above by ∆k gives +∆kbl,φ(xi +tk+1, tk+1) += +� N +� +n=1 +T(ε),i,n(xn +tk + fl−1,θ(xn +tk, tk)) +� +− +xi +tk+1 +Cε,i +− ∆kfl−1,θ(xti +k+1, tk). +(30) +Thus we may set the objective for finding the optimal backward drift bl,φ as +ℓi +k+1,no obs = ∥bl,φ(xi +tk+1, tk+1)∆k − +xi +tk+1 +Cε,i +− fl−1,θ(xi +tk+1, tk)∆k ++ 1 +Cε,i +�N +n=1 T(ε),i,n +� +xn +tk + fl−1,θ(xn +tk, tk)∆k +� +∥2. +(31) +Notice that if the weights before resampling are uniform, then T(ε) = IN, and for all i ∈ 1, 2, . . . , N it holds that Cε,i = 1, +since all but one of the terms in the sum +1 +g(tk+1)2 Var +��N +n=1 T(ε),i,n˜xn +tk+1 +� +vanish. Similarly, for one-hot weights Cε,i = 1. +In practice, we set the constant Cε,i = 1 as in Eq. (10) and observe good empirical performance with the simplified loss +function. +A.2. Connection to Hamilton–Jacobi Equations +We connect the backward drift function bl,φ(xtk+1, tk+1) = fl−1,θ(xtk+1, tk)−g(tk+1)2∇ ln ptk+1(xtk+1) to the Hamilton– +Jacobi equations for stochastic control through following the setting of Maoutsa & Opper (2021), which applies the drift +fl−1,θ(xt, t) − g(t)2∇ ln pt(xt) for a backwards SDE initialized at πT . +Consider a stochastic control problem with a path constraint U(xt, t), optimizing the following loss function, +J = 1 +N +N +� +i=1 +� T +t=0 +1 +2g(t)2 ∥fθ(xi +t, t) − f(xi +t, t)∥2 + U(xi +t, t) dt − ln χ(xi +T ), +(32) + +Transport with Support: Data-Conditional Diffusion Bridges +with the paths, xi +t sampled as trajectories from the SDE +x0 ∼ π0, +dxt = fl−1,θ(xt, t) dt + g(t) dβt, +(33) +and the loss ln χ(xi +T ) measures distance from the distribution πT . Since we set the path constraint via observational data, +our method resembles setting U(xi +t, t) = 0 when t is not an observation time, and U(xi +t) = − log p(y | xi +t), where p(y | xi +t) +is the observation model. +Let qt(x) denote the marginal density of the controlled (drift fθ) SDE at time t. In Maoutsa & Opper (2021), the marginal +density is decomposed as +qt(x) = ϕt(x)pt(x), +(34) +where ϕt(x) is a solution to a backwards Fokker-Planck-Kolmogorov (FPK) partial differential equation starting from +ϕT (x) = πT , and the density evolves as in +dϕt(x) +dt += −L† +fϕt(x) + U(x, t)ϕt(x), +(35) +where L† +f is the adjoint FPK operator to the uncontrolled system. The density pt(x) corresponds to the forward filtering +problem, initialized with π0, +dpt(x) +dt += Lf(pt(x)) − U(x, t)pt(x), +(36) +where Lf is the FPK operator of the uncontrolled SDE (with drift f). The particle filtering trajectories {xtk}i generated in +our method are samples from the density defined by Eq. (36). In the context of our method, the path constraint matches the +log-weights of particle filtering at observation times and is zero elsewhere. +In Maoutsa & Opper (2021), a backward evolution for qt is applied, using the backwards time ˜qT −τ(x) = qτ(x), yielding a +backwards SDE starting from ˜q0(x) = {xi +T }N +i=1, reweighted according to πT . The backward samples from ˜q are generated +following the SDE dynamics +dxi +τ = (f(xi +τ, T − τ) + g(t)2∇ ln pT −τ(xi +τ) dt + g(t) dβτ. +(37) +We have thus selected the backward drift bl,φ to match the drift of ˜qt(x), the backward controlled density. Intuitively, our +choice of bl,φ is a drift which generates the smoothed particles when initialized at {xi +T }N +i=1, the terminal state of the forward +SDE. The discrepancy between πT and the distribution induced by {xi +T }N +i=1 then motivates the use of an iterative scheme +after learning to simulate from qt(x). +A.3. Observing the Full Marginal Density +Suppose that at time tk, we let the number of observations grow unbounded. We analyse the behaviour of our model at the +resampling step, at the limit M → ∞ for the number of observations and σ → 0 for the observation noise. When applying +the bootstrap proposal, recall that we combined the multiple observations to compute the log-weights as +log wi +tk = − 1 +2σ2 +� +yj∈DH +i,tk +∥xi +tk − yj∥2, +(38) +which works well in practice for the sparse-data settings we have considered. Below we analyse the behaviour of an +alternative way to combine the weights and show that given an infinite number of observations, it creates samples from the +true underlying distribution. +Proposition 2. Let {xi +tk}N +i=1 be a set of particles and {yj}M +j=1 the observations at time tk. Assume that the observations +have been sampled from a density ρtk and that for all i it holds that xi +tk ∈ supp(ρtk). Define the particle weights as +log wi +tk,σ,M = log +� +1 +Z|DH(M) +i,tk +| +� +yj∈DH(M) +i,tk +exp(−∥xi +tk − yj∥2/2σ2) +� +, +(39) +where Z is the normalization constant of the observation model Gaussian p(y | xi +tk). Then for each particle xi +tk, its weight +satisfies +lim +σ→0 lim +M→∞ wi +tk,σ,M = ρtk(xtk) +(40) + +Transport with Support: Data-Conditional Diffusion Bridges +Time, t +π0 ∼ N(0, I) +πT ∼ N((10, 0)⊤, I) +(a) Unconstrained transport +(Schrödinger bridge) +Sparse observations +π0 ∼ N(0, I) +πT ∼ N((10, 0)⊤, I) +(b) Constrained transport +(Iterative smoothing bridge) +Figure 6. Illustrative example transport between an initial unit Gaussian and a shifted unit Gaussian at the terminal time T. Unconstrained +transport on the left and the solution constrained by sparse observations ( ) on the right. Colour coding of the initial points is only for +distinguishing the paths. +Proof sketch. We drop the σ and H(M) from the weight notation for simplicity of notation, but remark that the particle +filtering weights are dependent on both quantities. Consider the number of particles N fixed, and denote the d-dimensional +sphere centered at xi +tk as B(xi +tk, r). Since each particle xi +tk lies in the support of the true underlying marginal density ρtk, +then for any radius r > 0 such that B(xi +tk, r) ∈ supp(ρtk), and H > 0, we may choose M high enough so that the points +yj ∈ DH +i,tk satisfy yj ∈ B(xi +tk, r). It follows from Eq. (39) that +wi +tk = +1 +Z|DH(M) +i,tk +| +� +yj∈DH(M) +i,tk +exp(−∥xi +tk − yj∥2/2σ2). +(41) +For any r > 0 and with observation noise σ = cr, we may set c, H(M) so that the sum above approximates the integral +wi +r,tk ≈ +1 +|B(xi +tk, r)| +� +B(xi +tk ,r) +p(y | xi +tk)ρt(y) dy. +(42) +By applying the Lebesque differentiation theorem, we obtain that for almost every xi +tk, we have limr→0 wi +tk,r = ρtk(xi +tk), +since as σ → 0, the density p(y | xi +tk) collapses to the Dirac delta of xi +tk. +Prop. 2 can be interpreted as the infinite limit of a kernel density estimate of the true underlying distribution. Resampling +accurately reweights the particles so that the probability of resampling particle xi +tk is proportional to the density ρtk +compared to the other particles. Notice that the result does not guarantee that the particles will cover the support of ρtk, +since we did not assume that the drift initialization generates a marginal density at time tk covering its support. +B. Experimental Details +B.1. 2D Toy Data Sets +For the constrained transport problem for two-dimensional scikit-learn, the observational data we chose to use was different +for each of the three data sets presented; two moons, two circles and the S-shape. All three experiments had the same +discretization (t ∈ [0, 0.99]), ∆k = 0.01), learning rate 0.001, and differentiable resampling regularization parameter +ε = 0.01. The process noise g(t)2 follows a linear schedule from 0.001 to 1, with low noise at time t = 0 and high noise at +t = 0.99, and each iteration of the ISB method trains the forward and backward drift networks each for 5000 iterations, with +batch size 256. Other hyperparameters are explained below. +Two moons The observational data consists of 10 points selected from the Schrödinger bridge trajectories, all observed +at t ∈ [0.25, 0.5, 0.75] with an exponential observation noise schedule κ(l) = 1.25l−1. The ISB was run for 6 epochs and +initialized with a drift from the pre-trained Schrödinger bridge model from the unconstrained problem. + +1 +一 +QQ +1 +--- +口 +口 +1Transport with Support: Data-Conditional Diffusion Bridges +(a) t = 0.00 +(b) t = 0.25 +(c) t = 0.50 +(d) t = 0.75 +(e) t = 0.99 +Figure 7. Baseline for comparison: The IPFP result for the experiment in Fig. 3 in the main paper. 2D toy experiments, where observations +(red markers) were not used during training but included in the figure for reference. The dynamics learned by IPFP are clearly different +from the ISB learned dynamics. +Two circles The observational data consists of 10 points which lie evenly distributed on a circle, observed at t = 0.5 with +an exponential observational noise schedule κ(l) = 0.5 · 1.25l−1. The ISB was run for 6 epochs and initialized with a drift +from the pre-trained Schrödinger bridge model from the unconstrained problem. +S-shape +The observational data consists of 6 points, with pairs being observed at times t ∈ [0.4, 0.5, 0.6]. We used a +bilinear observational noise schedule with a linear decay for the first half of the iterations from κ(0)2 = 4 to κ(L/2)2 = 1 +and a linear ascend for the second half of the iterations from κ(L/2)2 = 1 to κ(L)2 = 4. The ISB ran for 6 epochs, with a +zero drift initialization. +0 +4 +8 +True πT +ISB +Schrödinger +Figure 8. A kernel density estimate of the Beneš SDE terminal state. We compare πT to the Schrödinger bridge and ISB terminal states. +Both unconstrained Schrödinger bridge and ISB terminal states succeed in representing πT well, with the Schrödinger bridge terminal +state more closely matching πT near its mean. + +Transport with Support: Data-Conditional Diffusion Bridges +0 +4 +8 +12 +−10 +0 +10 +Time, t +(a) Trajectories of the Beneš SDE +0 +4 +8 +12 +−10 +0 +10 +Time, t +(b) SBP trajectories +0 +4 +8 +12 +−10 +0 +10 +Time, t +(c) ISB trajectories (ours) +Figure 9. Comparison of the solution for the SBP (with Beneš SDE reference drift) and the ISB (with zero initial drift) on the Beneš SDE +under sparse observations ( ). The target distribution πT is slightly shifted and scaled from the Beneš SDE. Even if the SBP has the true +model as reference drift, its trajectories degenerate into a unimodal distribution, while the ISB manages to cover both modes even if only +sparse observations are available. +B.2. The Beneš SDE +In the Beneš SDE experiment, we obtain the sparse observational data from sampled Beneš SDE trajectories while the +terminal state is a shifted and scaled (3 + 5xT ) version of a Beneš marginal density. As the Beneš trajectories were first +generated by simulating the SDE until t = 6 and then in reverse from t = 6 to t = 0, we set T = 11.97. We apply the +analytical expression for the Beneš marginal density for computing log pt(x), +pt(x) = +1 +√ +2πt +cosh(x) +cosh(x0) exp +� +− 1 +2t +� +exp +� +− 1 +2t(x − x0)2 +� +. +(43) +See the Beneš SDE trajectories in Fig. 9a. As expected, the transport model with no observations performs well in the +generative task, but its trajectories cover also some low-likelihood space around t = 6 (in the middle part in Fig. 9b). The +observations for the ISB model were sampled from the generated trajectories, 10 observations at 10 random time-instances +(see Fig. 9c) +Both the unconstrained Schrödinger bridge model and the ISB model were run for 3 iterations, using a learning rate of +0.001 for the neural networks. Likely due to the fact that the problem was only one-dimensional, the convergence of the +Schrödinger bridge to a process which matches the desired terminal state was fast, and we chose not to run the model +for a higher number of ISB iterations, see Fig. 8 for a comparison of the trained model marginal densities and the true +terminal distribution πT . We set the observation noise schedule to the constant 0.7, and at each iteration of the ISB or +the unconstrained Schrödinger bridge the drift neural networks were trained for 5000 iterations each with the batch size +256, and the trajectories were refreshed every 500 iterations with a cache size of 1000 particles. The number of nearest +neighbours to compare to was H = 10. +B.3. The Bird Migration Data Set +The ISB model learned bird migration trajectories which transport the particles from the Northern Europe summer habitats +to the southern winter habitats, see Fig. 11 for a comparison of a Schrödinger bridge and ISB. Since the problem lies on +a sphere, Schrödinger bridge methods adjusted for learning on Riemannian manifolds could have been applied here. For +simplicity, we mapped the problem to a two-dimensional plane using a Mercator projection and solved the problem on a +[0, 5] × [0, 5] square. The SDE had the discretization t ∈ [0, 0.99], ∆k = 0.01 and a constant process noise g(t)2 = 0.05. +The model was trained for 12 iterations, and initialized with a zero drift, while the observational data was chosen by the +authors to promote learning trajectories clearly different from the unconstrained transport trajectories. The observation noise +schedule was piecewise linear (starting at 2, going to 0.1 at iteration 6, then rising linearly to reach 2 at iteration 12). At +each ISB iteration, the neural networks were trained for 5000 iterations each, and the trajectories were refreshed every 1000 +iterations. We used a batch size of 256 and a learning rate of 0.001. + +xTransport with Support: Data-Conditional Diffusion Bridges +π0 = noise +T = 0.5 +Observation +biasing the lower half of images +Figure 10. Model trajectories for MNIST digit ‘8’ conditioned on a lower-loop of a single ‘8’ at t = 0.38 to bias the lower half of the +digits to look alike, with the effect still visible at terminal time T. +B.4. The MNIST Generation Task +Applying state-space model approaches such as particle filtering and smoothing to generative diffusion models directly +in the observation space (that is, not in a lower-dimensional latent space) has to our knowledge not been explored before. +Some experimental design choices had a great impact on the training objectives sensibility, as the observational data is +completely artificial and its timing during the process modifies the filtering distribution significantly. As the MNIST +conditional generative model was trained to display the scalability of our method beyond low-dimensional toy examples, we +did not further explore optimizing the hyperparameters or the observation model. To avoid the background noise in MNIST +images in the middle of the generative process impacting the particle filtering weights excessively, the observation model +is a Gaussian with masked inputs equal to zero in pixels where the observation image is black, see Fig. 10 for sampled +trajectories. The figure shows the progression of seven samples, where the lower half of the eight resemble the observation +target. +The SDE was run for time t ∈ [0, 0.5], with the digit eight observed at t = 0.38. The ISB method was applied for 10 +iterations, with a discretization t ∈ [0, 0.495], ∆k = 0.005, and the process noise g(t)2 followed a linear schedule from +0.0001 to 1. At each iteration of the method, the forward and backward drift neural networks were trained for 5000 iterations +with a batch size of 256, and the trajectory cache regenerated every 1000 iterations. The observational data consisted of a +single sample of a lower half of the digit eight, observed at time t = 0.38. The observation noise schedule was a constant +κ(l) = 0.3. +B.5. Single-Cell Data Set +We directly use the preprocessed data from the TrajectoryNet (Tong et al., 2020) repository. A major difference between our +implementation and Vargas et al. (2021) is the reference drift. We set the reference drift to zero, which means that we utilize +the intermediate data only as observations in the state-space model. On the contrary, Vargas et al. (2021) fits a mixture +model of 15 Gaussians on the combined data set (across all measurement times) and sets the reference drift to the gradient +of the log-likelihood of the mixture model. Effectively, such a reference drift aids in keeping the SDE trajectories within +the support of the combined data set. We remark that if the intermediate observed marginals had clearly disjoint support, +combining all the data would cause the mixture model to have ‘gaps’ and could cause an unstable reference model drift. +Thus we consider our approach of setting the reference drift to zero as more generally applicable. +As in Vargas et al. (2021), we set the process noise to g(t) = 1 and model the SDE between time t ∈ [0, 4]. The learning +rate is set to 0.001 with a batch size of 256 and the number of neural network training iterations equal to 5000. We apply the +ISB for 6 iterations. We perform filtering using 1000 points from the intermediate data sets, but compute the Earth mover’s +distance by comparing it to all available data. As the observational data at T = 1, 2, 3 consists of a high number of data +points, the parameters H (number of nearest neighbours) and σ (observation noise) need to be carefully set. We set H = 10 + +Transport with Support: Data-Conditional Diffusion Bridges +to only include the close neighbourhood of each particle and set the observation noise schedule as constant 0.7. +C. Computational Considerations +In Sec. 3.2, we raised a number of important computational considerations for the constrained transport problem. Below +we discuss them in detail, analyzing the limit L → ∞ from the perspective of setting the observation noise schedule in +App. C.1, and presenting ablation results on modifying the initial drift in the bird migration experiment in App. C.2. +C.1. Discussion on Observation Noise +We briefly mentioned in Sec. 3.2 that when letting L → ∞, the choice of observation noise should be carefully planned in +order for the ISB procedure to have a stationary point. Here we explain why an unbounded observation noise schedule κ(l) +implies convergence to the IPF method for uncontrolled Schrödinger bridges (De Bortoli et al., 2021), when using a nearest +neighbour bootstrap filter as the proposal density. +Proposition 3. Let Ω ∈ Rd be a bounded domain where both the observations and SDE trajectories lie, and let the particle +filtering weights {wi +l,tk}N +i=1 at ISB iteration l be +log wi +l,tk = − +1 +2κ(l)2 +� +yj∈DH +tk +∥xi +tk − yj∥2. +(44) +If the schedule κ(l) is unbounded with respect to l, then for any δ there exists l′ such that for the normalized weights it holds +| ˆwi +l′,tk − 1 +N | ≤ δ. +(45) +Proof sketch. Since κ(l) is unbounded, for any S > 0 ∃ l′ such that κ(l′) ≥ S. We choose the value of S so that the +following derivation yields Eq. (45). +Let S = +� +0.5R−1|DH +tk| diam(Ω)2, and apply the property that ∥xi +tk − yj∥2 ≤ diam(Ω)2 to Eq. (44), +log wi +l′,tk ≥ − 1 +2S2 +� +yj∈DH +tk +∥xi +tk − yj∥2 +≥ − +� +yj∈DH +tk ∥xi +tk − yj∥2 +R−1|DH +tk| diam(Ω)2 +≥ − +� +yj∈DH +tk diam(Ω)2 +R−1|DH +tk| diam(Ω)2 ≥ −R. +(46) +The bound above is for the unnormalized weights, and the normalized log-weights are defined as +log ˆwi +l′,tk = log wi +l′,tk − log +� N +� +j=1 +exp(log wj +l′,tk) +� +, +(47) +where for the normalizing constant it holds that +log +� N +� +j=1 +exp(log wj +l′,tk) +� +≤ log +� N +� +j=1 +1 +� += log(N), +(48) +since wj +l′,tk is the value of a probability density and thus always wj +l′,tk ≤ 1. Combining Eq. (47), Eq. (46) and Eq. (48), it +follows that +log ˆwi +l′,tk − (− log(N) ≥ −R, +(49) +where taking exponentials on both sides gives +ˆwi +l′,tk − 1 +N ≥ −(1 − exp(−R)) 1 +N . +(50) + +Transport with Support: Data-Conditional Diffusion Bridges +Since the weights are normalized, even the largest particle weight ˆwj +l′,tk can differ from 1 +N as much as every smaller weight +in total lies under 1 +N , +ˆwj +l′,tk ≤ 1 +N + (N − 1) +� +(1 − exp(−R)) 1 +N +� +, +(51) +implying that for any weight ˆwj +l′,tk, it holds that +| ˆwj +l′,tk − 1 +N | ≤ (N − 1) +� +(1 − exp(−R)) 1 +N +� +≤ 1 − exp(−R), +(52) +and selecting R = − log(1 − δ) is sufficient for δ < 1. +Effectively, the above derivation implies that for an unbounded observation noise schedule κ(l), the particle weights will +converge to uniform weights. Since performing differentiable resampling on uniform weights implies that T(ε) = IN, +the ISB method trajectory generation step and the objective in training Nthe backward drift converge to those of the IPF +method for solving unconstrained Schrödinger bridges. Intuitively, this means that at the limit L → ∞, our method will +focus on reversing the trajectories and matching the terminal distribution while not further utilizing information from the +observations. +C.2. Ablation on Initial Drift +We conducted an ablation study on drift initialization for the bird migration problem. As the distributions π0 and πT (as +pictured in Fig. 11) are complex, we consider the problem setting to be interesting for setting f0 as the unconstrained +transport problem drift. To this end, we trained a Schrödinger bridge model for 10 epochs, and trained an ISB model with +the same hyperparameter selections as explained in App. B.3, using the Schrödinger bridge as the initialization. Compare +the two bottom rows of Fig. 11 to see a selection of marginal densities of the two processes. Based on a visual analysis of +the densities, it seems that the zero drift and pre-trained diffusion model initializations produce similar results around the +observations, although the Schrödinger bridge initialization gave slightly sharper results at the terminal time. +D. Differentiable Resampling +In the ISB model steps 1 and 3 presented in Sec. 3.1, we applied differentiable resampling (see Corenflos et al., 2021). +Resampling itself is a basic block of particle filtering. A differentiable resampling step transports the particles and weights +(˜xi +tk, wi +tk) to a uniform distribution over a set of particles through applying the differentiable ensemble transport map T(ε), +that is +(˜xi +tk, wi +tk) → ( ˜X⊤ +tk T(ε),i, 1/N) = (xi +tk, 1/N), +(53) +where ˜Xtk ∈ RN×d denotes the stacked particles {˜xi +tk}N +i=1 at time tk before resampling and xi +tk denotes the particles post +resampling. Here we give the definition of the map T(ε) and review the regularized optimal transport problem which has to +be solved to compute it. We partly follow the presentation in Sections 2 and 3 of Corenflos et al. (2021), but directly apply +the notation we use for particles and weights and focus on explaining the transport problem rather than the algorithm used to +solve it. +The standard particle filtering resampling step consists of sampling N particles from the categorical distribution defined +by the weights {wi +tk}N +i=1, resulting in the particles with large weights being most likely to be repeated multiple times. +A result from Reich (2013) gives the property that the random resampling step can be approximated by a deterministic +ensemble transform T. In heuristic terms, the ensemble transform map will be selected so that the particles {xi +tk}N +i=1 will +be transported with minimal cost, while allowing all the weights to be uniform. +Let µ and ν be atomic measures, µ = �N +i=1 wi +tkδ˜xi +tk and ν = �N +i=1 N −1δ˜xi +tk , where δx is the Dirac delta at x. Then µ is +the particle filtering distribution before resampling. Define the elements of a cost matrix C ∈ RN×N as Ci,j = ∥˜xi +tk −˜xj +tk∥2, +and the 2-Wasserstein distance between two atomic measures as +W2 +2(µ, ν) = +min +P ∈S(µ,ν) +N +� +i=1 +N +� +j=1 +Ci,jPi,j. +(54) + +Transport with Support: Data-Conditional Diffusion Bridges +Above the optimal matrix P is to be found within S(µ, ν), which is a space consisting of mixtures of N particles to N +particles such that the marginals coincide with the weights of µ and ν, formally +S(µ, ν) = +� +� +�P ∈ [0, 1]N×N | +N +� +i=1 +Pi,j = wi +tk, +N +� +j=1 +Pi,j = 1 +N +� +� +� . +(55) +The entropy-regularized Wasserstein distance with regularization parameter ε is then +W2 +2,ε = +min +P∈S(µ,ν) +N +� +i=1 +N +� +j=1 +Pi,j +� +Ci,j + ε log +Pi,j +wi +tk · 1 +N +� +. +(56) +The unique minimizing transport map of the above Wasserstein distance is denoted by POPT +ε +, and the ensemble transport +map is then set as T(ε) = NPOPT +ε +. This means that we can find the matrix T(ε) via minimizing the regularized Wasserstein +distance, which is done by applying the iterative Sinkhorn algorithm for entropy-regularized optimal transport (Cuturi, +2013). + +Transport with Support: Data-Conditional Diffusion Bridges +Summer +Winter +Intermediate observations during migration +Bird observations +Unconditional +Conditional (zero drift) +Conditional (transport drift) +Figure 11. Top row: The first map image on the left describes the initial position of the birds, and the final one on the right depicts their +position after migration. The observational data in the middle are bird observations during migration, at given timestamps. Second row: +Marginal densities of a Schrödinger bridge model from the initial to terminal distribution, without using the observations. Third row: +Marginal densities of our model, using both initial and terminal distributions and observational data and a zero drift initialization. Bottom +row: Same as the third row, but with the second-row dynamics as initialization. + diff --git a/CNFRT4oBgHgl3EQfvzii/content/tmp_files/load_file.txt b/CNFRT4oBgHgl3EQfvzii/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4a733e00248c2ec7bbb0ed2b62c8e39f1a36a07 --- /dev/null +++ b/CNFRT4oBgHgl3EQfvzii/content/tmp_files/load_file.txt @@ -0,0 +1,1179 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf,len=1178 +page_content='Transport with Support: Data-Conditional Diffusion Bridges Ella Tamir 1 Martin Trapp 1 Arno Solin 1 Abstract The dynamic Schrödinger bridge problem pro- vides an appealing setting for solving optimal transport problems by learning non-linear diffu- sion processes using efficient iterative solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Re- cent works have demonstrated state-of-the-art re- sults (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are limited to learning bridges with only initial and terminal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' We integrate Bayesian filtering and optimal control into learning the diffusion pro- cess, enabling constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' We assess the effectiveness of our method on synthetic and real- world data and show that the ISB generalises well to high-dimensional data, is computationally ef- ficient, and provides accurate estimates of the marginals at intermediate and terminal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Introduction Generative diffusion models have gained increasing popular- ity and achieved impressive results in a variety of challeng- ing application domains, such as computer vision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Dhariwal & Nichol, 2021), reinforcement learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2022), and time series modelling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', Rasul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Tashiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Recent works have explored connections between denoising diffusion models and the dynamic Schrödinger bridge problem (SBP, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' De Bortoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=', 2022) to adopt iterative schemes for solving the dynamic optimal transport problem more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' The solution of the SBP then acts as a denoising diffusion model in finite time and is the closest in Kullback–Leibler (KL) divergence to the for- ward noising process of the noising model under marginal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Data may then be generated by time reversal of 1Department of Computer Science, Aalto University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Espoo, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFRT4oBgHgl3EQfvzii/content/2301.13636v1.pdf'} +page_content=' Correspondence to: Ella Tamir +10yr) around several dozens of stars. Yet, the true nature of these companions remains unclear because of the uncertainty as to the +inclination of the companion orbital plane. +Aims. We wish to constrain the orbital inclination and the true mass of long-period single companions. +Methods. We used a Markov Chain Monte Carlo (MCMC) fitting algorithm to combine RV measurements with absolute astrometry +and, when available, relative astrometry data. +Results. We have lifted the sin(i) indetermination for seven long-period companions. We find true masses in the planetary mass range +for the candidate planets detected in the following systems: Epsilon Indi A, HD 13931, HD 115954, and HD 222155. The mass of +HD 219077 b is close to the deuterium-burning limit and its nature is uncertain because of the imprecise mass of the host star. Using +additional RV measurements, we refine the orbital parameters of HIP 70849 b and find a mass in the planetary range. By combining +RV data with absolute and relative astrometry, we significantly improve the characterization of HD 211847 B and properly determine +its mass, which appears to be in the low-mass star range. This work illustrates how Gaia and Hipparcos allow for the orbital properties +and masses of long-period RV companions to be further constrained. +Key words. Techniques: radial velocities – Techniques: high angular resolution – Proper motions – Stars: planetary systems – Stars: +brown dwarfs – Stars: low-mass +1. Introduction +In the last decade, several long-period giant planets have been +detected using the radial velocity (hereafter RV) method thanks +to the increasing temporal baselines of different surveys (Mayor +et al. (2011), Wittenmyer et al. (2020), Rosenthal et al. (2021)). +Yet, a precise determination of the orbital parameters and mass +of the planets is very difficult when the orbital period is much +larger than the RV time baseline. As a consequence, the radial +distribution of planets beyond 8-10 au – such as those found by +Fernandes et al. (2019) and Fulton et al. (2021) based on the +results of the two long RV surveys of Mayor et al. (2011) and +Rosenthal et al. (2021), respectively – are questionable. This +unfortunately prevents an accurate comparison with formation +model outputs from being made. +Combining RV data with other methods such as relative or +absolute astrometry can, in principle, improve the orbital charac- +terization of these companions. Furthermore, it can also remove +the uncertainty of the orbital inclination and then allow us to de- +termine the true mass of the planets. +Coupling RV data with relative astrometry from direct imag- +ing (hereafter DI) or interferometry has been, however, limited +to very few cases since high-contrast imaging (hereafter HCI) +or interferometry observations favor young systems to minimize +the flux contrast between the star and its companion while RV +⋆ Please send any requests to florian.philipot@obspm.fr +observations favor old and inactive stars which produce low RV +jitters. However, when possible, such a coupling is very efficient. +An illustration is the HD 7449 system for which the outer com- +panion was first reported as a planet candidate using only RV +data (Mayor et al. (2011), Wittenmyer et al. (2019)), and it was +then identified as a low-mass star by combining RV data with +HCI observations (Rodigas et al. 2016). +In the 2000s, the combination of RV data and absolute +astrometry, thanks to the Fine-Guidance-Sensor onboard the +Hubble Space Telescope, also allowed for the inclination of +a few stellar systems to be constrained and a few candidate +planets to be confirmed (Benedict et al. (2002), Benedict et al. +(2006)), while others were finally identified as brown dwarfs +or low-mass stars (Bean et al. (2007); Benedict et al. (2010)). +Today, the position and proper motion measurements obtained +with the telescopes Hipparcos (Perryman et al. (1997), van +Leeuwen (2007)) and Gaia (Gaia Collaboration 2020) allow us +to combine the RV data and more precise absolute astrometry +for a large number of systems. Since the publication of the first +Gaia data release (DR1), a few studies have proven the efficiency +of combining RV data with absolute and/or relative astrometry +to improve the constraints on the orbital parameters and mass +of a companion (Grandjean et al. (2019), Brandt et al. (2019), +Damasso et al. (2020), Lagrange et al. (2020), Nielsen et al. +(2020), Venner et al. (2021), Brandt et al. (2021a), Brandt et al. +Article number, page 1 of 17 +arXiv:2301.01263v1 [astro-ph.EP] 3 Jan 2023 + +A&A proofs: manuscript no. 45396corr +(2021b), Kiefer et al. (2021), Li et al. (2021), Feng et al. (2022)). +In this paper, we focus on seven long-period single com- +panions detected by the RV method, and combine the available +RV data with Hipparcos and Gaia early data release 3 (hereafter +EDR3) absolute astrometry and, when available, relative astrom- +etry, to improve the orbital parameters and determine the true +mass of these companions. In Section 2, we describe our target +selection method and present the RV, HCI, and astrometric data +used in our study. Section 3 presents the method used to per- +form the orbital fitting and, there, we provide the new orbital +parameters and mass found for each target. Finally, we discuss +the results in Section 4. +2. Target selection and data +2.1. Target selection +We first selected the planetary systems in the exoplanet.eu cat- +alog (Schneider et al. 2011) for which a single companion has +been reported with a semi-major axis greater than 5 au using the +RV method. Twenty-five companions were found with such cri- +teria. For nine of them (HD 13724 B, HD 25015 b, HD 181234 +b, and HD 219828 B (Feng et al. 2022); HD 92987 B (Venner +et al. 2021); HIP 36985 B (Biller et al. 2022); and HD 98649 b, +HD 196067 b, and HD 221420 B (Li et al. 2021)), the orbital +parameters and the true mass have already been properly deter- +mined in previous studies. We, therefore, do not consider them +in the present study. +We first discarded the HD 95872 system because no Hip- +parcos data were available. We then discarded four systems for +which the available RV time series did not cover both extrema +of the RV variations and the orbital period could not be prop- +erly determined (HD 26161, HD 120066, HD 150706, and HD +213472). In those four cases, the combination of RV and abso- +lute astrometry did not allow us to constrain the orbital param- +eters, the orbital inclination, or the true mass of the companion. +Finally, we discarded four systems for which the orbital period +was well covered by the RV data, but the coupling with absolute +astrometry did not allow us to constrain the orbital inclination +(HD 136925, HD 190984, HD 220773, and HD 238914). Indeed, +the variations in position and acceleration of the proper motion +of these stars were too small to constrain the orbital inclination +of the companion due to the low mass of the companion (< ∼2 +MJup) and/or the distance of the system. +Thus, we were left with seven systems for which the addi- +tion of absolute astrometry and/or new RV measurements and/or +relative astrometry measurements allowed us to determine the +exact nature of the companion: Epsilon Indi Ab, HD 13931, HD +1159554, HD 211847, HD 219077, HD 222155, and HIP 70849. +For three of these companions (Epsilon Indi Ab, HD 211847 B, +and HD 219077 b), a first estimation of their orbital inclination +and true mass has been obtained by combining RV data and ab- +solute astrometry. Yet, thanks to additional data or more precise +astrometric measurements, we obtained more precise and sig- +nificantly different results from those reported in the previous +studies for six of these companions. In the case of HD 219077 b, +the differences were mainly found for the mass of the compan- +ion. They are mainly due to the uncertainties as to the host star’s +mass. +2.2. RV data +The RV data used in this study were obtained with different spec- +trographs between 1997 and 2021. The HARPS (Mayor et al. +2003) data were taken from the ESO archives; the ELODIE +(Baranne et al. 1996) and SOPHIE (Perruchot et al. 2008) +data were retrieved from the OHP archives; and the CORALIE +(Queloz et al. 1999), the HIRES (Vogt et al. 1994), the UVES +(Dekker et al. 2000), the AAT (Diego et al. 1990), the CES +(Enard 1982) long camera (LC), and the CES very long camera +(VLC) data were taken from the literature. +As instrument upgrades can lead to new RV offsets, the same +instrument before and after a major upgrade is considered as two +different instruments. Consequently, HARPS data obtained be- +fore and after the optical fiber upgrade in 2015 (Lo Curto et al. +2015) are referred to as H03 and H15, respectively. The SO- +PHIE data obtained before and after the spectrograph upgrade in +2011 (Bouchy et al. 2013) are referred to as SOPHIE and SO- +PHIE+, respectively. The HIRES data obtained before and af- +ter the upgrade of the spectrograph in 2004 (Tal-Or et al. 2019) +are referred to as Hir94 and Hir04, respectively. Finally, the +CORALIE spectrograph had two major upgrades in 2007 (Sé- +gransan et al. 2010) and in 2014. The data obtained before 2007 +and after 2014 are referred to as C98 and C14, respectively, and +the data obtained between 2007 and 2014 are referred to as C07. +2.3. Direct imaging data +In three cases, HCI data are available in the SPHERE archive +and can provide relative astrometry. The three targets were ob- +served in angular (and spectral) differential imaging (A(S)DI, +Marois et al. (2006)) using the telescope in pupil tracking mode. +The standard observing mode of SPHERE was used, with IRDIS +(Dohlen et al. 2008) dual band images at H2 and H3 (K1 and +K2, respectively) and IFS (Claudi et al. 2008) data covering the +YJ (YJH, respectively) bands. The observing log is given in Ta- +ble.1. Whenever possible, the robust PACO A(S)DI algorithm +(Flasseur et al. (2020a), Flasseur et al. (2020b)) was used. The +processing step takes advantage of the developments made to the +COBREX data center pipeline (the prereduction improvement as +well as the improvement of the detection capability of PACO). If +a dataset did not sufficiently cover the field-of-view rotation to +apply ADI-based algorithms, the SPECAL (Galicher et al. 2018) +No-ADI algorithm was used. +In those three systems, only one companion was detected +(HD 211847 B). The detected companion is characterized in Ta- +ble.2. No detection above 5σ was found around HD 219077. Six +sources were detected around HIP 70849 but, given their posi- +tion in a color-magnitude diagram and their separations, they are +likely background sources. +2.4. Absolute astrometry +We used measurements from Hipparcos obtained around epoch +1991.25 and from Gaia EDR3 (Gaia Collaboration et al. (2016), +Gaia Collaboration et al. (2021)) obtained around epoch 2016.0. +For each target, the stellar acceleration was determined from the +proper motion and the position values were measured by Hip- +parcos and Gaia with an interval of about 25 years. We con- +sidered the proper motion values published by Brandt (2021) in +the Hipparcos-Gaia Catalog of Accelerations (HGCA). More- +over, a more accurate tangential proper motion (µHip−EDR3) was +estimated by the difference between the position measurements +obtained by Hipparcos and Gaia divided by the time interval be- +Article number, page 2 of 17 + +F. Philipot et al.: Updated characterisation of long-period single companion +Table 1: Observing logs. +STAR +DATE OBS +FILTER +DIT(s)×Nframea +∆ PA (°)a +Seeing (")b +Airmassb +τ0 (ms)a,b +Program ID +HIP 70849 +2015-05-05 +DB_H23 +64x64 +40.4 +1.12 +1.08 +0.0012 +095.C-0298(A) +HD 211847 +2015-06-10 +DB_K12 +64x8 +5.8 +1.31 +1.05 +0.0025 +095.C-0476(A) +HD 219077 +2015-06-09 +DB_K12 +8x64 +3.3 +1.25 +1.31 +0.0026 +095.C-0476(A) +Notes : a: DIT is the detector integration time per frame. ∆ PA is the amplitude of the parallactic rotation. τ0 corresponds to the +coherence time. b are values extracted from the updated DIMM information, averaged over the sequence. +Table 2: Relative astrometry for HD 211847 companion. +Sources +JD - 2400000 +IRDIS filter +SEP (mas) +PA (deg) +HD 211847 B +57183.39 +K12 +220±4.73 +194.5±2.23 +Notes : The errors displayed are 1σ. The relative astrometry +combines the astrometry measured in the dual bands. +tween the two measurements (∼25 years). The proper motion +values used for each star are given in table A.1. +3. Updated orbital parameters and mass +3.1. Orbit fitting +Orbits were fitted using a custom MCMC tool, based on the em- +cee 3.0 python package (Foreman-Mackey et al. 2013). It uses +a mixture of move functions (such as the differential evolution +move function) to alleviate potential multimodality issues. The +Hipparcos/Gaia data processing uses the HTOF package (Brandt +et al. 2021) and borrows large sections of the orvara code (Brandt +et al. 2021) for the likelihood computation. The HTOF package +(Brandt et al. 2021) was used to fit the intermediate astrometric +data (IAD) from Hipparcos, based on the 1997 (Esa 1997) and +2007 (van Leeuwen 2007) reductions and from Gaia, thanks to +the Gaia Observation Forecast Tool (GOST) which allowed us +to obtain the estimated Gaia observations and scan angles for +each target, in order to reproduce proper motion and position of +each observation. Using the Hipparcos and Gaia positions and +the temporal baseline, the algorithm derived a tangential proper +motion value that allowed us to better constrain the orbital fit +when combined with RV data. +We considered ten free parameters for each system: the semi- +major axis (a), the eccentricity, the orbital inclination (i), the host +star mass, the companion mass, the longitude of ascending node +(Ω), the argument of periastron (ω), the phase, a stellar jitter, and +the distance of the system. In addition, to combine data from +different instruments, we added an instrumental offset for each +instrument as a free parameter of the model (see above). Finally, +we considered uniform priors for all fitting parameters, except +for the host star mass and the distance of the system for which +we considered Gaussian priors. +3.2. Results +3.2.1. Epsilon Indi A +Epsilon Indi is a triple system with a 0.76 ± 0.04 M⊙, K2V star +(Epsilon Indi A) and a binary composed of a 75.0 ± 0.8 MJup, +T1.5 brown dwarf (Epsilon Indi B) and a 70.1 ± 0.7 MJup, T6 +brown dwarf (Epsilon Indi C) separated by about 2.6 au (Di- +eterich et al. 2018). The projected separation between the binary +brown dwarfs and the star is about 1460 au. Combining RV data +and absolute astrometry based on Hipparcos and the Gaia data +release 2 (DR2) measurements, Feng et al. (2019) reported a gi- +ant planet with a semi-major axis of 11.55+0.98 +−0.86 au, a mass of +3.25+0.39 +−0.65 MJup, an inclination of 64.25+13.80 +−6.09 °, and an eccentric- +ity of 0.26+0.07 +−0.03. Yet, the Gaia EDR3 proper motion and posi- +tion measurements are significantly more precise compared to +the Gaia DR2 measurement and they significantly improve the +characterization of Epsilon Indi Ab. +We used 4278 RV measurements obtained with the HARPS +spectrograph between 2003 and 20161, 163 RV measurements +obtained with the UVES spectrograph between 1996 and 2017, +72 RV measurements obtained with the LC spectrograph be- +tween 1992 and 1997, and 53 RV measurements obtained with +the VLC spectrograph between 2000 and 2006. We also com- +bined these RV data with absolute astrometry based on Hippar- +cos and the Gaia EDR3 measurements (Fig.1). We found sig- +nificantly different orbital parameters with a semi-major axis of +8.8+0.2 +−0.1 au, a mass of 3.0 ± 0.1 MJup, an inclination of 91+4 +−5°, and +an eccentricity of 0.48 ± 0.01. It is important to note that if we +consider only the 539 HARPS RV data with a signal-to-noise +ratio greater than 110 and thus remove the high cadence obser- +vations made in August 2011, we find similar solutions. +3.2.2. HD 13931 +HD 13931 is a 1.02 ± 0.05 M⊙(Rosenthal et al. 2021), G0V +star. Based on 66 RV measurements obtained with the HIRES +spectrograph between 1998 and 2019, Rosenthal et al. (2021) +reported a giant planet with a semi-major axis of 5.323 ± 0.091 +au, a minimum mass of 1.911+0.077 +−0.076 MJup , and an eccentricity of +0.02+0.021 +−0.014. +We combined these RV data with absolute astrometry +(Fig.2). As the RV baseline is much larger than the orbital pe- +riod, the orbital parameters are well-constrained. As expected, +we found a semi-major axis and an eccentricity very close to +those reported by Rosenthal et al. (2021) with a = 5.33 ± 0.09 +au and e < 0.04. Using, in addition, the absolute astrometry, we +found an orbital inclination of either 39+13 +−8 ° or 141+9 +−18° and a true +mass of 3.1+0.8 +−0.7 MJup. +3.2.3. HD 115954 +HD 115954 is a 1.18 ± 0.06 M⊙, G0V star (Demangeon et al. +2021). Based on four RV measurements obtained with the +ELODIE spectrograph between 2004 and 2005 and 45 RV mea- +surements obtained with the SOPHIE spectrograph between +2009 and 2018, Demangeon et al. (2021) reported a giant planet +1 The 3636 RV data obtained between Julian days 2455790 and +2455805 were obtained to study high-frequency oscillations of the star. +These data were measured with high cadence, which led to a signifi- +cantly lower signal-to-noise ratio compared to the other data. +Article number, page 3 of 17 + +A&A proofs: manuscript no. 45396corr +Fig. 1: Orbital fits for Epsilon Indi Ab. Top: Fit of the Epsilon +Indi A RV data corrected from the instrumental offset (V0). Bot- +tom: Fit of the Epsilon Indi A astrometric acceleration in right +ascension (left) and declination (right). The black points corre- +spond to the measurements obtained with Hipparcos (1991.25) +and Gaia EDR3 (2016.0). In each plot, the black curve shows the +best fit. The color bar indicates the log likelihood of the different +fits plotted. +with a semi-major axis of 5.00+1.3 +−0.36 au, a minimum mass of +8.29+0.75 +−0.58 MJup , and an eccentricity of 0.487+0.095 +−0.041. +We combined these RV data with the absolute astrometry +data (Fig.3). We found a semi-major axis compatible with De- +mangeon et al. (2021) with a = 4.5+0.2 +−0.1 au and an eccentricity of +0.46±0.03. Using, in addition, the absolute astrometry, we found +an orbital inclination of 92+17 +−16° and a true mass of 8.5+0.6 +−0.4 MJup. +3.2.4. HD 211847 +HD 211847 is a 0.94±0.04 M⊙, G5V star (Sahlmann et al. 2011). +Sahlmann et al. (2011) reported a brown dwarf candidate orbit- +ing around HD 211847 based on 31 RV measurements obtained +with the CORALIE spectrograph between 2002 and 2009. How- +ever, only one minimum of the HD 211847 B RV curve was +covered by the dataset. Thus, the orbital parameters and min- +imum mass reported in this study are poorly constrained. Us- +Fig. 2: Orbital fits for HD 13931 b. Top: Fit of the HD 13931 +RV data corrected from the instrumental offset (V0). Bottom: +Fit of the HD 13931 astrometric acceleration in right ascen- +sion (left) and declination (right). The black points correspond to +the measurements obtained with Hipparcos (1991.25) and Gaia +(2016.0). In each plot, the black curve shows the best fit. The +color bar indicates the log likelihood of the different fits plotted. +ing the Levenberg-Marquardt method, they found ranges corre- +sponding to a 3σ confidence interval for the semi-major axis, +the eccentricity, and the minimum mass of 4.6–42 au, 0.48–0.95, +and 16.3–24.3 MJup , respectively. Moutou et al. (2017) obtained +one HCI detection with SPHERE of HD 211847 B for a pro- +jected separation of 11.3 au. Using the BT-Settl models (Allard +2014), they fit the HD 211847 B spectrum and found a low stel- +lar mass of 155 ± 9 MJup assuming an age of 3 Gyr for the host +star. Based on the result of Sahlmann et al. (2011), Moutou et al. +(2017) estimated the inclination of the companion orbit to be +around seven°. Recently, combining the CORALIE RV measure- +ment and the absolute astrometry, Feng et al. (2022) reported HD +211847 B as a brown dwarf with a semi-major axis of 4.514+0.458 +−0.287 +au, a mass of 55.32+1.335 +−18.48 MJup, an inclination of 163.649+36.239 +−5.017 °, +and an eccentricity of 0.419+0.035 +−0.064. +Article number, page 4 of 17 + +RadialVelocities +LLH +best +100 +H03, VO-corrected +H15, V0-corrected +-8524 +UVES, Vo-corrected +LC, Vo-corrected +[s/w] +50 +VLC, VO-corrected +8526 +Radial velocity [ +0 +-8528 +50 +-8530 +-100 +8532 +100 +0 +上 +100 +-6000 +-4000-2000 +0 +2000 +4000 +Time (JD-2454000)ProperMotionRA +ProperMotionDec +best +best +2535.0 +-8524 +3968 +2535.5 +μα*[mas/yr] +-8526 +3967 +2536.0 +8528 +3966 +2536.5 +2537.0 +-8530 +3965 +2537.5 +8532 +3964 +LLH +1 +0-0 +0 +0 +-2 +-1 +6000-4000-2000 +0 +2000 +4000 +6000-4000-2000 +0 +2000 +4000 +Epoch[jD-2454000) +Epoch[jD-2454000)Radial Velocities +LLH +30 +best +Hir94, V0-corrected +138 +Hir04, VO-corrected +20 +[s/w] +-140 +10 +I velocity +-142 +0 +Radial +-10 +-144 +-20 +-146 +-30 +10 +O-C +0 +-10 +-2000 +0 +2000 +4000 +Time(JD-2454000)ProperMotionRA +ProperMotionDec +99.6 +best +best +-138 +183.2 +99.4 +99.2 +-183.4 +-140 +μα*[mas/yr] +99.0 +183.6 +masi +98.8 +-142 +183.8 +98.6 +98.4 +184.0 +-144 +98.2 +-184.2 +98.0 +146 +LLH +U +-4000-2000 +0 +2000 +4000 +6000 +-4000-2000 +0 +2000 +4000 +6000 +EpochfiD-2451544.5 +Epoch[JD-2451544.5)F. Philipot et al.: Updated characterisation of long-period single companion +Fig. 3: Orbital fits for HD 115954 b. Top: Fit of the HD 115954 +RV data corrected from the instrumental offset (V0). Bottom: +Fit of the HD 115954 astrometric acceleration in right ascen- +sion (left) and declination (right). The black points correspond to +the measurements obtained with Hipparcos (1991.25) and Gaia +(2016.0). In each plot, the black curve shows the best fit. The +color bar indicates the log likelihood of the different fits plotted. +We combined the RV dataset used by Sahlmann et al. (2011), +the relative astrometry observation obtained with SPHERE in +June 2015, and the absolute astrometry (Fig.4). Adding one rel- +ative astrometry observation allowed us to properly constrain the +orbital parameters and the mass of HD211847 B with results sig- +nificantly different from those reported by Feng et al. (2022). We +found a semi-major axis of 6.78 ± 0.08 au and an eccentricity of +0.59+0.01 +−0.02. Using, in addition, the absolute astrometry, we found +an orbital inclination of 172.3+0.05 +−0.04° and a true mass of 148 ± 5 +MJup. We note that by taking only RV data and absolute astrom- +etry into account, we found very poorly constrained solutions +with large uncertainties as to the semi-major axis (16 - 30 au) +and mass (80 - 140 MJup). Moreover, the solutions found are not +in agreement with those reported by Feng et al. (2022) or with +the solutions found when adding the HCI data. +Fig. 4: Orbital fits for HD 211847 B. Top left: Fit of the HD +211847 RV data corrected from the instrumental offset (V0). +Top right: Fit of HD 211847 relative astrometry data. The red +cross corresponds to the measurement obtained with SPHERE. +Bottom: Fit of the HD 211847 astrometric acceleration in right +ascension (left) and declination (right). The black points corre- +spond to the measurements obtained with Hipparcos (1991.25) +and Gaia (2016.0). In each plot, the black curve shows the best +fit. The color bar indicates the log likelihood of the different fits +plotted. +Article number, page 5 of 17 + +Radial Velocities +LLH +100 +156 +50 +[s/w] +-158 +0 +Radial velocity [ +50 +-160 +-100 +162 +150 +best +ELODIE,vO-corrected +SoPHIE, VO-corrected +164 +SOPHIE+,VO-corrected +200 +50 +-1000 +0 +1000 +2000 +3000 +4000 +Time (JD-2454000)ProperMotionRA +ProperMotionDec +21.8 +best +best +74.0 +21.6 +-156 +74.2 +157 +21.4 +μα*[mas/yr] +74.4 +-158 +μs[mas/yr] +21.2 +74.6 +-159 +21.0 +160 +74.8 +20.8 +-75.0 +-161 +20.6 +162 +75.2 +20.4 +163 +-75.4 +LLH +0.5 +0-0 +0 +0 +0.0 +0.5 +0 +2000 +4000 +6000 +-1 +-4000 +0-2000 +二4000-2000 +0 +2000 +4000 +6000 +Epoch[jD-2451544.5) +Epoch[JD-2451544.5】RadialVelocities +LLH +best +-128 +100 +C98, V0-corrected +-129 +0 +-130 +[s/w] +velocity +100 +131 +132 +200 +Radial +-133 +-300 +-134 +-400 +135 +-136 +25 +0 +25 +100015002000250030003500 +Time (JD-2451544.5)Sky plane +LLH +50 +best b +127 +-128 +129 +[mas] +-50 +-130 +DEC +-100 +-131 +-150 +-132 +-133 +-200 +134 +100 +50 +0 +-50 +-100 +-150 +RA [mas]Proper MotionRA +ProperMotionDec +17.5 +62.5 +best +best +15.0 +-127 +60.0 +12.5 +-128 +57.5 +μs[mas/yr] +10.0 +-129 +55.0 +7.5 +-130 +52.5 +131 +5.0 +50.0 +-132 +2.5 +47.5 +0.0 +-133 +45.0 +134 +2.5 +LLH +5 +L +0-0 +0 +.5 +-5 +-4000-2000 +0 +2000 +4000 +6000 +-4000-2000 +0 +2000 +4000 +6000 +Epoch[D-2451544.5) +Epoch[D-2451544.5)A&A proofs: manuscript no. 45396corr +3.2.5. HD 219077 +Based on 63 CORALIE RV measurements obtained between +1999 and 2012 and 30 HARPS RV measurements obtained be- +tween 2003 and 2012, Marmier et al. (2013) reported a very ec- +centric giant planet with a semi-major axis of 6.22 ± 0.09 au +and a minimum mass of 10.39 ± 0.09 MJup. It is important to +note that the RV data used by Marmier et al. (2013) are not pub- +licly available. Based on 72 pieces of RV data obtained with the +AAT spectrograph between 1998 and 2015, Kane et al. (2019) +reported slightly different properties for HD 219077 with a semi- +major axis of 7.03+0.20 +−0.21 au and a minimum mass of 13.40+0.76 +−0.78 +MJup. These differences are probably due to the different assump- +tions on the mass of the star. Indeed, Marmier et al. (2013) used +the values reported by Hipparcos (M⋆= 1.05 ± 0.02 M⊙) while +Kane et al. (2019) used the values reported in Valenti & Fischer +(2005) (M⋆= 1.51±0.13 M⊙). Recently, Feng et al. (2022) com- +bined the AAT RV measurements used by Kane et al. (2019) +and 33 HARPS RV measurements obtained between 2003 and +2012 with the absolute astrometry based on Hipparcos and the +Gaia EDR3 measurements and found a semi-major axis close to +Marmier et al. (2013), an orbital inclination of 90.178+9.527 +−9.462°, and +a true mass of 9.620+1.001 +−0.733 MJup. The prior on the mass of the star +is not given. +For this study, as the data used by Marmier et al. (2013) +are not available, we considered the HARPS and AAT RV mea- +surements used on Feng et al. (2022) and the 65 CORALIE RV +measurements available on the DACE archive2 obtained between +1999 and 20123. For the mass of the star, we considered the value +given by Kervella et al. (2022) based on the Gaia DR3 results +(M⋆= 1.15 ± 0.06 M⊙). We combined the RV data with the ab- +solute astrometry (Fig.5). We found a semi-major axis and an +eccentricity close to those reported in the previous studies with +a = 6.4 ± 0.1 au and e = 0.769 ± 0.002 and an orbital inclination +close to that of Feng et al. (2022) with either i = 83 ± 3° or i = +97±3°. Considering the star mass found by Kervella et al. (2022), +we found a planetary mass at 11.3 ± 0.4 MJup. However, consid- +ering that the mass used in Valenti & Fischer (2005) would lead +to a mass close to the deuterium-burning limit, Mb = 13.6 ± 0.5 +MJup. Due to the uncertainties on the mass of the host star, it is +not possible to determine the exact nature of HD 219077 b. +3.2.6. HD 222155 +HD 222155 is a 1.13 ± 0.11 M⊙, G2V star (Boisse et al. 2012). +Based on 44 RV measurements obtained with the ELODIE spec- +trograph between 1997 and 2005 and 67 RV measurements ob- +tained with the SOPHIE spectrograph between 2007 and 2011, +Boisse et al. (2012) reported a giant planet with a semi-major +axis of 5.1+0.6 +−0.7 au, a minimum mass of 1.90+0.67 +−0.53 MJup , and an +eccentricity of 0.38+0.28 +−0.32. +We considered 31 additional pieces of SOPHIE RV data ob- +tained between 2011 and 2016. We combined the RV data with +the absolute astrometry (Fig.6). We found orbital parameters +within the error bars associated with the values found by Boisse +et al. (2012) with a = 4.7±0.1 au and e = 0.34±0.09. As the RV +baseline is now much larger than the orbital period, the orbital +parameters are better constrained. Using, in addition, the abso- +2 https://dace.unige.ch +3 The CORALIE RV data available on DACE and the HARPS RV data +available on the ESO archive cover the same time base as those used by +Marmier et al. (2013). +Fig. 5: Orbital fits for HD 219077 b. Top: Fit of the HD 219077 +RV data corrected from the instrumental offset (V0). Bottom: +Fit of the HD 219077 astrometric acceleration in right ascen- +sion (left) and declination (right). The black points correspond to +the measurements obtained with Hipparcos (1991.25) and Gaia +(2016.0). In each plot, the black curve shows the best fit. The +color bar indicates the log likelihood of the different fits plotted. +lute astrometry, we found an orbital inclination of either 66+14 +−11° +or 115+13 +−16° and a true mass of 2.1+0.3 +−0.2 MJup. +3.2.7. HIP 70849 +HIP 70848 is a 0.63±0.03 M⊙, K7V star (Ségransan et al. 2011). +Ségransan et al. (2011) reported the first detection of HIP 70849 +b based on 18 RV measurements obtained with the HARPS spec- +trograph between 2006 and 2010. However, only one minimum +of the HIP 70849 b RV curve was covered by the dataset. The +observations carried out by Ségransan et al. (2011) led to poorly +constrained orbital parameters and minimum mass. Using a ge- +netic algorithm followed by MCMC simulations, they reported +a semi-major axis between 4.5 and 36 au, a minimum mass be- +tween 3 and 15 MJup , and an eccentricity between 0.47 and 0.96 +with ranges corresponding to a 3σ confidence interval. +Article number, page 6 of 17 + +RadialVelocities +LLH +250 +best +C98.o-corrected +-386 +200 +Co7.vo-corrected +[m +Ho3,VO-corrected +150 +AAT, VO-corrected +388 +velocity +100 +390 +50 +0 +-392 +R +-50 +-100 +-394 +20 +0-0 +0 +-20 +-2000 +0 +2000 +Time（iD-2454000)ProperMotionRA +ProperMotionDec +-422 +best +best +480 +386 +-423 +μα*[mas/yr] +479 +-388 +μs[mas/yr] +-424 +-390 +478 +-425 +-392 +477 +-426 +394 +476 +2.5 +LLH +0.0 +0.0 +2.5 +-6000-4000-2000 +0 +2000 +4000 +6000-4000-2000 +0 +2000 +4000 +Epoch[jD-2454000) +Epoch[jD-2454000)F. Philipot et al.: Updated characterisation of long-period single companion +Fig. 6: Orbital fits for HD 222155 b. Top: Fit of the HD 222155 +RV data corrected from the instrumental offset (V0). Bottom: +Fit of the HD 222155 astrometric acceleration in right ascen- +sion (left) and declination (right). The black points correspond to +the measurements obtained with Hipparcos (1991.25) and Gaia +(2016.0). In each plot, the black curve shows the best fit. The +color bar indicates the log likelihood of the different fits plotted. +We considered 39 additional pieces of HARPS RV data ob- +tained between 2011 and 2021. We combined the RV data with +the absolute astrometry (Fig.7). With these additional observa- +tions, the dataset then covered two minimum and one maximum +of the RV curve of HIP 70849 b and this allowed us to prop- +erly constrain the properties of the companion. We found a semi- +major axis of 3.99+0.06 +−0.07 au and an eccentricity of 0.65+0.02 +−0.01. Using, +in addition, the absolute astrometry, we found an orbital inclina- +tion of 96 ± 16° and a true mass of 4.5+0.4 +−0.3 MJup. +4. Summary and concluding remarks +Combining RV measurements from various spectrographs with +absolute astrometry based on Hipparcos and Gaia EDR3 data +and, when available, relative astrometry, we determined the or- +bital parameters and, in particular, the orbital inclination and the +Fig. 7: Orbital fits for HIP 70849 b. Top: Fit of the HIP 70849 +RV data corrected from the instrumental offset (V0). Bottom: +Fit of the HIP 70849 astrometric acceleration in right ascen- +sion (left) and declination (right). The black points correspond to +the measurements obtained with Hipparcos (1991.25) and Gaia +(2016.0). In each plot, the black curve shows the best fit. The +color bar indicates the log likelihood of the different fits plotted. +true mass of seven long-period single companions detected by +the RV method. Figure 8 summarizes the true mass and the semi- +major axis of the companions and compares them with the previ- +ous estimations. Clearly, Gaia EDR3 data allow for a better de- +termination of these companions’ orbital parameters and mass. +All of these companions have true masses of 2 MJup or more and +orbit between 3.9 - 9 au from their stars. Absolute astrometry +would probably help to determine the true mass of planets with +a period larger than the duration of Gaia DR3 observations (P +> ∼1000 d) down to 1 MJup , provided the RV variations are +well covered and the variations in position and acceleration of +the proper motion of the star are large enough. In practice, in +most cases, when the period is not well constrained by the RV +data, the impact of the coupling of RV data with absolute as- +trometry is more limited. An illustration of this is the case of +HD 211847 B for which, by combining RV data that cover only +a minimum of the RV variations with absolute astrometry, Feng +Article number, page 7 of 17 + +Radial Velocities +LLH +60 +best +ELODIE,VO-corrected +40 +SOPHIE, VO-corrected +464 +SOPHIE+, VO-corrected +Radial velocity [m/s] +20 +-466 +-468 +-470 +20 +-472 +-40 +474 +-60 +50 + 0 4n +O-C +0 +50 +4000 +-2000 +0 +2000 +4000 +Time (JD-2454000)ProperMotionRA +ProperMotionDec +195.8 +best +-117.2 +best +-464 +-117.4 +. +195.6 +-466 +*[mas/yr] +uolmas/yr +117.6 +195.4 +117.8 +-468 +μa* +195.2 +118.0 +-470 +195.0 +118.2 +-472 +194.8 +118.4 +474 +0.5 F +LLH +0-0 +0 +0.0 +-0.5 +6000-4000-2000 +0 +2000 +4000 +6000-4000-2000 +0 +2000 +4000 +Epoch[jD-2454000) +Epoch[jD-2454000)Radial Velocities +LLH +-172 +50 +-173 +Radial velocity [m/s] +0 +-174 +-50 +-175 +100 +-176 +-150 +best +177 +H03, VO-corrected +H15, V0-corrected +-200 +25 +O-C +0 +25 +0 +10002000300040005000 +Time (JD-2454000)ProperMotionRA +ProperMotionDec +-172 +best +-201 +best +-44 +-173 +45 +202 +-174 +μα*[mas/yr] +[mas/yr] +46 +-175 +-203 +47 +176 +-204 +-48 +-177 +-178 +-49 +205 +LLH +0.0 +0.0 +2.5 +-2.5 +-4000-2000 +0 +2000 +4000 +6000 +-4000 +-2000 +0 +2000 +4000 +6000 +Epoch[jD-2451544.5) +Epoch[iD-2451544.5)A&A proofs: manuscript no. 45396corr +Table 3: Summary of posteriors obtained with our MCMC algorithm. +Parameter +Eps Ind A +HD 13931 +HD 115954 +HD 211847 +HD 219077 +HD 222155 +HIP 70849 +a (au) +8.8+0.2 +−0.1 +5.33 ± 0.09 +4.5+0.2 +−0.1 +6.78 ± 0.08 +6.4 ± 0.1 +4.7 ± 0.1 +3.99+0.06 +−0.07 +Period (days) +10932+266 +−228 +4442+49 +−46 +3258+179 +−190 +6199+52 +−46 +5514+44 +−39 +3470+102 +−106 +3649 ± 18 +Eccentricity +0.48 ± 0.01 +< 0.04 +0.46 ± 0.03 +0.769 ± 0.002 +0.59+0.01 +−0.02 +0.34 ± 0.09 +0.65+0.02 +−0.01 +Inclination (°) +91+4 +−5 +39+13 +−8 or 141+9 +−18 +59+5 +−4 or 127 ± 4 +172.3+0.5 +−0.4 +83 ± 3 or 97 ± 3 +66+14 +−11 or 115+13 +−16 +96 ± 16 +Mass (MJup ) +3.0 ± 0.1 +3.1+0.8 +−0.7 +8.5+0.6 +−0.4 +148 ± 5 +11.3 ± 0.4 +2.1+0.3 +−0.2 +4.5+0.4 +−0.3 +Ω (°) +58 ± 5 +343+17 +−19 or 110+19 +−24 +211+25 +−28 +184 ± 5 +135+38 +−21 or 347+28 +−30 +264+34 +−33 or 180+34 +−35 +35 ± 6 +ω (°) +85 ± 3 +74 - 227 +173+7 +−8 +168+5 +−4 +56.2 ± 0.4 +153 - 217 +182 ± 1 +Phase +0.37 ± 0.01 +0.38 - 0.86 +0.40+0.09 +−0.06 +0.420 ± 0.004 +0.354+0.002 +−0.003 +0.98+0.04 +−0.13 +0.745+0.007 +−0.006 +Jitter (m/s) +3.37 ± 0.04 +2.9 ± 0.3 +6.6+1.3 +−1.1 +11.0+2.8 +−2.0 +4.3+0.3 +−0.2 +12.7+0.9 +−0.8 +6.7+0.8 +−0.7 +H03 = −39972 ± 1 +Hir94 = −13 ± 1 +ELODIE = −14757+52 +−41 +C98 = 6907 ± 17 +C98 = −30867+2 +−1 +ELODIE = −3999 ± 3 +H03 = 53 ± 1 +Instrumental +H15 = −39952+2 +−1 +Hir04 = −8.0 ± 0.5 +SOPHIE = −14768+9 +−8 +C07 = 6849 ± 22 +C07 = −30864+8 +−9 +SOPHIE = −3950+3 +−2 +H15 = 65+2 +−1 +offset (m/s) +UVES = −5 ± 1 +SOPHIE+ = −14743+4 +−3 +H03 = −30830+1 +−2 +SOPHIE+ = −3923+4 +−3 +LC = −39978+2 +−1 +AAT = −68 ± 1 +VLC = −39976+2 +−1 +Notes: The results were obtained by combining RV, absolute astrometry, and, when available, relative astrometry. We provide 68% +confidence intervals for each parameter and the median is only given when the probability distribution has a profile close to a +Gaussian distribution. +Fig. 8: Update of the orbital parameters and masses of the seven +analyzed systems thanks to the combination of absolute astro- +metric and RV data and, when available, absolute astrometry +data. For each system, a dotted line between two solutions was +drawn to allow for the different solutions obtained to be com- +pared. +et al. (2022) reported a mass of about 55 MJup, corresponding to +a brown dwarf. Yet, HCI revealed a companion and the fit of the +RV and the relative and absolute astrometry leads to a mass of +about 150 MJup instead. +We conclude that Gaia/Hipparcos can help to further con- +strain the orbital parameters of long-period RV planets, provid- +ing good coverage of the RV variations is available. Otherwise, +additional information is needed, such as relative astrometry, +provided by DI or interferometry. +Acknowledgements. This study was funded by a grant from PSL/OCAV. This +project has also received funding from the European Research Council (ERC) +under the European Union’s Horizon 2020 research and innovation programme +(COBREX; grant agreement n° 885593). This work presents results from +the European Space Agency (ESA) space mission Gaia. Gaia data are being +processed by the Gaia Data Processing and Analysis Consortium (DPAC). +Funding for the DPAC is provided by national institutions, in particular the +institutions participating in the Gaia MultiLateral Agreement (MLA). The +Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia archive +website is https://archives.esac.esa.int/gaia. This publications makes use of +the The Data & Analysis Center for Exoplanets (DACE), which is a facility +based at the University of Geneva (CH) dedicated to extrasolar planets +data visualisation, exchange and analysis. DACE is a platform of the Swiss +National Centre of Competence in Research (NCCR) PlanetS, federating the +Swiss expertise in Exoplanet research. The DACE platform is available at +https://dace.unige.ch. This research has made use of the SIMBAD database and +VizieR catalogue access tool, operated at CDS, Strasbourg, France. Based on +data retrieved from the SOPHIE archive at Observatoire de Haute-Provence +(OHP), available at atlas.obs-hp.fr/sophie. Based on spectral data retrieved +from the ELODIE archive at Observatoire de Haute-Provence (OHP). 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A., Wang, S., Horner, J., et al. 2020, MNRAS, 492, 377 +Article number, page 9 of 17 + +A&A proofs: manuscript no. 45396corr +Appendix A: Proper motion values +Table A.1: Proper motion values from HGCA. +Star +Eps ind A +HD 13931 +HD 115954 +HD 211847 +HD 219077 +HD 222155 +HIP 70849 +µα +Hip (mas/yr) +3964.6 ± 0.4 +98.8 ± 0.8 +−74.6 ± 0.7 +56.1 ± 1.1 +477.5 ± 0.4 +195.3 ± 0.5 +−47.0 ± 2.1 +µδ +Hip (mas/yr) +−2537.1 ± 0.4 +−183.7 ± 0.6 +20.9 ± 0.6 +14.1 ± 0.8 +−424.9 ± 0.4 +−117.8 ± 0.6 +−203.3 ± 1.9 +µα +EDR3 (mas/yr) +3966.7 ± 0.1 +98.57 ± 0.04 +−74.77 ± 0.02 +44.43 ± 0.03 +478.30 ± 0.03 +195.31 ± 0.02 +−44.05 ± 0.02 +µδ +EDR3 (mas/yr) +−2536.2 ± 0.1 +−183.41 ± 0.04 +21.49 ± 0.02 +9.66 ± 0.04 +−424.43 ± 0.04 +−117.34 ± 0.02 +−201.58 ± 0.3 +µα +Hip−EDR3 (mas/yr) +3965.02 ± 0.01 +98.45 ± 0.03 +−74.79 ± 0.02 +47.90 ± 0.04 +478.36 ± 0.01 +195.25 ± 0.02 +−44.43 ± 0.06 +µδ +Hip−EDR3 (mas/yr) +−2537.25 ± 0.01 +−183.51 ± 0.02 +−21.41 ± 0.02 +−10.44 ± 0.03 +−424.40 ± 0.01 +−117.39 ± 0.02 +−202.05 ± 0.04 +Notes: µHip corresponds to the proper motion obtained by Hipparcos. µEDR3 corresponds to the proper motion obtained by Gaia +EDR3. µHip−EDR3 corresponds to the proper motion obtained by the Hipparcos-Gaia EDR3 positional difference. +Appendix B: MCMC priors +Table B.1: Priors considered for each free parameter. +Parameter +Eps ind A +HD 13931 +HD 115954 +HD 211847 +HD 219077 +HD 222155 +HIP 70849 +a (au) +[1,20] +[1,10] +[1,10] +[1,100] +[1,10] +[1,10] +[1,10] +Eccentricity +[0,0.95] +[0,0.95] +[0,0.95] +[0,0.95] +[0,0.95] +[0,0.95] +[0,0.95] +Inclination (°) +[0,180] +[0,180] +[0,180] +[0,180] +[0,180] +[0,180] +[0,180] +Mass (MJup ) +[1,20] +[1,20] +[1,20] +[1,500] +[1,20] +[1,20] +[1,20] +Ω (°) +[0,360] +[0,360] +[0,360] +[0,360] +[0,360] +[0,360] +[0,360] +ω (°) +[0,360] +[0,360] +[0,360] +[0,360] +[0,360] +[0,360] +[0,360] +Phase +[0,1] +[0,1] +[0,1] +[0,1] +[0,1] +[0,1] +[0,1] +Jitter (m/s) +[0,10] +[0,10] +[0,10] +[0,20] +[0,10] +[0,20] +[0,10] +Star mass (M⊙) +0.76 ± 0.04 +1.02 ± 0.05 +1.18 ± 0.06 +0.94 ± 0.05 +1.15 ± 0.06 +1.13 ± 0.11 +0.63 ± 0.03 +Distance (pc) +3.622 ± 0.004 +44.2 ± 1.4 +218 ± 2 +50.6 ± 3.3 +29.3 ± 0.2 +49.1 ± 0.9 +24.0 ± 0.7 +H03: [-41,-39] +Hir94: [-1,1] +ELODIE: [-15,-13] +C98: [5,7] +C98: [-31,-29] +ELODIE: [-5,-3] +H03: [-1,1] +Instrumental +H15: [-41,-39] +Hir04: [-1,1] +SOPHIE: [-15,-13] +C07: [5,7] +C07: [-31,-29] +SOPHIE: [-5,-3] +H03: [-1,1] +offset (km/s) +UVES: [-1,1] +SOPHIE+: [-15,-13] +H03: [-31,-29] +SOPHIE+: [-5,-3] +LC: [-41,-39] +AAT: [-1,1] +VLC: [-41,-39] +Article number, page 10 of 17 + +F. Philipot et al.: Updated characterisation of long-period single companion +Appendix C: MCMC results +Fig. C.1: Corner plot of the posteriors’ fit of Epsilon Indi A combined RV and absolute astrometry. An offset of 39.9 km/s was added +to V0, V1, V3, and V4 to improve readability. +Article number, page 11 of 17 + +75.64±1:4g +3.62±8:88 + 84.763:28 +O +[yr] = 29.93±:72 +28. +O +Ob [deg] = +O +O +O +O +smpa +%心 +mps +V2 [mps] +pargb [deg] +Ib [deg] +Ob [deg]A&A proofs: manuscript no. 45396corr +Fig. C.2: Corner plot of the posteriors’ fit of HD 13931 combined RV and absolute astrometry. +Article number, page 12 of 17 + +vo[mps +V1[mps] +[sw] +3 +jitter +1.02±8:85 +[pc] +0.02±8:83 +151.65±74:69 +mb [Mi +:3.07±8:65 +[Mjup] +39.799.12 +lb [deg] +% +Ob[deg] +公心 + 0? +Q: +vo [mps] +V1 [mps] + jitter [ms] +mi [Msun] +dstar [pc] +smab [au] +pargb [deg] +ptimeb [yr] +perb [yr] +mb [Mjup] +Ib [deg] +Ob [deg]F. Philipot et al.: Updated characterisation of long-period single companion +Fig. C.3: Corner plot of the posteriors’ fit of HD 115954 combined RV and absolute astrometry. An offset of 14 km/s was added to +V0, V1, and V2 to improve readability. +Article number, page 13 of 17 + +/2[mps +742.683:19 +jitter [ms] = 6.571:33 +[s a! +.18±0.03 +2 +[ne] ew +00.52 +.72.55±9.59 +6 [yr] = 3.55±8.5] +1? +3.92±0.59 +perb [yr] +: 8.48±8:4 +b [deg] = 92.39t15.81 +Ob [deg] : +:211.522 +[deg] +%%%% +115 +Vo [mps] +Vi [mps] +V2 [mps] + jitter [ms] +m1 [Msun] +dstar Ipc' +smab [au] +pargb [deg] +ptimeb [yr] +perb [yr] +mb [Mjup] +Ib [deg] +Ob [deg]A&A proofs: manuscript no. 45396corr +Fig. C.4: Corner plot of the posteriors’ fit of HD 211847 combined RV, relative astrometry, and absolute astrometry. An offset of 6 +km/s was subtracted to V0 and V1 to improve readability. +Article number, page 14 of 17 + +vo[mps +: 907.20±19:36 +18 +: 0.94±8.83 +0.60-3.05 +ab [au] = 6.78±8:89 +oargb[deg] +120 +ptimeb [yr] = 7.12±8:05 +[yr] = 16.97±8:13 +LE +O[deg]: +? +多 +vo [mps] +V1 [mps] +jitter [ms] +m1 [Msun] +dstar [pc] +smab [au] +pargb [deg] +ptimeb [yr] +perb [yr] +mb [Mjup] +[6ap] +Ob [deg]F. Philipot et al.: Updated characterisation of long-period single companion +Fig. C.5: Corner plot of the posteriors’ fit of HD 219077 combined RV and absolute astrometry. An offset of 30 km/s was added to +V0, V1, and V2 to improve readability. +Article number, page 15 of 17 + +93±1.51 +-829.61±1:43 +4.27±83 +Msun +jitter [ms] +smab [au] +ptime, lyr? +I [deg] +O, [deg]A&A proofs: manuscript no. 45396corr +Fig. C.6: Corner plot of the posteriors’ fit of HD 222155 combined RV and absolute astrometry. An offset of 3.9 km/s was added to +V0, V1, and V2 to improve readability. +Article number, page 16 of 17 + +673.号 +.0 +[pc] = 49.108:93 +V2 [mps] +jitter [ms] +m1 [Msun +dstar [pc] +smab[au +pargb [deg] +ptimeb [yr] +perb [yr] +Ib [deg] +Ob [deg]F. Philipot et al.: Updated characterisation of long-period single companion +Fig. C.7: Corner plot of posteriors fit of HIP 70849 combined RV and absolute astrometry. +Article number, page 17 of 17 + +3.19±1:19 +心 +3.99±0.09 +0.65±8.82 +otimeb [yr] = 7.44±8:83 +ptimep [yr] +6 +4.40±8.2 +95.85-15.76 +120 +[6a p] 9] +Ob [deg] +25o +2 +s? +?q? +0% +心 +8 +Vo [mps] +vi [mps] + jitter [ms] +mi [Msun] +dstar [pc] +smab [au] +pargbdeg] +ptimeb [yr] +perplyr +mp [Mjup] +Ib [deg] +Ob [deg] \ No newline at end of file diff --git a/EtAzT4oBgHgl3EQfUPx6/content/tmp_files/load_file.txt b/EtAzT4oBgHgl3EQfUPx6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca39033dec282d4b6fa5175e788543a849052dd7 --- /dev/null +++ b/EtAzT4oBgHgl3EQfUPx6/content/tmp_files/load_file.txt @@ -0,0 +1,1412 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf,len=1411 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr ©ESO 2023 January 4, 2023 Updated characterization of long-period single companion by combining radial velocity, relative astrometry, and absolute astrometry F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot1,⋆, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Lagrange1,2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Rubini3, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Kiefer1, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Chomez1 1 LESIA, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen, 92190 Meudon, France 2 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 3 Pixyl S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' La Tronche, France Received XXX / Accepted XXX ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Thanks to more than 20 years of monitoring, the radial velocity (RV) method has detected long-period companions (P > 10yr) around several dozens of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Yet, the true nature of these companions remains unclear because of the uncertainty as to the inclination of the companion orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We wish to constrain the orbital inclination and the true mass of long-period single companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We used a Markov Chain Monte Carlo (MCMC) fitting algorithm to combine RV measurements with absolute astrometry and, when available, relative astrometry data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We have lifted the sin(i) indetermination for seven long-period companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We find true masses in the planetary mass range for the candidate planets detected in the following systems: Epsilon Indi A, HD 13931, HD 115954, and HD 222155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The mass of HD 219077 b is close to the deuterium-burning limit and its nature is uncertain because of the imprecise mass of the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using additional RV measurements, we refine the orbital parameters of HIP 70849 b and find a mass in the planetary range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' By combining RV data with absolute and relative astrometry, we significantly improve the characterization of HD 211847 B and properly determine its mass, which appears to be in the low-mass star range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This work illustrates how Gaia and Hipparcos allow for the orbital properties and masses of long-period RV companions to be further constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Techniques: radial velocities – Techniques: high angular resolution – Proper motions – Stars: planetary systems – Stars: brown dwarfs – Stars: low-mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Introduction In the last decade, several long-period giant planets have been detected using the radial velocity (hereafter RV) method thanks to the increasing temporal baselines of different surveys (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011), Wittenmyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2020), Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Yet, a precise determination of the orbital parameters and mass of the planets is very difficult when the orbital period is much larger than the RV time baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' As a consequence, the radial distribution of planets beyond 8-10 au – such as those found by Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019) and Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021) based on the results of the two long RV surveys of Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011) and Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021), respectively – are questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This unfortunately prevents an accurate comparison with formation model outputs from being made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Combining RV data with other methods such as relative or absolute astrometry can, in principle, improve the orbital charac- terization of these companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Furthermore, it can also remove the uncertainty of the orbital inclination and then allow us to de- termine the true mass of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Coupling RV data with relative astrometry from direct imag- ing (hereafter DI) or interferometry has been, however, limited to very few cases since high-contrast imaging (hereafter HCI) or interferometry observations favor young systems to minimize the flux contrast between the star and its companion while RV ⋆ Please send any requests to florian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='philipot@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='fr observations favor old and inactive stars which produce low RV jitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' However, when possible, such a coupling is very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An illustration is the HD 7449 system for which the outer com- panion was first reported as a planet candidate using only RV data (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011), Wittenmyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019)), and it was then identified as a low-mass star by combining RV data with HCI observations (Rodigas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In the 2000s, the combination of RV data and absolute astrometry, thanks to the Fine-Guidance-Sensor onboard the Hubble Space Telescope, also allowed for the inclination of a few stellar systems to be constrained and a few candidate planets to be confirmed (Benedict et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2002), Benedict et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2006)), while others were finally identified as brown dwarfs or low-mass stars (Bean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Benedict et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Today, the position and proper motion measurements obtained with the telescopes Hipparcos (Perryman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (1997), van Leeuwen (2007)) and Gaia (Gaia Collaboration 2020) allow us to combine the RV data and more precise absolute astrometry for a large number of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Since the publication of the first Gaia data release (DR1), a few studies have proven the efficiency of combining RV data with absolute and/or relative astrometry to improve the constraints on the orbital parameters and mass of a companion (Grandjean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019), Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019), Damasso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2020), Lagrange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2020), Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2020), Venner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021), Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021a), Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 1 of 17 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01263v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='EP] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr (2021b), Kiefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021), Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021), Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In this paper, we focus on seven long-period single com- panions detected by the RV method, and combine the available RV data with Hipparcos and Gaia early data release 3 (hereafter EDR3) absolute astrometry and, when available, relative astrom- etry, to improve the orbital parameters and determine the true mass of these companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In Section 2, we describe our target selection method and present the RV, HCI, and astrometric data used in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Section 3 presents the method used to per- form the orbital fitting and, there, we provide the new orbital parameters and mass found for each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Finally, we discuss the results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Target selection and data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Target selection We first selected the planetary systems in the exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='eu cat- alog (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2011) for which a single companion has been reported with a semi-major axis greater than 5 au using the RV method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Twenty-five companions were found with such cri- teria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' For nine of them (HD 13724 B, HD 25015 b, HD 181234 b, and HD 219828 B (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HD 92987 B (Venner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HIP 36985 B (Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' and HD 98649 b, HD 196067 b, and HD 221420 B (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021)), the orbital parameters and the true mass have already been properly deter- mined in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We, therefore, do not consider them in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We first discarded the HD 95872 system because no Hip- parcos data were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We then discarded four systems for which the available RV time series did not cover both extrema of the RV variations and the orbital period could not be prop- erly determined (HD 26161, HD 120066, HD 150706, and HD 213472).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In those four cases, the combination of RV and abso- lute astrometry did not allow us to constrain the orbital param- eters, the orbital inclination, or the true mass of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Finally, we discarded four systems for which the orbital period was well covered by the RV data, but the coupling with absolute astrometry did not allow us to constrain the orbital inclination (HD 136925, HD 190984, HD 220773, and HD 238914).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Indeed, the variations in position and acceleration of the proper motion of these stars were too small to constrain the orbital inclination of the companion due to the low mass of the companion (< ∼2 MJup) and/or the distance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Thus, we were left with seven systems for which the addi- tion of absolute astrometry and/or new RV measurements and/or relative astrometry measurements allowed us to determine the exact nature of the companion: Epsilon Indi Ab, HD 13931, HD 1159554, HD 211847, HD 219077, HD 222155, and HIP 70849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' For three of these companions (Epsilon Indi Ab, HD 211847 B, and HD 219077 b), a first estimation of their orbital inclination and true mass has been obtained by combining RV data and ab- solute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Yet, thanks to additional data or more precise astrometric measurements, we obtained more precise and sig- nificantly different results from those reported in the previous studies for six of these companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In the case of HD 219077 b, the differences were mainly found for the mass of the compan- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' They are mainly due to the uncertainties as to the host star’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' RV data The RV data used in this study were obtained with different spec- trographs between 1997 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The HARPS (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2003) data were taken from the ESO archives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' the ELODIE (Baranne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 1996) and SOPHIE (Perruchot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2008) data were retrieved from the OHP archives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' and the CORALIE (Queloz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 1999), the HIRES (Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 1994), the UVES (Dekker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2000), the AAT (Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 1990), the CES (Enard 1982) long camera (LC), and the CES very long camera (VLC) data were taken from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' As instrument upgrades can lead to new RV offsets, the same instrument before and after a major upgrade is considered as two different instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Consequently, HARPS data obtained be- fore and after the optical fiber upgrade in 2015 (Lo Curto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2015) are referred to as H03 and H15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The SO- PHIE data obtained before and after the spectrograph upgrade in 2011 (Bouchy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2013) are referred to as SOPHIE and SO- PHIE+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The HIRES data obtained before and af- ter the upgrade of the spectrograph in 2004 (Tal-Or et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2019) are referred to as Hir94 and Hir04, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Finally, the CORALIE spectrograph had two major upgrades in 2007 (Sé- gransan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2010) and in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The data obtained before 2007 and after 2014 are referred to as C98 and C14, respectively, and the data obtained between 2007 and 2014 are referred to as C07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Direct imaging data In three cases, HCI data are available in the SPHERE archive and can provide relative astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The three targets were ob- served in angular (and spectral) differential imaging (A(S)DI, Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2006)) using the telescope in pupil tracking mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The standard observing mode of SPHERE was used, with IRDIS (Dohlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2008) dual band images at H2 and H3 (K1 and K2, respectively) and IFS (Claudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2008) data covering the YJ (YJH, respectively) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The observing log is given in Ta- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Whenever possible, the robust PACO A(S)DI algorithm (Flasseur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2020a), Flasseur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2020b)) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The processing step takes advantage of the developments made to the COBREX data center pipeline (the prereduction improvement as well as the improvement of the detection capability of PACO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' If a dataset did not sufficiently cover the field-of-view rotation to apply ADI-based algorithms, the SPECAL (Galicher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2018) No-ADI algorithm was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In those three systems, only one companion was detected (HD 211847 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The detected companion is characterized in Ta- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' No detection above 5σ was found around HD 219077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Six sources were detected around HIP 70849 but, given their posi- tion in a color-magnitude diagram and their separations, they are likely background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Absolute astrometry We used measurements from Hipparcos obtained around epoch 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25 and from Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2016), Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021)) obtained around epoch 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' For each target, the stellar acceleration was determined from the proper motion and the position values were measured by Hip- parcos and Gaia with an interval of about 25 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We con- sidered the proper motion values published by Brandt (2021) in the Hipparcos-Gaia Catalog of Accelerations (HGCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' More- over, a more accurate tangential proper motion (µHip−EDR3) was estimated by the difference between the position measurements obtained by Hipparcos and Gaia divided by the time interval be- Article number, page 2 of 17 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Table 1: Observing logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' STAR DATE OBS FILTER DIT(s)×Nframea ∆ PA (°)a Seeing (")b Airmassb τ0 (ms)a,b Program ID HIP 70849 2015-05-05 DB_H23 64x64 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0012 095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='C-0298(A) HD 211847 2015-06-10 DB_K12 64x8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0025 095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='C-0476(A) HD 219077 2015-06-09 DB_K12 8x64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0026 095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='C-0476(A) Notes : a: DIT is the detector integration time per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' ∆ PA is the amplitude of the parallactic rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' τ0 corresponds to the coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' b are values extracted from the updated DIMM information, averaged over the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Table 2: Relative astrometry for HD 211847 companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Sources JD - 2400000 IRDIS filter SEP (mas) PA (deg) HD 211847 B 57183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='39 K12 220±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='73 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='23 Notes : The errors displayed are 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The relative astrometry combines the astrometry measured in the dual bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' tween the two measurements (∼25 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The proper motion values used for each star are given in table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Updated orbital parameters and mass 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Orbit fitting Orbits were fitted using a custom MCMC tool, based on the em- cee 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 python package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' It uses a mixture of move functions (such as the differential evolution move function) to alleviate potential multimodality issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The Hipparcos/Gaia data processing uses the HTOF package (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021) and borrows large sections of the orvara code (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021) for the likelihood computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The HTOF package (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021) was used to fit the intermediate astrometric data (IAD) from Hipparcos, based on the 1997 (Esa 1997) and 2007 (van Leeuwen 2007) reductions and from Gaia, thanks to the Gaia Observation Forecast Tool (GOST) which allowed us to obtain the estimated Gaia observations and scan angles for each target, in order to reproduce proper motion and position of each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using the Hipparcos and Gaia positions and the temporal baseline, the algorithm derived a tangential proper motion value that allowed us to better constrain the orbital fit when combined with RV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We considered ten free parameters for each system: the semi- major axis (a), the eccentricity, the orbital inclination (i), the host star mass, the companion mass, the longitude of ascending node (Ω), the argument of periastron (ω), the phase, a stellar jitter, and the distance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In addition, to combine data from different instruments, we added an instrumental offset for each instrument as a free parameter of the model (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Finally, we considered uniform priors for all fitting parameters, except for the host star mass and the distance of the system for which we considered Gaussian priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Epsilon Indi A Epsilon Indi is a triple system with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 M⊙, K2V star (Epsilon Indi A) and a binary composed of a 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 MJup, T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 brown dwarf (Epsilon Indi B) and a 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 MJup, T6 brown dwarf (Epsilon Indi C) separated by about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 au (Di- eterich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The projected separation between the binary brown dwarfs and the star is about 1460 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Combining RV data and absolute astrometry based on Hipparcos and the Gaia data release 2 (DR2) measurements, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019) reported a gi- ant planet with a semi-major axis of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='86 au, a mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='65 MJup, an inclination of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25+13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='80 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 °, and an eccentric- ity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Yet, the Gaia EDR3 proper motion and posi- tion measurements are significantly more precise compared to the Gaia DR2 measurement and they significantly improve the characterization of Epsilon Indi Ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We used 4278 RV measurements obtained with the HARPS spectrograph between 2003 and 20161, 163 RV measurements obtained with the UVES spectrograph between 1996 and 2017, 72 RV measurements obtained with the LC spectrograph be- tween 1992 and 1997, and 53 RV measurements obtained with the VLC spectrograph between 2000 and 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We also com- bined these RV data with absolute astrometry based on Hippar- cos and the Gaia EDR3 measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We found sig- nificantly different orbital parameters with a semi-major axis of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 au, a mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 MJup, an inclination of 91+4 −5°, and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' It is important to note that if we consider only the 539 HARPS RV data with a signal-to-noise ratio greater than 110 and thus remove the high cadence obser- vations made in August 2011, we find similar solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HD 13931 HD 13931 is a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 M⊙(Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021), G0V star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Based on 66 RV measurements obtained with the HIRES spectrograph between 1998 and 2019, Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021) reported a giant planet with a semi-major axis of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='323 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='091 au, a minimum mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='911+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='077 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='076 MJup , and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='021 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We combined these RV data with absolute astrometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' As the RV baseline is much larger than the orbital pe- riod, the orbital parameters are well-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' As expected, we found a semi-major axis and an eccentricity very close to those reported by Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021) with a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 au and e < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using, in addition, the absolute astrometry, we found an orbital inclination of either 39+13 −8 ° or 141+9 −18° and a true mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HD 115954 HD 115954 is a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 M⊙, G0V star (Demangeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Based on four RV measurements obtained with the ELODIE spectrograph between 2004 and 2005 and 45 RV mea- surements obtained with the SOPHIE spectrograph between 2009 and 2018, Demangeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021) reported a giant planet 1 The 3636 RV data obtained between Julian days 2455790 and 2455805 were obtained to study high-frequency oscillations of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' These data were measured with high cadence, which led to a signifi- cantly lower signal-to-noise ratio compared to the other data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 3 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 1: Orbital fits for Epsilon Indi Ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top: Fit of the Epsilon Indi A RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bot- tom: Fit of the Epsilon Indi A astrometric acceleration in right ascension (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points corre- spond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia EDR3 (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' with a semi-major axis of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='00+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='36 au, a minimum mass of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='58 MJup , and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='487+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='095 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We combined these RV data with the absolute astrometry data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We found a semi-major axis compatible with De- mangeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2021) with a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 au and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using, in addition, the absolute astrometry, we found an orbital inclination of 92+17 −16° and a true mass of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HD 211847 HD 211847 is a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 M⊙, G5V star (Sahlmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Sahlmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011) reported a brown dwarf candidate orbit- ing around HD 211847 based on 31 RV measurements obtained with the CORALIE spectrograph between 2002 and 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' How- ever, only one minimum of the HD 211847 B RV curve was covered by the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Thus, the orbital parameters and min- imum mass reported in this study are poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Us- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2: Orbital fits for HD 13931 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top: Fit of the HD 13931 RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bottom: Fit of the HD 13931 astrometric acceleration in right ascen- sion (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points correspond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' ing the Levenberg-Marquardt method, they found ranges corre- sponding to a 3σ confidence interval for the semi-major axis, the eccentricity, and the minimum mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6–42 au, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='48–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95, and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 MJup , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Moutou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2017) obtained one HCI detection with SPHERE of HD 211847 B for a pro- jected separation of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using the BT-Settl models (Allard 2014), they fit the HD 211847 B spectrum and found a low stel- lar mass of 155 ± 9 MJup assuming an age of 3 Gyr for the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Based on the result of Sahlmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011), Moutou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2017) estimated the inclination of the companion orbit to be around seven°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Recently, combining the CORALIE RV measure- ment and the absolute astrometry, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) reported HD 211847 B as a brown dwarf with a semi-major axis of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='514+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='458 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='287 au, a mass of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='32+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='335 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='48 MJup, an inclination of 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='649+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='239 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='017 °, and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='419+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 4 of 17 RadialVelocities LLH best 100 H03, VO-corrected H15, V0-corrected 8524 UVES, Vo-corrected LC, Vo-corrected [s/w] 50 VLC, VO-corrected 8526 Radial velocity [ 0 8528 50 8530 100 8532 100 0 上 100 6000 4000-2000 0 2000 4000 Time (JD-2454000)ProperMotionRA ProperMotionDec best best 2535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 8524 3968 2535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 μα*[mas/yr] 8526 3967 2536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 8528 3966 2536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 2537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 8530 3965 2537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 8532 3964 LLH 1 0-0 0 0 2 1 6000-4000-2000 0 2000 4000 6000-4000-2000 0 2000 4000 Epoch[jD-2454000) Epoch[jD-2454000)Radial Velocities LLH 30 best Hir94, V0-corrected 138 Hir04, VO-corrected 20 [s/w] 140 10 I velocity 142 0 Radial 10 144 20 146 30 10 O-C 0 10 2000 0 2000 4000 Time(JD-2454000)ProperMotionRA ProperMotionDec 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 best best 138 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 140 μα*[mas/yr] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 masi 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 142 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 144 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 146 LLH U 4000-2000 0 2000 4000 6000 4000-2000 0 2000 4000 6000 EpochfiD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 Epoch[JD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3: Orbital fits for HD 115954 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top: Fit of the HD 115954 RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bottom: Fit of the HD 115954 astrometric acceleration in right ascen- sion (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points correspond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We combined the RV dataset used by Sahlmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011), the relative astrometry observation obtained with SPHERE in June 2015, and the absolute astrometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Adding one rel- ative astrometry observation allowed us to properly constrain the orbital parameters and the mass of HD211847 B with results sig- nificantly different from those reported by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We found a semi-major axis of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='08 au and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using, in addition, the absolute astrometry, we found an orbital inclination of 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04° and a true mass of 148 ± 5 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We note that by taking only RV data and absolute astrom- etry into account, we found very poorly constrained solutions with large uncertainties as to the semi-major axis (16 - 30 au) and mass (80 - 140 MJup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Moreover, the solutions found are not in agreement with those reported by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) or with the solutions found when adding the HCI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 4: Orbital fits for HD 211847 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top left: Fit of the HD 211847 RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top right: Fit of HD 211847 relative astrometry data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The red cross corresponds to the measurement obtained with SPHERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bottom: Fit of the HD 211847 astrometric acceleration in right ascension (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points corre- spond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 5 of 17 Radial Velocities LLH 100 156 50 [s/w] 158 0 Radial velocity [ 50 160 100 162 150 best ELODIE,vO-corrected SoPHIE, VO-corrected 164 SOPHIE+,VO-corrected 200 50 1000 0 1000 2000 3000 4000 Time (JD-2454000)ProperMotionRA ProperMotionDec 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 best best 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 156 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 157 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 μα*[mas/yr] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 158 μs[mas/yr] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 159 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 160 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 161 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 162 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 163 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 LLH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 0-0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 0 2000 4000 6000 1 4000 0-2000 二4000-2000 0 2000 4000 6000 Epoch[jD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5) Epoch[JD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5】RadialVelocities LLH best 128 100 C98, V0-corrected 129 0 130 [s/w] velocity 100 131 132 200 Radial 133 300 134 400 135 136 25 0 25 100015002000250030003500 Time (JD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5)Sky plane LLH 50 best b 127 128 129 [mas] 50 130 DEC 100 131 150 132 133 200 134 100 50 0 50 100 150 RA [mas]Proper MotionRA ProperMotionDec 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 best best 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 127 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 128 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 μs[mas/yr] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 129 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 130 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 131 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 132 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 133 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 134 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 LLH 5 L 0-0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 5 4000-2000 0 2000 4000 6000 4000-2000 0 2000 4000 6000 Epoch[D-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5) Epoch[D-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HD 219077 Based on 63 CORALIE RV measurements obtained between 1999 and 2012 and 30 HARPS RV measurements obtained be- tween 2003 and 2012, Marmier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2013) reported a very ec- centric giant planet with a semi-major axis of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 au and a minimum mass of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' It is important to note that the RV data used by Marmier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2013) are not pub- licly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Based on 72 pieces of RV data obtained with the AAT spectrograph between 1998 and 2015, Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019) reported slightly different properties for HD 219077 with a semi- major axis of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='21 au and a minimum mass of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='78 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' These differences are probably due to the different assump- tions on the mass of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Indeed, Marmier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2013) used the values reported by Hipparcos (M⋆= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 M⊙) while Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019) used the values reported in Valenti & Fischer (2005) (M⋆= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='13 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Recently, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) com- bined the AAT RV measurements used by Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2019) and 33 HARPS RV measurements obtained between 2003 and 2012 with the absolute astrometry based on Hipparcos and the Gaia EDR3 measurements and found a semi-major axis close to Marmier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2013), an orbital inclination of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='178+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='527 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='462°, and a true mass of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='620+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='733 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The prior on the mass of the star is not given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' For this study, as the data used by Marmier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2013) are not available, we considered the HARPS and AAT RV mea- surements used on Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) and the 65 CORALIE RV measurements available on the DACE archive2 obtained between 1999 and 20123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' For the mass of the star, we considered the value given by Kervella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) based on the Gaia DR3 results (M⋆= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We combined the RV data with the ab- solute astrometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We found a semi-major axis and an eccentricity close to those reported in the previous studies with a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 au and e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='769 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='002 and an orbital inclination close to that of Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) with either i = 83 ± 3° or i = 97±3°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Considering the star mass found by Kervella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022), we found a planetary mass at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' However, consid- ering that the mass used in Valenti & Fischer (2005) would lead to a mass close to the deuterium-burning limit, Mb = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Due to the uncertainties on the mass of the host star, it is not possible to determine the exact nature of HD 219077 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HD 222155 HD 222155 is a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='11 M⊙, G2V star (Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Based on 44 RV measurements obtained with the ELODIE spec- trograph between 1997 and 2005 and 67 RV measurements ob- tained with the SOPHIE spectrograph between 2007 and 2011, Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2012) reported a giant planet with a semi-major axis of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 au, a minimum mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='53 MJup , and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='38+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We considered 31 additional pieces of SOPHIE RV data ob- tained between 2011 and 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We combined the RV data with the absolute astrometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We found orbital parameters within the error bars associated with the values found by Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2012) with a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 au and e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' As the RV baseline is now much larger than the orbital period, the orbital parameters are better constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using, in addition, the abso- 2 https://dace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='ch 3 The CORALIE RV data available on DACE and the HARPS RV data available on the ESO archive cover the same time base as those used by Marmier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 5: Orbital fits for HD 219077 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top: Fit of the HD 219077 RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bottom: Fit of the HD 219077 astrometric acceleration in right ascen- sion (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points correspond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' lute astrometry, we found an orbital inclination of either 66+14 −11° or 115+13 −16° and a true mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' HIP 70849 HIP 70848 is a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 M⊙, K7V star (Ségransan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Ségransan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011) reported the first detection of HIP 70849 b based on 18 RV measurements obtained with the HARPS spec- trograph between 2006 and 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' However, only one minimum of the HIP 70849 b RV curve was covered by the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The observations carried out by Ségransan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2011) led to poorly constrained orbital parameters and minimum mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using a ge- netic algorithm followed by MCMC simulations, they reported a semi-major axis between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 and 36 au, a minimum mass be- tween 3 and 15 MJup , and an eccentricity between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='47 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='96 with ranges corresponding to a 3σ confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 6 of 17 RadialVelocities LLH 250 best C98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='o-corrected 386 200 Co7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='vo-corrected [m Ho3,VO-corrected 150 AAT, VO-corrected 388 velocity 100 390 50 0 392 R 50 100 394 20 0-0 0 20 2000 0 2000 Time（iD-2454000)ProperMotionRA ProperMotionDec 422 best best 480 386 423 μα*[mas/yr] 479 388 μs[mas/yr] 424 390 478 425 392 477 426 394 476 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 LLH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 6000-4000-2000 0 2000 4000 6000-4000-2000 0 2000 4000 Epoch[jD-2454000) Epoch[jD-2454000)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 6: Orbital fits for HD 222155 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top: Fit of the HD 222155 RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bottom: Fit of the HD 222155 astrometric acceleration in right ascen- sion (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points correspond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We considered 39 additional pieces of HARPS RV data ob- tained between 2011 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We combined the RV data with the absolute astrometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' With these additional observa- tions, the dataset then covered two minimum and one maximum of the RV curve of HIP 70849 b and this allowed us to prop- erly constrain the properties of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We found a semi- major axis of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='07 au and an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Using, in addition, the absolute astrometry, we found an orbital inclina- tion of 96 ± 16° and a true mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Summary and concluding remarks Combining RV measurements from various spectrographs with absolute astrometry based on Hipparcos and Gaia EDR3 data and, when available, relative astrometry, we determined the or- bital parameters and, in particular, the orbital inclination and the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 7: Orbital fits for HIP 70849 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Top: Fit of the HIP 70849 RV data corrected from the instrumental offset (V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Bottom: Fit of the HIP 70849 astrometric acceleration in right ascen- sion (left) and declination (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The black points correspond to the measurements obtained with Hipparcos (1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25) and Gaia (2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In each plot, the black curve shows the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The color bar indicates the log likelihood of the different fits plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' true mass of seven long-period single companions detected by the RV method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Figure 8 summarizes the true mass and the semi- major axis of the companions and compares them with the previ- ous estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Clearly, Gaia EDR3 data allow for a better de- termination of these companions’ orbital parameters and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' All of these companions have true masses of 2 MJup or more and orbit between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 - 9 au from their stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Absolute astrometry would probably help to determine the true mass of planets with a period larger than the duration of Gaia DR3 observations (P > ∼1000 d) down to 1 MJup , provided the RV variations are well covered and the variations in position and acceleration of the proper motion of the star are large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' In practice, in most cases, when the period is not well constrained by the RV data, the impact of the coupling of RV data with absolute as- trometry is more limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An illustration of this is the case of HD 211847 B for which, by combining RV data that cover only a minimum of the RV variations with absolute astrometry, Feng Article number, page 7 of 17 Radial Velocities LLH 60 best ELODIE,VO-corrected 40 SOPHIE, VO-corrected 464 SOPHIE+, VO-corrected Radial velocity [m/s] 20 466 468 470 20 472 40 474 60 50 0 4n O-C 0 50 4000 2000 0 2000 4000 Time (JD-2454000)ProperMotionRA ProperMotionDec 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 best 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 best 464 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 466 [mas/yr] uolmas/yr 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 468 μa* 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 470 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 472 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 F LLH 0-0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 6000-4000-2000 0 2000 4000 6000-4000-2000 0 2000 4000 Epoch[jD-2454000) Epoch[jD-2454000)Radial Velocities LLH 172 50 173 Radial velocity [m/s] 0 174 50 175 100 176 150 best 177 H03, VO-corrected H15, V0-corrected 200 25 O-C 0 25 0 10002000300040005000 Time (JD-2454000)ProperMotionRA ProperMotionDec 172 best 201 best 44 173 45 202 174 μα*[mas/yr] [mas/yr] 46 175 203 47 176 204 48 177 178 49 205 LLH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 4000-2000 0 2000 4000 6000 4000 2000 0 2000 4000 6000 Epoch[jD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5) Epoch[iD-2451544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr Table 3: Summary of posteriors obtained with our MCMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Parameter Eps Ind A HD 13931 HD 115954 HD 211847 HD 219077 HD 222155 HIP 70849 a (au) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='07 Period (days) 10932+266 −228 4442+49 −46 3258+179 −190 6199+52 −46 5514+44 −39 3470+102 −106 3649 ± 18 Eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='769 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 Inclination (°) 91+4 −5 39+13 −8 or 141+9 −18 59+5 −4 or 127 ± 4 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 83 ± 3 or 97 ± 3 66+14 −11 or 115+13 −16 96 ± 16 Mass (MJup ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 148 ± 5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 Ω (°) 58 ± 5 343+17 −19 or 110+19 −24 211+25 −28 184 ± 5 135+38 −21 or 347+28 −30 264+34 −33 or 180+34 −35 35 ± 6 ω (°) 85 ± 3 74 - 227 173+7 −8 168+5 −4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 153 - 217 182 ± 1 Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='38 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='420 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='354+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='745+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='006 Jitter (m/s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 H03 = −39972 ± 1 Hir94 = −13 ± 1 ELODIE = −14757+52 −41 C98 = 6907 ± 17 C98 = −30867+2 −1 ELODIE = −3999 ± 3 H03 = 53 ± 1 Instrumental H15 = −39952+2 −1 Hir04 = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 SOPHIE = −14768+9 −8 C07 = 6849 ± 22 C07 = −30864+8 −9 SOPHIE = −3950+3 −2 H15 = 65+2 −1 offset (m/s) UVES = −5 ± 1 SOPHIE+ = −14743+4 −3 H03 = −30830+1 −2 SOPHIE+ = −3923+4 −3 LC = −39978+2 −1 AAT = −68 ± 1 VLC = −39976+2 −1 Notes: The results were obtained by combining RV, absolute astrometry, and, when available, relative astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We provide 68% confidence intervals for each parameter and the median is only given when the probability distribution has a profile close to a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 8: Update of the orbital parameters and masses of the seven analyzed systems thanks to the combination of absolute astro- metric and RV data and, when available, absolute astrometry data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' For each system, a dotted line between two solutions was drawn to allow for the different solutions obtained to be com- pared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' (2022) reported a mass of about 55 MJup, corresponding to a brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Yet, HCI revealed a companion and the fit of the RV and the relative and absolute astrometry leads to a mass of about 150 MJup instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' We conclude that Gaia/Hipparcos can help to further con- strain the orbital parameters of long-period RV planets, provid- ing good coverage of the RV variations is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Otherwise, additional information is needed, such as relative astrometry, provided by DI or interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This study was funded by a grant from PSL/OCAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (COBREX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' grant agreement n° 885593).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This work presents results from the European Space Agency (ESA) space mission Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The Gaia mission website is https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='int/gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The Gaia archive website is https://archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='int/gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This publications makes use of the The Data & Analysis Center for Exoplanets (DACE), which is a facility based at the University of Geneva (CH) dedicated to extrasolar planets data visualisation, exchange and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' DACE is a platform of the Swiss National Centre of Competence in Research (NCCR) PlanetS, federating the Swiss expertise in Exoplanet research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' The DACE platform is available at https://dace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' This research has made use of the SIMBAD database and VizieR catalogue access tool, operated at CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Based on data retrieved from the SOPHIE archive at Observatoire de Haute-Provence (OHP), available at atlas.' metadata={'source': 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T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=', Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2019, MNRAS, 484, 5859 Wittenmyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=', Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=', Horner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 2020, MNRAS, 492, 377 Article number, page 9 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr Appendix A: Proper motion values Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1: Proper motion values from HGCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Star Eps ind A HD 13931 HD 115954 HD 211847 HD 219077 HD 222155 HIP 70849 µα Hip (mas/yr) 3964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 −74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5 −47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 µδ Hip (mas/yr) −2537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 −183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 −424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 −117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 −203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 µα EDR3 (mas/yr) 3966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 −74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 µδ EDR3 (mas/yr) −2536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 −183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 −424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 −117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 µα Hip−EDR3 (mas/yr) 3965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 −74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 µδ Hip−EDR3 (mas/yr) −2537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 −183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 −424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='01 −117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 −202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 Notes: µHip corresponds to the proper motion obtained by Hipparcos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' µEDR3 corresponds to the proper motion obtained by Gaia EDR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' µHip−EDR3 corresponds to the proper motion obtained by the Hipparcos-Gaia EDR3 positional difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Appendix B: MCMC priors Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1: Priors considered for each free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Parameter Eps ind A HD 13931 HD 115954 HD 211847 HD 219077 HD 222155 HIP 70849 a (au) [1,20] [1,10] [1,10] [1,100] [1,10] [1,10] [1,10] Eccentricity [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] [0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='95] Inclination (°) [0,180] [0,180] [0,180] [0,180] [0,180] [0,180] [0,180] Mass (MJup ) [1,20] [1,20] [1,20] [1,500] [1,20] [1,20] [1,20] Ω (°) [0,360] [0,360] [0,360] [0,360] [0,360] [0,360] [0,360] ω (°) [0,360] [0,360] [0,360] [0,360] [0,360] [0,360] [0,360] Phase [0,1] [0,1] [0,1] [0,1] [0,1] [0,1] [0,1] Jitter (m/s) [0,10] [0,10] [0,10] [0,20] [0,10] [0,20] [0,10] Star mass (M⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 Distance (pc) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='622 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='004 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4 218 ± 2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7 H03: [-41,-39] Hir94: [-1,1] ELODIE: [-15,-13] C98: [5,7] C98: [-31,-29] ELODIE: [-5,-3] H03: [-1,1] Instrumental H15: [-41,-39] Hir04: [-1,1] SOPHIE: [-15,-13] C07: [5,7] C07: [-31,-29] SOPHIE: [-5,-3] H03: [-1,1] offset (km/s) UVES: [-1,1] SOPHIE+: [-15,-13] H03: [-31,-29] SOPHIE+: [-5,-3] LC: [-41,-39] AAT: [-1,1] VLC: [-41,-39] Article number, page 10 of 17 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Appendix C: MCMC results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='1: Corner plot of the posteriors’ fit of Epsilon Indi A combined RV and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An offset of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 km/s was added to V0, V1, V3, and V4 to improve readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 11 of 17 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='64±1:4g 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='62±8:88 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='763:28 O [yr] = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='93±:72 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' O Ob [deg] = O O O O smpa %心 mps V2 [mps] pargb [deg] Ib [deg] Ob [deg]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2: Corner plot of the posteriors’ fit of HD 13931 combined RV and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 12 of 17 vo[mps V1[mps] [sw] 3 jitter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02±8:85 [pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='02±8:83 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='65±74:69 mb [Mi :3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='07±8:65 [Mjup] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='12 lb [deg] % Ob[deg] 公心 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Q: vo [mps] V1 [mps] jitter [ms] mi [Msun] dstar [pc] smab [au] pargb [deg] ptimeb [yr] perb [yr] mb [Mjup] Ib [deg] Ob [deg]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='3: Corner plot of the posteriors’ fit of HD 115954 combined RV and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An offset of 14 km/s was added to V0, V1, and V2 to improve readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 13 of 17 /2[mps 742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='683:19 jitter [ms] = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='571:33 [s a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='03 2 [ne] ew 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='52 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='55±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='59 6 [yr] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='55±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5] 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='59 perb [yr] : 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='48±8:4 b [deg] = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='39t15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='81 Ob [deg] : :211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content="522 [deg] %%%% 115 Vo [mps] Vi [mps] V2 [mps] jitter [ms] m1 [Msun] dstar Ipc' smab [au] pargb [deg] ptimeb [yr] perb [yr] mb [Mjup] Ib [deg] Ob [deg]A&A proofs: manuscript no." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='4: Corner plot of the posteriors’ fit of HD 211847 combined RV, relative astrometry, and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An offset of 6 km/s was subtracted to V0 and V1 to improve readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 14 of 17 vo[mps : 907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='20±19:36 18 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='94±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='60-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='05 ab [au] = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='78±8:89 oargb[deg] 120 ptimeb [yr] = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='12±8:05 [yr] = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='97±8:13 LE O[deg]: ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 多 vo [mps] V1 [mps] jitter [ms] m1 [Msun] dstar [pc] smab [au] pargb [deg] ptimeb [yr] perb [yr] mb [Mjup] [6ap] Ob [deg]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='5: Corner plot of the posteriors’ fit of HD 219077 combined RV and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An offset of 30 km/s was added to V0, V1, and V2 to improve readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 15 of 17 93±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='51 829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='61±1:43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='27±83 Msun jitter [ms] smab [au] ptime, lyr?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' I [deg] O, [deg]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 45396corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='6: Corner plot of the posteriors’ fit of HD 222155 combined RV and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' An offset of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='9 km/s was added to V0, V1, and V2 to improve readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 16 of 17 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='号 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='0 [pc] = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='108:93 V2 [mps] jitter [ms] m1 [Msun dstar [pc] smab[au pargb [deg] ptimeb [yr] perb [yr] Ib [deg] Ob [deg]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Philipot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' : Updated characterisation of long-period single companion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='7: Corner plot of posteriors fit of HIP 70849 combined RV and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' Article number, page 17 of 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='19±1:19 心 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='65±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='82 otimeb [yr] = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='44±8:83 ptimep [yr] 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='40±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='85-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='76 120 [6a p] 9] Ob [deg] 25o 2 s?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content='q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} +page_content=' 0% 心 8 Vo [mps] vi [mps] jitter [ms] mi [Msun] dstar [pc] smab [au] pargbdeg] ptimeb [yr] perplyr mp [Mjup] Ib [deg] Ob [deg]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfUPx6/content/2301.01263v1.pdf'} diff --git a/EtE0T4oBgHgl3EQfQwBZ/content/tmp_files/2301.02197v1.pdf.txt b/EtE0T4oBgHgl3EQfQwBZ/content/tmp_files/2301.02197v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15ad7cdbb4d1e4835210b41a7cd3b51711a72c48 --- /dev/null +++ b/EtE0T4oBgHgl3EQfQwBZ/content/tmp_files/2301.02197v1.pdf.txt @@ -0,0 +1,5486 @@ +Virtual Node Graph Neural Network for Full Phonon +Prediction +Ryotaro Okabe1,2,†, Abhijatmedhi Chotrattanapituk1,3,†, Artittaya Boonkird1,4, Nina +Andrejevic5, Xiang Fu3, Tommi S. Jaakkola3, Qichen Song6, Thanh Nguyen1,4, Nathan +Drucker1,7, Sai Mu8, Bolin Liao9, Yongqiang Cheng10, and Mingda Li1,4,* +1Quantum Measurement Group, Massachusetts Institute of Technology, Cambridge, MA, USA +2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, USA +3Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, +MA, USA +4Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA +5Argonne National Laboratory, Lemont, IL, USA +6Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA +7Applied Physics, School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA +8Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina, USA +9Department of Materials, University of California, Santa Barbara, Santa Barbara, CA, USA +10Chemical Spectroscopy Group, Spectroscopy Section, Neutron Scattering Division Oak Ridge National +Laboratory, Oak Ridge, TN, USA +†These authors contributed equally +*e-mail: mingda@mit.edu +ABSTRACT +The structure-property relationship plays a central role in materials science. Understanding the structure-property relationship +in solid-state materials is crucial for structure design with optimized properties. The past few years witnessed remarkable +progress in correlating structures with properties in crystalline materials, such as machine learning methods and particularly +graph neural networks as a natural representation of crystal structures. However, significant challenges remain, including +predicting properties with complex unit cells input and material-dependent, variable-length output. Here we present the virtual +node graph neural network to address the challenges. By developing three types of virtual node approaches - the vector, +matrix, and momentum-dependent matrix virtual nodes, we achieve direct prediction of Γ-phonon spectra and full dispersion +only using atomic coordinates as input. We validate the phonon bandstructures on various alloy systems, and further build +a Γ-phonon database containing over 146,000 materials in the Materials Project. Our work provides an avenue for rapid +and high-quality prediction of phonon spectra and bandstructures in complex materials, and enables materials design with +superior phonon properties for energy applications. The virtual node augmentation of graph neural networks also sheds light +on designing other functional properties with a new level of flexibility. +Introduction +The structure-property relationship defines one of the most fundamental questions in materials science16,21. The ubiquitous +presence of structure-property relationships profoundly influences almost all branches of materials sciences, such as structural +materials3, energy harvesting and conversion and energy storage materials5,17,19, catalysts37 and polymers13, and quantum +materials15. However, despite its central importance to materials design, building an informative structure-property relationship +can be nontrivial. On the one hand, the number of stable structures grows exponentially with unit cell size22, and the structure +design efforts have been largely limited to crystalline solids with relatively small unit cells. On the other hand, certain material +properties are challenging to acquire due to experimental or computational complexities. +In the past few years, data-driven and machine-learning methods play an increasingly important role in materials science +and significantly boost the research on building structure-property relationships6,24,38. Complex structures such as porous +materials1,27, nanoalloys10,36, and grain boundaries34 are becoming more feasible to handle, and properties ranging from +mechanical strength to quantum ordering can be learned with increased confidence9,29. One particular powerful approach +is the graph neural networks (GNNs)7. By representing atoms as graph nodes and interatomic bonds as graph edges, GNNs +arXiv:2301.02197v1 [cond-mat.dis-nn] 5 Jan 2023 + +provide a natural representation of molecules and materials. For crystalline solids, crystallographic symmetry offers a further +boost on the GNN performance, with a few symmetry-augmented GNNs being proposed8,30,35. A few fundamental challenges +still exist. For one thing, many materials properties are not naturally represented as a weighted aggregation of each atom in +real space, such as reciprocal and energy space properties. For another thing, the output property length is usually fixed, like +the heat capacity4 as a single scalar. In contrast, many materials’ properties have unique degrees of dimensions, such as the +number of electronic and phononic bands2, frequency ranges with optical responses, and the features of magnetic structures +like propagation vectors. +In this work, we propose Virtual Node Graph Neural Network (VGNN) as a generically applicable approach to augment +GNN. In contrast to symmetry-augmented GNN which focuses on reducing the input data volume, VGNN focuses on handling +the output properties with variable or even arbitrary dimensions. We study materials’ phonon spectra and dispersion relations, +given that phonons bands are challenging to compute or measure with high computational cost and limited experimental +resources. By using the phonon spectra as examples, we present three versions of VGNN: the vector virtual nodes (VVN), +the matrix virtual nodes (MVN), and the momentum-dependent matrix virtual nodes (k-MVN). All three VGNN models take +atomic structures as input without prior knowledge of interatomic forces. The VVN is the simplest VGNN that takes in a crystal +structure with m atoms and outputs 3m branches Γ-phonon energies. The MVN is a more involved VGNN that shows higher +accuracy for complex materials with slightly higher computational cost. Finally, the k-MVN is a VGNN that can predict full +phonon band structure at arbitrary k points in the Brillouin zone. To achieve so, the crystal graphs contain "virtual-dynamical +matrices", which are matrix structures that resemble phonon dynamical matrices14. Instead of performing direct ab initio +calculations on each material, all matrix elements are learned from the neural network optimization process using training data +comprised of all other materials. Our work offers an efficient technique that can compute zone-center phonon energies and +full phonon band structures directly from atomic structures in complex materials and enables phonon property optimization +within a larger structure design space. The prediction methods has enabled us to acquire relevant information of materials such +as group velocities, heat capacities, density of states as by-products. Meanwhile, the virtual node structures also shed light +on future flexible GNN design, that to put intermediate crucial quantities (e.g. dynamical matrix) as key learning parameters +without having to put target properties (e.g. phonon band structures) as output. +Results +Virtual node augmentation for graph neural networks. Figure 1 gives an overview of the VGNN method as a generic +approach to augment GNN. For a crystal with m atoms per unit cell (Figure 1a), a typical GNN model converts the crystal into +a crystal graph, where each graph node represents an atom, and each graph edge represents the interatomic bonding as shown in +Figure 1b. The node features associated with each atomic node (gray arrays in Figure 1b) are updated by neighborhood nodes +and edges connecting the nodes (gray arrows in Figure 1b). After iterative layers of graph convolutions, m final-layer node +features are obtained that represent the atomic features from each of the m atoms. The final graph output can be obtained by +aggregating the final-layer node features into one fixed-sized output. +Figures 1c,d describe the general idea of VGNN that endows a GNN with greater flexibility for prediction. On top +of the conventional, real-node GNN, virtual atoms are added into crystal (yellow nodes in Figure 1c), which become the +virtual nodes in the corresponding GNN (yellow nodes in Figure 1d). As Figure 1d illustrates, just like the bi-directional +message passing between real atomic nodes (double-arrow gray lines), the message passing (double-arrow yellow lines) +between virtual nodes is also bi-directional. On the other hand, to preserve the structure of the conventional GNN, the +messages from real nodes to virtual (single-arrow gray-to-yellow gradient lines) are uni-directional. Given the flexibility of the +choice of the virtual nodes, a VGNN gains huge flexibility to predict materials-dependent outputs with arbitrary lengths and in +spaces. We will introduce three VGNN methods for phonon prediction with increased levels of predictive power and complexity. +Vector virtual nodes for Γ-phonon prediction. As illustrated in Figure 1, VGNN makes it possible to adjust output dimension +based on input information with flexibility. We first introduce the vector virtual node (VVN) method, which is the simplest +approach to acquire 3m phonon branches when inputting a crystal with m atoms per unit cell. (See Methods for more detail) +Figure 2 shows the VVN approach to predict Γ-phonon spectra. Since the virtual nodes do not pass information to real nodes, +there is additional flexibility in choosing the position of the virtual node. Without loss of generality, we assign the position of +the virtual nodes evenly spaced along the diagonal line of the unit cell. The crystal graph is constructed with virtual and real +nodes (Figure 2a). After updating node features in each convolution layer, the feature vectors pass a linear layer so that virtual +node features Vi,i ∈ [1,3m] are converted to 3m scalars, which represent the predicted Γ-phonon energies. Throughout this +work, the GNN part is implemented through the Euclidean neural networks8 that are aware of the crystallographic symmetry. +Data preparation, neural network architectures, and optimizations are described in Supplementary Information I-III. +The main results using the VVN for Γ-phonon prediction are shown in Figure 2b. The three-row spectral comparison plots +2/11 + +Figure 1. Overview of virtual node graph neural network (VGNN). a. Atomic structure of a crystalline material with m +atoms per unit cell. b. A GNN converts the atomic structures into a crystal graph. After layers of graph convolutions (omitted +for simplicity), the final node features are aggregated into a single fixed-sized output feature. c. A flexible of n virtual atoms are +added into the crystal structure. d. After forming the crystal graph with both real and virtual nodes, the flexibility of virtual +nodes enables the choices of output not necessarily from real-node aggregation but can have variable length and in different +spaces. +are randomly selected samples from the test set within each error tertile (top-to-bottom rows are top-to-bottom performance +tertiles, respectively). The first four columns are taken from the same database as the training set from high-quality density- +functional perturbation theory (DFPT) calculations25, and the fifth column contains additional test examples with much larger +unit cells from a frozen phonon database31. It is worthwhile mentioning that although for very complicated materials, the +VNN-predicted phonons tend to have higher frequencies (e.g., third row, fourth column of Ba12I36Y4), the resulted phonon +density-of-states over the entire Brillouin zone can still be largely comparable. However, the prediction loss becomes larger +and distributed broader as the input materials are more complicated (Figure 2c). From the correlation plot of predicted and +ground-truth phonon frequencies (Figure 2d), most data points are along the diagonal line, indicating good prediction between +VNN prediction and ground-truth from DFPT calculations with the number of atoms per unit cell m ≤ 24 (blue dots). For +complex materials, the correlation performance could be degraded (orange dots). More test results are shown in Supplementary +Information IV. +Matrix virtual nodes for Γ-phonon with enhanced performance. In this section, we introduce another type of virtual nodes +approach, the matrix virtual nodes (MVN). The MVN approach performs better Γ-phonon prediction than VVN, especially for +complex materials, with a slightly higher computational cost. Moreover, the structure of MVN lays the groundwork for the full +phonon band structures to be discussed in the next section. In MVN, m copies of virtual crystals are generated for material with +m atoms per unit cell, and each copy contains m virtual nodes that share the same crystal structure as the real crystal (Figure 3a). +This results in a total of m2 virtual nodes Vi j, i, j ∈ [1,m] with more involved node connectivity. (See Methods for more detail). +With this graph construction scheme, after the neural network training, the virtual nodes Vij would capture the essence +of the connection between Ri, and R j. Hence, after the message passes in each convolutional layer, each virtual node feature +3/11 + +realfeature +build crystal +c +graph +... +R2 +1 (fixed) +aggregate +feature +R +a +Add n virtual +上 +atoms +build crystal +virtual feature +graph +n (variable) +features +real-to-real +real-to-virtual +virtual-to-virtual +message passing +message passing +message passingFigure 2. The vector virtual node (VVN) method to predict Γ-point phonons. a. Schematic of VVN model construction +and prediction. For material with m atoms per unit cell, 3m Virtual nodes are augmented along the diagonal vector +⃗v =⃗a+⃗b+⃗c of the unit cell. We embedded the components of the crystal when building the GNN model. For instance, atomic +numbers of the mth real atom (ARm) and that of the 3mth virtual atom (AV3m) are embedded as the attributes of each nodes. The +atomic mass of mth real atom (ZRm) is set as the initial feature of that node. The relative position of the node V1 with respect to +Rm is⃗rV1Rm, which is used to embed the edge attribute between the two nodes. The model predicts Γ-phonon spectra by sorting +the scalar output features from virtual nodes. b. Spectral prediction samples in the test set within each error tertile compared +with ground truth (black): Test from the same database as the training set (blue), and a different database containing complex +materials (orange). c-d. Evaluation of the test accuracy through the distribution of loss function and correlation plot between +ground-truth and predicted average phonon frequencies, respectively. The heavy distribution at low loss regime of the +distribution plot and the agreement along the diagonal line of the correlation plot for the test set (blue) indicates a high-quality +phonon prediction at least for relatively simple materials with the number of atoms per unit cell m ≤ 24. The loss becomes +higher with reduced performance for complex materials (orange). +is further converted into a three-by-three matrix. Each of Vij is assembled to form (i, j) block of a supermatrix ˜D of shape +(3m,3m). Given the structural similarity of this matrix and the dynamical matrix expressed in Equation (2) with⃗k = 0, we +predict Γ-point phonon energies by solving for 3m eigenvalues of the matrix ˜D. It is still worthwhile mentioning that although +the matrix shares a similar feature with the dynamical matrix, the matrix elements are learned from neural network training and +are not necessarily the matrix elements from the real dynamical matrix. An intuitive comparison is that the edge of GNN does +not necessarily reflect true chemical bonding, but is more like an atomic neighbor connection. +The predicted phonons using MVN are summarized in Figure 3b, which shares the same structure with Figure 2b as error +tertile plots from the high-quality DFPT database (blue) and database for complex materials (orange). MVN shows comparable +performance with VVN for simple materials (blue curves in Figure 2c and Figure 3c), but shows significant performance +improvement for complex materials. The prediction loss distribution of MVN shows a heavier distribution toward a lower loss +regime compared to VVN (orange curves in Figure 2c and Figure 3c), and the average phonon frequencies in the correlation +plot align better toward ground truth (orange dots in Figure 2d and Figure 3d). More results and correlation plots are shown in +Supplementary Information IV. +Momentum-dependent matrix virtual nodes for predicting full phonon band structures. The structure of MVN inspires +4/11 + +O.SizZr2 +F202Y2 +Cs2F.KRh +GaO,P +Br1:OPb16Tl2 +b +600 +1000 +600 +1000 +a +400 +750 +400- +ZRm +400 +500 +000 +500 +200 +200 +200 +250 +ARm +0. +0 +0. +0 +Ag2Cl4CS2 +Cl-F2Sr2 +FePaSi4 +Cs2F.Pt +Lu2O3s Ta14 +600- +1000 +400- +200 +400 +300 +400- +200 +500 +100 +200 +200 +R: +100 +XV3m +R1 +0: +0 +0 +0 +口 +CMgN2 +Li,N +HgK,O2 +BraLizRb2 +Ba12l36Y4 +Add 3m +600- +virtual atoms +build crystal graph +2000 +Avgm +750 +300 +200 +1500 +400 +500 +200 +000 +1000 +100 +TViRm +250 +200 +100 +500 +0 +TV, Rm +0. +0 +C +d +3500 +FePSi4 +40 +3000 +Probability Density +30 +2500 +sort +Predicted w +2000 +CMgN2 +20 +1500 +O.SizZr2 +prediction +1000 +10 +Lu2O3sTa14 +Br1:O.Pb1.Tl2 +500 +Ba12l36Y4 +0.001 +0.010 +0.100 +1.000 +1000 +2000 +3000 +Loss +True w [cm-1]Figure 3. The matrix virtual node (MVN) method to predict Γ-point phonons. a. Augment m2 virtual nodes as m sets of +virtual crystals (left) and the message passing scheme and post-processing of virtual node features (right). The legends are the +same as Figure 2. In contrast to VVN, where each node Vj is a scalar, here, each node Vij is a 3×3 matrix. The phonon spectra +in MVN are obtained by solving the eigenproblems instead of direct output, as done in VVN. b. Selected test examples within +each error tertile. Tests from the same dataset as the training set and additional tests containing complex materials are predicted +in blue and orange, respectively. c. Comparison of prediction loss distribution with several examples of materials. d. The +correlation plots of average phonon frequencies with the graph y = x as reference. Better performance for MVN is achieved +than VVN for complex materials (orange color), which can be seen from both the loss distribution and the average phonon +frequencies. +us to take one step further and construct full momentum-dependent virtual dynamical matrices by taking into account the +unit cell translation, termed momentum-dependent matrix virtual nodes (k-MVN). We construct virtual-dynamical matrices +following Equation (2). In contrast to the MVN, which focuses on Γ-point phonons by taking⃗k = 0, here in k-MVN, we include +the phase factor ei⃗k·⃗T when defining the virtual dynamical matrices, where ⃗T is the relative unit-cell translation of a neighboring +unit cell origin relative to the chosen reference unit cell ⃗T0 (Figure 4a). If a total number of t neighboring unit cells are included, +each with translation ⃗Th,h ∈ [0,t −1] (reference cell included), then a total t copies of MVN-type virtual nodes matrices will +be generated, with a total number of tm2 virtual nodes V h +ij,h ∈ [0,t − 1],i, j ∈ [1,m] in k-MVN. To obtain the phonon band +structure, each set of virtual nodes at a given ⃗Th needs to multiply by the phase factor ei⃗k·⃗Th, and all virtual nodes at each ⃗Th +are summed in Equation 3. Thanks to the graph connectivity within the cutoff radius (see Methods), only a small number of t +is needed as long as crystal graph connectivity can be maintained. In practice, t is materials dependent, and t = 27 (nearest +neighbor unit cells) is sufficient for many materials and does not need to go beyond t = 125 (next-nearest neighbor unit cells) in +all cases. Intuitively, such a supercell approach resembles the ab initio band structure calculations with frozen phonons. To +facilitate the training, phonons from selected high-symmetry points are included in the training data, without the need to use +full phonon energies in the entire Brillouin zone. This significantly facilitates the training process while maintaining accuracy. +More details are discussed in Methods and Supplementary Information V. +Figure 4b shows the prediction results of phonon band structures. Here 12 materials are selected from the same dataset +for training (blue color) and the additional dataset for complex materials (orange color). Despite the complexity of a generic +phonon band structure, the k-MVN model could predict the positions and the shapes of the phonon bands, such as gaps between +different optical branches. The dispersion relations of the acoustic phonons are also well generated around the Γ-points on +5/11 + +Ag2Al.,Se4 +CINSr2 +Ba2Se6 +Ga2HgTe4 +Br18O:Pb16Tl2 +b +400 +400 +200 +三 +200 +400 +a +200 +200 +100- +100 - +200 +0 +0. +Cal.Sr2 +K.Se6 +As2SrZn2 +Al40. +Cd:O32Rb16Sis +200 +750- +200 +750- +500 - +100 - +500- +100- +100 +250- +250 +3m +0 +0 +0- +0 +0 +Add m virtual +B40O84Pr16 +La2N2 +NNaO2 +Ca2Cl4 +Br.Cs2Cu4 +copies +00 +build crystal graph +750 +200- +300- +1500 +1000- +500 - +200 +1000 +100 +500 - +100 +250- +500 +3m +0- +0 +0 +0. +C +100 +3500 + CdgO32Rb16Sig +Q ++0000 +80 +jth copy + Density +←B4oOg4Pr16 +60, +Predicted w [ +Ag2Al2Se4 +2000 + eigenvalues +Probability +40 +1500 +ZONN +1000 +Cal.Sr2 +20 +500 +Br1:O:Pb16Tl2 +prediction +-0 +0.001 +0.010 +2000 +3000 +1000 +Loss +True w [cm-1]the left three columns, even though we do not enforce that acoustic Γ-phonons have to be gapless with zero-energy known as +acoustic sum rule28. This may enable the prediction of crystal stability for future works. While there are risks that prediction +performance could be degraded for the phonon bands of at higher frequencies, most of the predicted phonons follow the +references, including the complex materials with more than 40 atoms per unit cell. +Figure 4. The momentum-dependent matrix virtual nodes (k-MVN) to predict full phonon band structures. a. (Top) +Augment m2 virtual nodes for each translation vector ⃗T, and with a total t neighboring unit cells, a total tm2 virtual nodes are +generated. (Bottom) By multiplying a phase factor by each translated unit cell, a full virtual dynamical matrix can be +constructed. b. Selected examples in the test set within each error tertile, for the high-quality DFPT database (blue) and +additional complex materials test (orange). Γ-point positions are labeled for each spectrum. +Discussion +We demonstrate the prediction of phonons directly from the materials’ atomic coordinates, using three different types of virtual +nodes – the VVN, the MVN, and the k-MVN – to augment the symmetry-aware Euclidean neural networks. The comparison +between the three virtual node approaches is summarized in Table 1. VVN directly acquires the phonon spectra from the virtual +nodes. The assignment of 3m virtual nodes ensures that the output phonon band number is always 3m for a crystal with a +primitive unit cell containing m atoms. In MVN, instead of computing phonon energies directly, a virtual dynamical matrix +(VDM) is constructed first, from which the phonon energies are solved as an eigenvalue problem. This step is crucial to gain +robustness for complex materials prediction since intermediate quantities like force constants and dynamical matrices are con- +sidered more “fundamental” than final phonon energies to reflect the interatomic interactions. The k-MVN goes one step further, +using the unit-cell translations to generate the momentum dependence that could be used to obtain the full phonon band structure. +Today, the ab initio calculations like frozen-phonon and DFPT remain the most accurate methods for phonon calculations. +Even so, since the VGNN-based phonon calculation skips the direct calculation of the material-by-material dynamical matrix, it +shows significantly faster computation speed while maintaining reasonable accuracy. Additional tests on SiGe alloys, FeCoNi +alloys, and other high-energy alloys are performed, which agree well with existing literature (Supplementary Information VI). +Finally, by using MVN, we build a database containing the Γ-phonon spectra for over 140,000 materials listed in Materials +6/11 + +a +b +Cs.Se +In16Rb16Sb24 +CaMg2Sb2 +Be2Li2Sb2 +175 +250 +500 +250 +150 +cell +200 +400 +200 +150 +300 +100 +150 +75 +100 +100 +200 +50 +50 +50 +100 +25 +T。 := 0 cell +Cs2FelnNa +Ag204Y2 +BrCsO3 +Hg4K:P:Se24 +500 +500 +800 +F009 +400 +400 +600 +300 +400 +400 +200 +200 +200 +200 +100 +100 +ild crystal graph +As4OS2 +Ge:O. +K.N2O6 +AugNd16O36 +1200 +700 +300 +prediction +1000 +1000 +600 +800 - +500 - +800 +200 +eigenvalues +600 +400 - +600 +300 +400 +DT +D(k) +100 +400 +200 +200 +200 +100 +DTi(k +SToProject (Supplementary Data and Supplementary Information VII). It took an eight-GPU system less than five hours to obtain +all results, even though some materials contain over 400 atoms per unit cell. Such efficiency enables the material design, +searching, and optimization in a much larger design space, including alloys, interfaces, and even amorphous solids, with superior +engineered phonon properties for thermal storage, energy conversion and harvesting, and superconductivity. In parallel, by +taking advantage of the flexibility endowed by virtual nodes, other properties that are challenging to predict for a conventional +GNN can be predicted similarly, such as electronic band structures and tight-binding and k · p effective Hamiltonian with +a variable number of bands, optical properties like flexible optical absorption peaks as in the Lorentz oscillator model, and +magnetic properties such as the number of propagation vectors. +Table 1. 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Charting lattice thermal conductivity for inorganic +crystals and discovering rare earth chalcogenides for thermoelectrics. Energy & Environmental Science, 14(6):3559–3566, +2021. +Methods +Phonon data preparation +We trained all of our models against an ab initio DFPT computational database for phonon dispersion in harmonic model25. +The data set contains material structures (the same as the primitive structure obtained from the Material Project12), second-order +derivatives of energies with respect to atomic perturbations for regular points inside the irreducible zone, and phonon dispersion +along highly symmetric paths of 1,521 crystalline inorganic materials. These materials have 2 to 40 atoms per unit cell, with an +average of 7.38. For this work, we only used the highly symmetric path phonon dispersion as our training data. The dispersion +is between wave vectors⃗k in the fractional reciprocal unit and response spectra in cm−1. All models randomly split the data +into 90% training (1,365 materials), and 10% testing (156 materials) sets. Furthermore, we trained our models with a 5-fold +cross-validation scheme. +We also got phonon dispersion of complex (more number of atoms per unit cell) materials from Atsushi Togo’s phonon +database31. We used seekpath11,33 module to get the highly symmetric path of each material. Then, we fed it alongside +POSCAR, FORCE_SET, and phonopy.config files from the database to Phonopy32’s python command to calculate the phonon +dispersion along such path. To quality control the data, we selected materials whose lowest Γ-phonon band is higher than +−0.07 cm−1. We also filtered the material to get only the ones with more than 40 atoms per unit cell. Finally, we randomly +selected 156 (the same as the number of data in the testing set for ease of comparison) out of 505 filtered materials. We used +them as our complex material data set. +Computation environments +We coded the models in Python 3.9.13 and trained them on our GPU cluster with CUDA version 10.2. To facilitate the model +implementation, and training, we used some important python modules: Pymatgen23 and ase20 for handling material structure +files (.CIF), PyTorch26 for managing model training framework, e3nn8 for implementing our neural network models in the +form that is equivariant for Euclidean group. +Virtual node graph neural network (VGNN) +We have developed a scheme for a graph neural network (GNN) for it to be able to have variable output dimensions depending +on the input size. For ease of understanding, we will explain the method with our work on phonon prediction. +Considering a material with m atoms per unit cell, we add n additional virtual atoms. We can adjust the number n depending +on the model architecture. Using both real and virtual atoms, we convert the crystal structures into periodic graphs with m real +nodes for the actual atoms and n virtual nodes for the added virtual atoms. Then, we connect nodes with edges indicating the +message-passing process. To preserve the structural information of the materials and limit the computational cost, we apply the +following rules for connections. First, if the distance between the two real nodes is within a specified cutoff radius rmax, the real +nodes are connected through bi-directed edges. We also set up an edge between a real node and a virtual node according to the +model description, but this edge is directed from real to virtual nodes. Lastly, we embed the information of radial distance +vector, e.g.,⃗rab from atom b to a, in the form of radial basis functions and spherical harmonics on the corresponding edge as +edge attributes, which represent the distance and the direction of⃗rab respectively. +Since each node represents an atom in the unit cell, we embedded the atomic numbers A information as node attributes A +by passing one-hot representation vectors of length 118 through an embedding layer. As for the model’s input, we embedded +the atomic masses Z information as input node features Z by passing the product of atomic mass and one-hot representation of +atomic number through an embedding layer. +The constructed graph is then passed through the model message passing that operates on the features with multiple +convolutions and gated activation layers18. After the final layer, which consists of only a convolution (no gated activation), each +9/11 + +of the n virtual node features is collected, and passed through the post-processing block, which output the 3m predicted phonon +branches. The post-processing block is different and will be explained in detail in the subsequent section of each model. +The model is optimized by minimizing the mean squared error (MSE) loss function between the phonon of the training data +set and the one predicted by the model after normalizing them by the maximum phonon frequency of each material. The full +network structure is provided in the supplementary Information. +Vector virtual node method (VVN) +VVN is a VGNN we designed for learning to predict Γ-phonon spectra from material structures. Since, for a material with m +atoms per unit cell, there are 3m phonon bands, one sensible choice of adding virtual atoms is to add 3m virtual atoms each +outputs the prediction of one of the bands. Hence, when there are m atoms in the unit cell of crystalline material, we assign +the position⃗rVi of the virtual nodes Vi,i ∈ [1,3m] following equation (1). We can set the atomic species of the virtual node as +anything, and we use Fe after optimization. +⃗rVi = i−1 +3m (⃗a+⃗b+⃗c). +(1) +Here⃗a,⃗b,⃗c indicates the unit cell vector of the material. In other words, 3m virtual atoms are placed along the diagonal line +from (0,0,0) to⃗a+⃗b+⃗c with equal spacing. By keeping the distances between the virtual nodes in the real space, it is possible +to give position dependencies to the feature updating process. In that sense, equation (1) can consistently keep virtual nodes +away from each other and enables us to use the virtual 3m virtual nodes as the output nodes of the network. To get information +from the whole structure, each of the 3m virtual nodes is connected to all of the real nodes via directed edges from real to +virtual nodes. After each convolution layer, the virtual node features are passed to a linear layer, converted to a scalar output, +and sorted based on their magnitudes. The outputted 3m scalars represent the predicted Γ-phonon. +Matrix virtual node method (MVN) +MVN is a VGNN we designed with the influence of the dynamic matrix representation of a periodic harmonic system for +learning to predict Γ-phonon spectra from material structures. Given the momentum vector⃗k, the dynamical matrix element +˜Dij(⃗k), which is a three-by-three matrix representing 3D harmonic interaction between atom Ri and R j, can be written as the +Fourier transform of the force constant matrix Φαβ +ij +following equation (2). Here, ZRi is Ri atom’s atomic mass, and ⃗Tα is +the αth unit cell position. Note that, for each k-vector, the system has 3m degrees of freedom and frequencies where m is the +number of atoms per unit cell. We can get the phonon dispersion relations ω(⃗k) by solving eigenvalues ω2(⃗k) of ˜D(⃗k), which is +a matrix with shape (3m,3m) that composed of m2 blocks of ˜Dij(⃗k) for i, j ∈ [1,m], +˜Di j(⃗k) = ∑ +α,β +Φαβ +i j +�ZRiZR j +ei⃗k·(⃗Tα−⃗Tβ ). +(2) +In the MVN method, we generate a matrix that could work like a dynamical matrix as is written in equation (2). Here, we focus +on the prediction of Γ-phonon, i.e. ⃗k =⃗0. So, the contributions of the same atom pair, e.g., Ri, and R j from every unit cell +separation ⃗Tα −⃗Tβ are summed without the⃗k-dependent exponential phase factor. Hence, the model needs to predict a matrix +with shape (3m,3m) representing such summation. In order to do that, while preserving the relation of each matrix element, +we generate m virtual crystals Cj, j ∈ [1,m] each of which has m virtual nodes Vij,i ∈ [1,m] of the same atomic species and at +the same positions as the real atoms Ri,i ∈ [1,m]. Here, a virtual node Vij represents the interaction term from a real node Rj +to another real node Ri by adding a directed edge from R j to Vij whenever there is an edge connecting Rj to Ri. After each +convolution layer, the virtual node features are passed to a linear layer and converted to complex-valued output vectors with +length 9. For each output feature, we reshape the output features into three-by-three matrices and arrange them such that Vij’s +matrix is the (i, j) block of ˜D supermatrix with shape (3m, 3m). Finally, we solve ˜D for its 3m eigenvalues, which work as the +Γ-phonon prediction. +Momentum-dependent matrix virtual node method (k-MVN) +k-MVN is a generalization of MVN model with non-zero⃗k. Unlike the MVN case, the k-MVN model needs to predict matrices +representing interactions between atoms from a unit cell, e.g., ⃗Tβ, to the different unit cell, e.g., ⃗Tα. Since the phase factor +depends only on the difference in unit cell positions, we can redefine ⃗T to be such a difference and simplify equation (2) into +˜Di j(⃗k) = ∑ +⃗T +Φ⃗T +i j +�ZRiZRj +ei⃗k·⃗T := ∑ +⃗T +D⃗T +i jei⃗k·⃗T. +(3) +10/11 + +With this simplification, for each ⃗T, we generate m virtual crystal C⃗T +j∈[1,m] the same way as in MVN. However, in this case, V⃗T +i j +represents the interaction term from a real node Rj to another real node Ri that is in the unit cell with unit cell position ⃗T with +respect to R j’s. In other words, we add a directed edge from R j to V⃗T +ij whenever there is an edge connecting Rj to Ri and that +edge represent⃗ri −⃗r j =⃗r′ +i +⃗T −⃗r′ +j. Here,⃗r′ is the atomic position relative to its unit cell. Since GNN only considers edges +with interatomic distance less than rmax, the model can generate, with this scheme, a non-zero matrix for only a finite number +of ⃗T that satisfy +min +i, j∈[1,m]|⃗r′ +i +⃗T −⃗r′ +j| ≤ rmax. +(4) +Hence, before the virtual crystal generations, the model also iterates through atom pairs to find all viable ⃗T. +Similar to the MVN model, we convert virtual node features into three-by-three matrices and merge them into a matrix with +shape (3m,3m) representing ˜D⃗T for each ⃗T. Finally, we weight sum these matrices with their phase factor to get ˜D and solve +for its 3m eigenvalues as phonon spectrum at wave vector⃗k. +Acknowledgements +RO and AC contribute equally to this work. RO, AC, AB, and ML thank M Geiger, S Fang, T Smidt, and K Persson for helpful +discussions, and acknowledge the support from the U.S. Department of Energy (DOE), Office of Science (SC), Basic Energy +Sciences (BES), Award No. DE-SC0021940, and National Science Foundation (NSF) Designing Materials to Revolutionize +and Engineer our Future (DMREF) Program with Award No. DMR-2118448. BL acknowledges the support of NSF DMREF +with Award No. DMR-2118523. TN, ND, and ML are partially supported by DOE BES Award No. DE-SC0020148. TN +acknowledges support from Mathworks Fellowship and Sow-Hsin Chen Fellowship. ML acknowledges the support from the +Class of 1947 Career Development Chair and discussions with S. Yip. +Competing interests +The authors declare no competing interests +Data Availability Statement +The data that support the findings of this study are openly available in GitHub at https://github.com/RyotaroOKabe/ +phonon_prediction. The Γ-phonon database generated with the MVN method is available at https://osf.io/ +k5utb/ +11/11 + +Virtual Node Graph Neural Network for Full Phonon +Prediction: Supplementary Information +Ryotaro Okabe1,2,†, Abhijatmedhi Chotrattanapituk1,3,†, Artittaya Boonkird1,4, Nina +Andrejevic5, Xiang Fu3, Tommi S. Jaakkola3, Qichen Song6, Thanh Nguyen1,4, Nathan +Drucker1,7, Sai Mu8, Bolin Liao9, Yongqiang Cheng10, and Mingda Li1,4,* +1Quantum Measurement Group, Massachusetts Institute of Technology, Cambridge, MA, USA +2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, USA +3Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, +MA, USA +4Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA +5Argonne National Laboratory, Lemont, IL, USA +6Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA +7Applied Physics, School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA +8Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina, USA +9Department of Materials, University of California, Santa Barbara, Santa Barbara, CA, USA +10Chemical Spectroscopy Group, Spectroscopy Section, Neutron Scattering Division Oak Ridge National +Laboratory, Oak Ridge, TN, USA +†These authors contributed equally +*e-mail: mingda@mit.edu +Contents +I +DATA PREPARATION +1 +II +VIRTUAL NODE GRAPH NEURAL NETWORKS +2 +II.1 Euclidean Neural Networks and Real Node Graph Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 +II.2 Virtual Nodes Augmentation and Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 +II.3 Neural Network Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 +III +NEURAL NETWORK TRAINING AND OPTIMIZATION +6 +IV +PERFORMANCE on Γ-PHONON PREDICTIONS +6 +IV.1 Additional Γ-Phonon Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 +IV.2 General Validity of Prediction on Unseen Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 +IV.3 Element-wise Prediction of Average Phonon Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 +V +FULL PHONON BAND STRUCTURES PREDICTIONS +20 +VI +Applications +24 +VI.1 Phonon Prediction on a Topological Weyl Semimetal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 +VI.2 Phonon Prediction in Alloy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 +VII Γ-PHONON DATABASE +26 +I DATA PREPARATION +The data set containing full phonon bands of 1521 semiconducting inorganic materials are calculated from density functional +perturbation theory (DFPT)?, which is the main dataset for training (“Main Database” for short). The phonon energy values at +Γ-points (for vector virtual nodes, VVN and matrix virtual nodes, MVN model), and other high symmetric points (k-MVN +model) were extracted and randomly split for training (90%) and testing set (10%). High symmetric points are different for +1 +arXiv:2301.02197v1 [cond-mat.dis-nn] 5 Jan 2023 + +each material depending on its structure, and the reduced fractional coordinates are implemented for both real and reciprocal +spaces for a primitive unit cell. A consistency check has been performed for all data, ensuring the intended k-points in the +fractional unit match the desired k-points in the Brillouin zone. Another set of data for testing the model was taken from the +Phonon database by Dr. Atsuhi Togo at Kyoto University (“Togo Database” for short)?, which contains the phonons of more +complex materials, but meanwhile contain more phonons with imaginary phonon energies. We use the Main Database for +training, given the more stringent convergence criteria. In addition, we randomly selected 156 materials in the Togo Database +with the lowest Γ phonons greater than -0.07 cm−1, and have at least 40 atoms per unit cell for testing the model trained from +the Main Database. The 156 materials from the Togo Database match the number as the testing set from the Main Database +(10% of the 1521 materials). Most elements appear in both the test set of the Main Database and the additional test in the Togo +Database. The distribution of the number of atoms per unit cell (m-value in the main text) for the Main Database and the Togo +Database are shown in Figure S1. The profiles of these data sets with respect to the elements are shown in Figures S2 - S3. +Figure S1. The distribution of atoms per unit cell. The number of atoms per unit cell in a. Training and test data set from +the Main Database b. Additional test data set of complex materials from the Togo Database. a. The total 1521 materials from +the Main database contain 2 to 40 atoms per unit cell with an average of 7.4 atoms per unit cell. b. The randomly selected 156 +complex materials with good ground-truth quality in the Togo Database with 42 to 174 atoms per unit cell (an average of 69.1 +atoms per unit cell). +II VIRTUAL NODE GRAPH NEURAL NETWORKS +II.1 Euclidean Neural Networks and Real Node Graph Convolutions +Our approaches were developed within the framework of a symmetry-aware graph convolutional neural network, the Euclidean +neural networks (E3NN)?. Figure S4 illustrates the overall architecture of the model. The model takes atomic mass and position +of atoms in a unit cell as an input, the same as the previous report?. This information is converted to a periodic graph with nodes +representing atoms and edges controlling the message passing between nodes. This input is then passed through a series of +convolution layers separated by nonlinear layers, which introduce the complexity to the model. The convolution layer computes +the tensor product of input features, and the convolution kernel is defined as: +f ′ +i = 1 +√z ∑ +j∈∂(i) +f j ⊗(h(||rij||))Y(rij/||rij||) +(1) +where f ′ +i is the output feature of atom i, and ∂(i) is the set of neighbouring atoms which rij ≤ rmax where rij is the relative +position from atom j to i. We sum the tensor product of the features of atom j and the convolution kernel consisting of the +learned radial function (h(||ri j||)) and the spherical harmonics Y(rij/||rij||). The normalizing factor z is the coordination +number, aka the number of neighboring atoms. After passing through the last convolution layer, the output features are +processed through the screening virtual node and processing layer. The mean square error (MSE) loss is calculated and used for +backpropagation to optimize the model. +2/28 + +a +250 +25 +200 +20 +S +counts +count +150 +15 +100 +10 +50 +5 +510 +20 +25 +40 +40 +60 +80 +100 +120 +140160 +180 +The number of atoms per unit cel +The number of atoms per unit cellFigure S2. Training and testing data by elements in the Main Database. The number of appearances by each chemical +element in the training (green) and the testing (blue) data sets. +II.2 Virtual Nodes Augmentation and Convolutions +With the advantages of parameter sharing of graph convolution and symmetry awareness of Euclidean neural networks, the +augmentation of the virtual node must satisfy the basic requirements for the model to still perform correct convolution that +is equivariant for Euclidean group; Virtual nodes must be embedded with the same dimensions of node attributes, and input +features as the real nodes and every edge connecting virtual nodes to any other nodes must be embedded with the same +dimensions of edge attributes as the edges connecting real nodes. +In our work on phonon prediction, we decided to augment virtual nodes by adding virtual atoms that represent virtual +nodes into the crystal structure and constructing the graph in the same way as when there are no virtual atoms. Because the +graph construction treats virtual atoms and real atoms equally, the results must already satisfy all requirements. Although this +augmentation method limits the freedom of virtual nodes to be constructed from some virtual atoms, there are still plenty of +degrees of freedom for us to engineer the model construction: numbers, types, and positions of added virtual atoms. In fact, +depending on the model, the virtual atoms do not need to be actual elements, e.g. virtual atoms with an atomic number of 4, but +with an atomic mass of 0. +Another degree of freedom that the virtual node method allows is node connectivity. However, since we want the model +to preserve the connection structure and message passing between real nodes, we decided to restrict the edges between real +and virtual nodes to be directed from a real to a virtual node only, while the connections among real nodes are the same as +previously described. Hence, with the restriction on the direction of message passing between real and virtual nodes, we can +design our model with any remaining connection combinations. +To sum up, all of our models (VVN, MVN, and k-MVN) augment virtual nodes by adding a certain number of virtual atoms +of certain types at certain locations in the original crystal cells. Then, they build a graph by treating virtual atoms as real atoms +to keep the functionality of convolution and symmetry awareness. Finally, they include all edges connecting real nodes, include +only some directed edges from real to virtual nodes, and include some edges connecting virtual nodes depending on the design +3/28 + +400 +350 +300 +250 +200 +150 +100 +50 +H Li Be B C NOF NaMg AI Si PS CI K Ca Sc Ti +Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br +400 +350 +300 +250 +200 +150 +100 +50 +Rb SrYZr Nb Mo Tc Ru Rh Pd Ag Cd In Sn SbTe +Cs Ba La Hf +Os +Pt Au Hg TI Pb BiFigure S3. Additional testing data by elements in the Togo Database. The number of appearances by each chemical +element in testing data of complex materials is shown among the 156 randomly selected materials. +of the model. +II.3 Neural Network Architectures +The generic neural network architectures is shown in Figure S4. +4/28 + +40 +Counts +30 +20 +10 +0 +LiBeBCNO +F Na Mg Al SiPS Cl +Cr Fe Ni Cu Zn Ga Ge As Se +H +KCa Sc +V +50 +40 +Counts +30 +20 +10 +0 +Zr Nb Ru Rh Pd Ag Cd In Sn Sb Te +Br Rb SräYä +Cs Ba La +Pt Au Hg TI Pb Bi +Hf +Ta +W +ReFigure S4. Overall architecture of the equivariant neural network and convolution layer. The model consists of n +layers of convolution layer separated by non-linear layers, screen virtual node layer, and post-processing layer. +5/28 + +Z +A +(m+n)×1 +(m+n) ×1 +One-hot +One-hot +embed +onehot(Z) +(m +n) x din +onehot(A) +(m+ n) Xdin +Irl +gn(rmax) × drad +Embedding +Embedding +gn(rmax) × Imax? +(o) Z +(m+n) × dem +Convolution +nconv × +Gated block +(u)Z +(m+n) ×mul +Convolution +(m+ n) Xmul +Screen n Virtual Nodes +n Xmul +Post Process +n X dout +WpredIII NEURAL NETWORK TRAINING AND OPTIMIZATION +We optimize the values of the training parameters. The set of parameters that gives the best results for VVN, MVN, k-MVN are +shown in Table S1. +Table S1. The parameter setting of VVN +Hyperparameter +VVN +MVN +k-MVN +Maximum cutoff radius (rmax[ ˚A]) +4 +4 +4 +Multiplicity of irreducible representation (mul) +16 +4 +4 +Number of pointwise convolution layer (nconv) +2 +3 +2 +Number of basis for radial filters (nrad) +10 +10 +10 +Maximum l of spherical harmonics (lmax) +2 +2 +2 +Length of embedding feature vector (dimem) +16 +32 +32 +Length of output feature vector (dimout) +1 +18 +18 +AdamW optimizer learning rate +(5·10−3)×0.96k +(5·10−3)×0.96k +(5·10−3)×0.96k +AdamW optimizer weight decay coefficient +0.05 +0.05 +0.05 +IV PERFORMANCE on Γ-PHONON PREDICTIONS +IV.1 Additional Γ-Phonon Predictions +We present additional Γ-phonon predictions. Figure S5-S10 shows the direct prediction and correlation plots of training, testing +data from the Main Database, and testing data from the Togo Database using the VVN model, MVN, and k-MVN model. +6/28 + +Figure S5. VVN model: Direct prediction results within the training set of the Main Database. a. Black lines indicate +the Γ-phonon from DFPT calculation (ground truth) in [cm−1]. The colored (green, yellow, red) lines represent predicted +Γ-phonon in 1st, 2nd, and 3rd tertiles, respectively. (Left) Loss distribution shows that it is heavily peaked in the 1st and 2nd +tertiles with lower error. b. The correlation plots (prediction Vs. ground truth) of all Γ-phonon within the training set in each +tertile. +7/28 + +a.) +FaK,Zn +Ag20,Sr2 +Br.Ca2 +Cl.RbzTi +Cs2F.KRh +LaRbS2 +400 +600- +400 +200 +400 +400 +200- +200 +200 +100 +200 +200 +100 +0.002 +0 +0 +As20.Y2 +LiO.Siz +F.NaRhTl2 +K.Sbz +AszMg3 +Br.CszNay +150 +300 +三 +1000 +750 +200 +400 +100 +200 +500 +500 +100 +50 +200 +250 +100 +0.005 +0: +0 +AuzLizO4 +MnzPe +P,SrZn2 +O.PdSr2 +Na2Nb204 +O.Ta2Y2 +750 +600 +600 +500 +300 +400 +0.010 +400 +400 +500 +200 +250 +250 +200 +200 +200 +100 +0 +0 +0. +0. +0 +CuzNa.P2 +K,PtTe2 +InzSe.Zn +Cl,La2z +AgsAs2Ks +CszSe +0.020 +400- +200 +200 +100 +200 +400 +200 +100 +100 +100 +50 +200 +04 +0. +PtSiZr +C.NaO. +O,RbSc +CozLa2Ne +Ba2O.Se2 +H,N.P2 +0.050 +2000 +600 +750 +300 +3000 +400 +500 +200 +2000- +1000 +1000 +100 +200 +250 +1000 +0.100 +0. +0 +0 +0- +0. +CNNaO +MgO +BaO2 +Cs202 +Cs2Ga,S4 +SSr +400 +300- +600- +2000- +750 +200- +400 +500 +500 +200 +1000 +200 +250 +100- +三 +b.) +0- +0 +0: +0 +2500 +2000 +1: +3000 +3000 +1500 +2000 +2000 +1000 +1000 +1000 +t +500 +0: +0 +1000 +2000 +3000 +500 +1000 +1500 +2000 +2500 +2000 +3000 +1000Figure S6. VVN model: Direct prediction results within the testing set of the Main Database. a. The predictions of +Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd, and 3rd tertiles (green, yellow, and red) in [cm−1]. The loss +distribution on the left shows a peak in the 1st and 2nd tertiles. b. The correlation plots (prediction Vs. ground truth) of all +Γ-phonon within the test set in each tertile. +8/28 + +Li,O,Sb +Cal.Sr2 +Baals +Na2O.Sb2 +FPd,Sr2 +MgzN,Sr +a.) +750+ +600 +600 +100 +500 +400- +400 +400 +100 +50 +250 +200 +200 +200 +0.002 +OgP2RbTa +Au2Na.Sb2 +BBe2CsF,Oa +HaMg2Ru +Cl.Cs2Pb +C2MgNa20 +2000 +200 +1500 +1000 +1000 +200 +1000 +100 +1000 +500 +500 +100 +500 +0 +0 +0 +0 +0.005 +K,O.Sb2 +Ge.N:Zn4 +BazBiO.Y +BazSes +AlzF. +O,Znz +750 +750 +750 +600 +200 +500 +400 +500 +500 +400 +250 +100 +200 +200 +250 +250 +0 +0.010 +Ag2Al,Ses +K,Li,Tez +As,Be2Gez +AgO,Sc +BiF.Rb +Ir.Sba +750 +600 +300 +400 +400- +200 +500 +200 +200 +250 +200 +200 +100 +100 +0.020 +0 +BaF.Ti +CITI +Bas +C2 +B1CraY4 +Ba2FN +1500 +400 +600 +150 +200 +1000 +400 +100 +500 +200 +100 +200 +50 +500 +三 +0 +0.050 +BizCazK2 +AuCs +C.Sia +HGaSnSr +H2O.Y2 +O.RbzRe2 +1000 +200 +1000 +1000 +100 +500 +100 +2000 +500 +500 +50 +0. +0 +0 +b.) +2000- +1500 +3000 +1250 +1500- +1000 +2000 +1000 +750 +500 +1000 +500 +250 +0. +0- +0 +500 +1000 +1500 +2000 +0 +500 +1000 +1500 +0 +1000 +2000 +3000Figure S7. VNN model: Direct prediction results within test set of the Togo Database using VVN. a. The predictions of +Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd, and 3rd tertiles (green, yellow, and red) in [cm−1]. The loss +distribution ranges broader in the higher values than those with training data (Figure S5) and testing data (Figure S6). b. The +correlation plots (prediction Vs. ground truth) of all Γ-phonon within the test set in each tertile. +9/28 + +Ca1:OseTa +Ge.Nd1:O32 +Ba,MgaO2aSic +AlsF4ePb12 +a.) +Na1-Oa2Rb16Sig +K1O20Sba +1000 +750 +600 +750 +600 +600 +500 +500 +400 +400 +400 +500 +0.010 +250 +250 +200 +200 +200 +0. +0- +0 +0- +BisS2oTl16 +GeaK2oLi4O28 +Al.O36Y16 +Bai2PsSeaz +CaeSiae +O24TegTl16 +0.020 +300 +400 +1000 +750 +600 +200 +500 +500 +400 +500 +200 +100 +250 +200 +0.050 +Oz.SmgZrg +Lu2O3Ta14 +I12NdO36 +O32Pr:Se:Ta12 +K40O32Tls +OzaPrgSis +750 +1000 +750 +1000 +400 +500 +500 +0.100 +500 +500 +500 +200 +250 +250 +-0 +0.200 +Br24HgzaP1&Sn4 +KeP1zSessTh4 +Be.Cd.O24Si.Tez +HgeN12OaaPb +OzePr1eSe12Sis +Ca12O36Te12 +3000 +1000 +750 +750 +1000 +400 +2000 +500- +500 +500 +200 +500 +250 +1000 +250 +0.500 +0- +-0 +0- +0- +Ag2Se54Ti12Tl10 +SeaSb24Sr24 +Sb,TeasTls4 +Bis2O112Te32 +CdizEr24Sese +1s6Mo24Pb4 +200 +750 +750 +1000 +600 +200 +1.000 +500 +400 +500 +100 +500 +100 +200 +250 +250 +0: +0- +2.000 +Ca12GagO4eSi.Sng +Al2004gSm12 +B24Ca12S48 +Bi4aBreK.O7 +As1oCl2aHg24ln4 +Er12Ga20O48 +3000 +3000 +3000 +1000- +1000 +2000 +2000 +2000 +2000 +500 +500 +1000 +1000 +1000 +1000 +5.000 +0: +0- +(q +7000 +5000- +20000- +6000 +4000 +5000 +15000- +3000 +4000 +3000 +10000 +2000 +2000 +5000 +1000 +1000 +40 +4000 +6000 +2000 +0 +2000 +4000 +0 +500010000 15000 20000Figure S8. MVN model: Direct prediction results within train set of the Main Database using MVN a. The predictions +of Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd and 3rd tertiles(green, yellow, and red) in [cm−1]. The loss +distribution on the left shows a peal in the 1st and 2nd tertiles. b. The correlation plots (prediction Vs. ground truth) of all +Γ-phonon within the test set in each tertile. +10/28 + +a.) +H,AICs2 TI +AISb +AsaK,Siz +PtSnTi +Ba2CICuO2 +Cu2F2Se2Sr2 +300 +400 +600 +300 +200 +1000 +200 +400 +200 +200 +100- +500 +100 +200 +100 +10 +0 +OaRbS,TI +PdsiZr +Ag4Ge2Se +In.Se.Zn +F.PtRb2 +Ag2GaSs +0.001 +400 +400 +600 +1000 +200 +200 +400 +200 +200 +500 +100 +100 +200 +0.002 +0. +F0 +0 +AIF.K,Na +F,Sc +As2Mgs +BazLizP2 +AgAsF6 +BeO.S +400 +750 +300 +600 +600 +1000 +500 +0.005 +400 +400 +200 +200 +250 +500 +200 +200 +100 +0. +0.010 +CaMg2Sb2 +CuaNbS4 +F,LiSb +As2OSr4 +BesF. +BeaN,Sr2 +400 +1000 +400 +600 +200 +1000 +400 +200 +500 +200 +100 +0.020 +500 +200 +Q. +0 +0 +BrRb +H2O2Rb2 +Mg2Mo2N4 +As4B2Rbe +LizNaSb +ClaK,Tl2 +1000 +300 +300 +750 +0.050 +3000 +100 +500 +200 +200 +2000 +500 +50 +250 +100 +100 +1000 +0. +0 +0.100. +BrCu +LiPZn +AlzP2 +CITI +SeSn +CICu +150- +150 +二 +150 +100- +400 +100 +0.200 +100 +100- +200- +50 - +200 +50 +50- +50 +0 +0- +b.) +2000 +2000 +3000 +1500 +1500 +2000 +1000 +1000 +1000 +500 +500 +0- +0 +0 +500 +1000 +1500 +500 +1000 +2000 +1500 +2000 +1000 +2000 +3000Figure S9. MVN model: Direct prediction results within the test set of the Main Database using MVN. a. The +predictions of Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd and 3rd tertiles(green, yellow, and red) in +[cm−1]. The loss distribution on the left shows a peak in the 1st and 2nd tertiles. b. The correlation plots (prediction Vs. +ground truth) of all Γ-phonon within the test set in each tertile. +11/28 + +a.) +BaBrCuO2 +BaCaO.Te +GeHg-Ses +Ge.NZn4 +Cs2F.NaSc +Ba,Se. +600 +750 +400 +200 +500 +200 +400 +500 +250 +100 +200 +100 +200 +250 +Br.CazK, +HzCl,Sr2 +12K.0 +Ga.Te. +Cs2S.Sn2 +AuaK.Sb2 +1000 +200 +200 +200 +400 +200 +500 +100 +0.005 +100 +100 +200 +100 +La2O2P2Znz +Ag2Na.Sbz +As2La2O,Zn2 +H.MgzRu +Baals +K.Ses +2000 +200 +400 +100 +200 +400 +100 +1000 +200 +0.010 +50 +200 +100 +0 +-0 +0 +Be,LizP2 +BazPd,S4 +CaP,Znz +LizNbOs +AlO. +IgSra +600 +150 +300 +750 +750 +100 +200 +400 +200 +500 +500 +100 +200 +250 +250 +K,Li,O.Znz +C.Si4 +CuGaO2 +CaCla +K.O4 +1000 +0.020 +600 +300 +600 +600 +750- +400 +200 +500 +400 +400 +500 +200 +100 +200 +250 +200 +0 +0 +0- +CuGal4 +NazO +Rb,Te +AuBiSrz +B1sCr4Y4 +200 +400 +100- +0.050 +600 +1000 +100 +400 +100 +200 +50 +500 +50 +200 +0 +b.) +1500 +2000 +1250 +3000 +1500 +: +1000 +2000 +750 +1000 +500 +1000 +500 +250 +Fo +0 +500 +1000 +1500 +0 +500 +1000 +1500 +2000 +0 +1000 +2000 +3000Figure S10. MVN model: Direct prediction results within test set of the Togo Database using MVN. a. The predictions +of Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd and 3rd tertiles(green, yellow, and red) in [cm−1]. The loss +distribution on the left shows a peak in the 1st and 2nd tertiles. b. The correlation plots (prediction Vs. ground truth) of all +Γ-phonon within the test set in each tertile. +12/28 + +a.) +FsOzoSb16 +Oa2Sn4Te12 +Er:OasTa14 +Ca12Ga:O48Si4Sna +Cl24Hf.Te24 +Bi4eClK.O72 +750 +1000 +750 +750 +300 +0.001 +500 +500 +500 +500 +200 +500 +250 +250 +100 +250- +0 +BagSa2Sb16 +BrsMo24Pba +PbaoSsaSb32 +Al.FsPb10 +K,O4oP12Th4 +CaBasF12024Y4 +400- +400 +1500 +0.002 +600 +1000 +400- +1000 +400 +200 +200 +500 +200 +200 +500 +0 +O +AlaF4gPb12 +I5Moz4Pba +Ba24N32W: +KNa.O2:P:Zn4 +112Nd,O36 +Bis2OseRb16 +400 +750- +750 +1000 +600 +500 +500 +500 +400 +200 +500- +250 +0.005 +250 +200 +Ca12O3eTe12 +Al20O4gY12 +InieRb1Sb24 +O2ReSm12 +Ba17Ho1:OsPt.Zng +B12Ca18N24 +1000 +750 +200 +1500 +500- +500 +1000 +500 +500 +100 +250 +500 +0.010- +01 +0 +FO +BaoOg4Pr16 +B,Ge,O4sPrs +Baa2SesoSn16 +B1-Pb16S40 +AuaBiasBrs6 +B12N12012Sr12 +1500 +1000 +1500 +400 +200 +1500 +1000 +1000 +1000 +200 +500 +100 +500 +005 +500 +0.020 +0 +0 +0 +Cds2l24Sb16 +B12Ba.O24 +B24Ca12048 +H1Li2020P2U2 +Hz2O4oSr4 +Cd12Er24Se4s +1500 +400 +200 +3000 +1000 +2000 +1000 +2000 +200 +100 +500 +1000 +1000 +b.) +1750 +3500- +1500 +. +1500 +3000 +1250 +1250 +2500 +1000 +1000 +2000 +750 +750 +1500 +500 +500 +1000- +250 +250 +500 +0. +-0 +0. +500 +1000 +1500 +0 +500 +1000 +1500 +0 +1000 +2000 +3000IV.2 General Validity of Prediction on Unseen Materials +In Figure S11, we plot the average phonon frequency against the quadratic mean of atomic mass in the unit cell. We apply +the predictive model on 5000 unseen crystal structures from the Materials Project with atomic site number N in each unit +cell (S11a,b) 1≤ N ≤ 20, (S11c,d) 21≤ N ≤ 40, (S11e,f) 41≤ N ≤ 80. For each material the average phonon frequency is +related to the mass by the hyperbolic relationship ¯ω = C ¯m−1/2, where a constant C represents the rigidity of the crystal. The +reasonable distribution of rigidity supports the physical validity of our model for unknown materials. Moreover, we characterize +the non-uniformity of atomic masses in each material by computing the ratio of the minimum mass mmin in a crystal to ¯m. The +materials with high mmin/ ¯m, containing both small and large atoms, tend to aggregate at lower ¯ω and higher ¯m. Whereas, the +materials with low mmin/ ¯m, containing atoms with similar mass, tend to aggregate at higher ¯ω and lower ¯m. These tendencies +suggest that small atoms, such as hydrogen and oxygen, give high-frequency phonons which agree with the results in the +previous study?. Note that the VVN’s plots show higher and broader ¯ω distribution in the low ¯m region as N gets larger, while +those of MVN keep narrower distribution in the same region. This corresponds to the results in Figure S7 that the Γ-phonon +predicted by VVN gets higher than the ground truth in the case the input materials are complicated. +13/28 + +Figure S11. Evaluation of model predictions on unseen crystal structures Average frequency of the predicted Γ-phonon +¯ω versus average atomic mass ¯m using (a.,c.,e.) VVN and (b.,d.,f.) MVN. 5000 structures with N atoms per unit cell are +randomly sampled; (a.,b.) 1≤ N ≤ 20 (c.,d.) 21 ≤ N ≤ 40 (e.,f.) 41 ≤ N ≤ 80. The black solid lines represent the least squares +that fit the hyperbolic relation ¯ω = C ¯m−1/2. The constant C is estimated from the fit as a. C=1743 b. C=1847 c. C=1760 d. +C=2073 e. C=2113 f. C=2157. The dot colors represent the magnitudes of mmin/ ¯m. +14/28 + +mmin/m +mmi/m +a +b +T1.0 +1.0 +4000 +1750 +3500 +0.8 +1500 +0.8 +3000 +1250 +2500 +0.6 +[cm-1] +0.6 +1000 +2000 +131500 +750 +0.4 +13 +0.4 +500 +1000 +0.2 +0.2 +500 +250 +0 +0 +50 +100 +150 +200 +250 +0 +50 +100 +150 +200 +250 +m [amu] +m [amu] +mmin/m +d +mmin/m +C +1.0 +1.0 +1200 +3500 +0.8 +3000 +0.8 +1000 +2500 +800 +0.6 +0.6 +2000 +131500 +13 +0.4 +0.4 +400 +1000 +0.2 +500 +200 +0.2 +0 +01 +0 +50 +75100125 +5150175200 +0 +25 +50 +75 +100125150175200 +m [amu] +m [amu] +mmn/m +mmin/m +e +1.0 +1.0 +4000 +1200 +3500 +0.8 +0.8 +1000 +3000 +0.6 +800 +0.6 +[cm-1 +[cm +2000 +600 +13 +13 +1500 +0.4 +-0.4 +400 +1000 +0.2 +200 +0.2 +500 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +m [amu] +m [amu]IV.3 Element-wise Prediction of Average Phonon Energies +In Figure S12-S17, we illustrate the correlation plots of the γ-phonons of all element appearing in the training and testing +data sets using VVN and MVN. The background color represents the value of the prediction loss (yellow to blue from low to +high loss). We could see that it is not strict, but there are dependencies of the prediction accuracy on the periodicity for some +parts of the elements (e.g. from 5B to 9F). Moreover, when compared to Figure S2-S3, it is generally found that elements +corresponding to high prediction loss in Figure S12-S17 tend to appear less frequently than those with lower errors. +Figure S12. Element-wise correlation plot of the prediction results within train set using VVN Correlation plots with +63 elements existing within the train set. +15/28 + +2000 +5B +1H +2000 +3Li +4Be +1500 +2000 +‘C +3000 +:i +9F +1500 +80 +0000 +3000 +2000 +2000 +1000 +2000 +1000 +1000 +1000 +1000 +. +1000 +1000 +1000 +500 +500 +2000 +2000 +1000 2000 +2000 +1000 +1000 +1000 +1000 +2000 +2000 +Avg. Loss +0 +0 +2000 +11Na +2000 +12Mg +13 Al +17CI +3000 +2000 +1500 +3000 +2000 +1500 +1000 +2000 +1000 +2000 +2000 +1000 +0.030 +1000 +: +1000 +500 + 500 +1000 +1000 +500 +1000 +0 +10 +10002000 +0 +2000 +1000 +2000 +1000 +2000 +10002000 +2000 +2000 +0 +1000 +0 +0 +0 +0 +0 +2000 +1500 +22Ti +25Mn + 27Co +210 +3000 +SC +1500 +1500 +1000 +400 +1000 +400 +400 +0000 +1000 +1000 + 500 +0.025 +500 +200 +1000 +200 +200 +500 +0 +0 +2000 +1000 +1000 +500 +2000 +1000 +250 +500 +1000 +0 +1500 +37Rb +1000 +750 +1000 +3000 +3000 +1000 +500 +2000 +2000 +500 +500 +500 +500 +500 +250 +1000 +1000 +0.020 +10 +500 +0 +1000 +5001000 +1000 +1000 +1000 +2000 +2000 +0 +0 +500 +0 +500 +500 +0 +0 +1000 +45Rh +1500 +41Nb + 42Mo +43- +Tc +1500 +0000 +600 +2000 +1000 +1000 +750 +1000 +2000 +1000 +400 + 500 +. +500 +1000 +500 +1000 + 500 +200 +0os +250 +0.015 +0 +01 +2000 +1000 +1000 +1000 +2000 +0 +0 +0 +500 +1000 +0 +500 +2000 +1500 +2000 +1000 +800 +46Pd +52- +53 +750 +Te +1500 +1000 +1500 +750 +1500 +600 +1000 + 500 +1000 +1000 +500 +1000 +400 +500 +500 +500 +250 +500 +250 +500 +200 +0.010 +O +1000 +500 +1000 +2000 +0 +1000 +2000 +0 +1000 +1000 +500 +1000 +0 +500 +0 +0 +0 +57 La +72Hf +2000 +2000 + 56Ba +1000 +74W +750 +3000 +Ta +1500 +750 +1000 +1500 +2000 +500 +2000 +500 +1000 +500 +1000 +500 +250 +1000 +1000 +500 +500 +250 +f.i. +0： +0 +0.005 +2000 +1000 +2000 +2000 +1000 +5001000 +5001000 +500 +1000 +0 +500 +2000 +0 +0 +1500 +77/r +80Hg +78pt +81TI +83Bi +600 +2000 +: +600 +400 +1000 +1000 +400 +400 +400 +: +1000 +500 +200 +200 +500 +200 +200 +200400 +1000 +2000 +0 +500 +0 +500 +0 +1000 +0 +500 +0 +1000 +0 +0Figure S13. Element-wise correlation plot of the prediction results within test set using VVN. Correlation plots with 59 +elements existing within the test set. +16/28 + +3Li +4Be +1H +9F +°℃ +1000 +80 +3000 +1500 +.. +: +0000 +1000 +1000 +1000 +1000 +2000 +1000 +2000 +500 +: +500 +500 +500 +500 +1000 +1000 +500 +2000 +1000 +0 +1000 +1000 +1000 +5001000 +2000 +1000 +0 +500 +0 +Avg. Loss +0 +0 +1000 +1000 +1000 +2000 +11Na +2000 +17 CI +12Mg +19 K. +14Si +15p +750 +1000 +750 +750 +1500 +1500 +1000 +1000 +. + 500 +500 + 500 +1000 +1000 +: +500 +500 +500 +250 +250 +250 +500 +500 +0.040 +0 +1000 +2000 +1000 +2000 +1000 +500 +1000 +500 +1000 +0 +1000 +500 +1000 +500 +0 +0 +1000 +0 +0 +0 +28Ni +800 +29 Cu +23V +600 +300 +750 +750 +200 +600 +300 + 500 +500 +400 +200 +500 +0.035 +1.: +400 +200 +100 +250 +250 +200 +100 +250 +200 +100 +0 +.. +0 +500 +500 +200 +500 +400 +500 +500 +200 +100 +200 +0 +0 +0 +800 +800 +800 +33 As +37Rb +0.030 +307 +Zn +1500 +750 +1500 +600 +600 +600 +400 +1000 +400 +500 +/ +400 +400 +1000 + 500 +/: +茶 +250 +200 +. +500 +200 +200 +200 +0.025 +500 +500 +500 +1000 +500 +500 +1000 +1000 +0 +0 +500 +0 +0 +0 +0 +0 +45Rh +48Cd +2000 +46pd +39V +3000 +600 +1500 +1500 +600 +1500 +1000 +2000 +400 +1000 +1000 +400 +500 +1000 +500 +1000 +0.020 +200: +500:1 +500 +200 +500 +2. +10 +0 +500 +2000 +1000 +0 +1000 +0 +2000 +1000 +0 +500 +0 +1000 +0 +0 +50Sn +1000 +600 +600 +400 +1000 +1000 +400 +1000 +0.015 +400 +400 +500 +200 + 500 +500 +500 +200 +200 +200 +0 +0 +FC +0 +1000 +0 +1000 +500 +0 +500 +0 +200 +400 +1000 +0 +1000 +500 +0 +200 +400 +0.010 +1000 +79 Au +80Hg +JHz +77/r +2000 +1000 +74W +78pt +600 +200 +1000 +200 +750 +200 +1500 +. +750 +400 +500 + 500 +1000 +100 +100 +500 +100 +200 +250: +250 + 500 +: +0.005 +70 +0 +70 +100 200 +5001000 +500 +1000 +1000 +200 +1000 +2000 +100 +200 +0 +500 +0 +800 +800 +82pb +83Bi +600 +600 +1000 +400 +400 +200 +500 +200 +0 +500 +500 +0 +1000 +0 +0Figure S14. Element-wise correlation plot of the prediction results within train set of the Togo Database using VVN. +Correlation plots with 64 elements existing within the test set of complicated materials. +17/28 + +4000 +20000 +4000 +6000 +20000 +20000 +15000 +4000 +3000 +: +15000 +3000 +15000 +15000 +10000 +4000 +10000 +2000 +2000 +10000 +10000 +ii +2000 +5000 +2000 +5000 +1000 +1000 +5000 +5000 +6C +1H +3Li +4Be +5B +7N +80 +9F +0 +10000 20000 +5000 +4000 +4000 +5000 +10000 20000 +10000 20000 +2500 +2000 +10000 +2000 +Avg. Loss +3000 +6000 +20000 +3000 +20000 +6000 +6000 +7500 +4000 +2000 +15000 +15000 +4000 +2000 +4000 +5000 +10000 +10000 +1000 +2000 +2000 +1000 +2500 +2000 +11Na +5000 +5000 +19K +12Mg +13 Al +25005000 +2000 +10000 20000 +5000 +5000 +2000 +5000 +1000020000 +0 +0 +0 +3000 +3000 +4000 +1500 +3000 +6000 +6000 +0.8 +3000 +2000 +2000 +2000 +1000 +2000 +4000 +4000 +2000 +1000 +1000 +1000 +2000 +2000 +500 +1000 +24Cr +1000 +32Ge +22Ti +31Ga + 33 As +70 +2000 +5000 +1000 +2000 +4000 +2000 +2000 +2000 +0 +5000 +1500 +1500 +1500 +: +2000 +3000 +4000 +i +1500 +10000 +1000 +1500 +1000 +1000 +2000 +0.6 +1000 +1000 +2000 +5000 +500 +500 +500 +1000 +500 +39 +500 +41Nb +37Rb +42Mo +70 +0 +0 +1000 +1000 2000 +1000 +20004000 +5000 10000 +2000 +1000 +1000 +0 +0 +0 +0 +0 +0 +2000 +3000 +3000 +3000 +1500 +2000 +15000 +1000 +1500 +: +2000 +2000 +1000 +2000 +10000 +1000 +1000 +500 +1000 +1000 + 500 +1000 +5000 +0.4 + 500 +47 Ag +48Cd +49/m +51Sb +52Te + 56Ba +0 +2000 +2000 +1000 +2000 +1000 2000 +2000 +5001000 +1000 +0 +10000 +0 +0 +0 +0 +3000 +3000 +3000 +6000 +3000 +6000 +1000 +2000 +2000 +2000 +2000 +4000 +4000 +2000 +1000 +500 +1000 +1000 +1000 +2000 +2000 +57 /a +1000 +PNo9 +62Sm +65Tb +59Pr +67 Ho +66Dy +68Er +0.2 +1000 +5000 +5000 +2000 +2000 +2000 +2000 +0 +0 +2000 +0 +0 +3000 +4000 +3000 +1500 +800 +3000 +3000 +2000 +3000 +600 +2000 +2000 +1500 +1000 +2000 +2000 +2000 +1000 +400 +1000 +1000 +500 +1000 +1000 +1000 +70Yb +73Ta +72Hf +500 +200 +69Tm +71Lu +74W +75Re +2000 +2000 +2000 +1000 +4000 +500 +2000 +0 +0 +2000 +0 +0 +3000 +1500 +3000 +3000 +1000 +1500 +1500 +4000 +2000 +2000 +1000 +2000 +. +1000 +1000 +500 +2000 +1000 +500 +1000 +500 +1000 +500 +90Th +92V +8oHg +82pb +83Bi +78 Pt +70 +0 +1000 +500 +0 +1000 +2000 +1000 +2000 +1000 +2000 +2500 +5000 +0 +0 +0 +0Figure S15. Element-wise correlation plot of the prediction results within train set using MVN. Correlation plots with +about 63 elements existing within the train set. The plotted phonon frequencies are in [cm−1]. +18/28 + +2000 +2000 +1H +2000 +3Li +5B +2000 +3000 +7N +80 +9F +4Be +1500 +3000 +1500 +3000 +2000 +2000 +1000 +1000 +2000 +1000 +1000 +1000 +1000 +1000 +1000 +500 +500 +0 +0.: +Avg. Loss +2000 +10002000 +1000 +2000 +1000 +1000 2000 +2000 +2000 +1000 2000 +0 +0 +2000 +12Mg +2000 +13 Al +17 Cl +19 K +3000 +3000 +2000 +1500 +2000 +3000 +1000 +1500 +2000 +1000 +2000 +2000 +1000 +1000 +1000 +500 +0.0200 +500 +1000 +1000 +1000 +500 +0 +1000 2000 +0 +2000 +2000 +2000 +2000 +2000 +1000 +1000 +5001000 +1000 2000 +0 +600 +2000 +600 + 20Ca +23V +25Mn +28Ni +21C +3000 +SC +1500 +1500 +1000 +400 +. +1000 +400 +400 +0.0175 +2000 +1000 +1000 +. +500 +500 +200 +200 +200 +1000 + 500 +500 +:0 +2000 +1000 +1000 +1000 +2000 +1000 +0 +0 +200 +400 +250 +500 +0 +250 +500 +0 +0 +0 +0 +800 +1500 +33 As +0.0150 +1000 +1000 +1000 +1000 +3000 +3000 +600 +1000 +2000 +2000 +400 +500 +500 +. +500 +500 +. +: +500 +1000 +200 +1000 +0 +C +?0 +500 +1000 +1000 +5001000 +0 +500 +1000 +2000 +2000 +0 +0 +0.0125 +39 +41Nb + 42Mo +1000 +600 +3000 +1500 +1000 +2000 +1000 +1000 +750 + 400 +2000 +1000 +500 +500 +1000 +500 +500 +200 +1000 +. +500 +250 +0.0100 +0 +0 +2000 +2000 +1000 +1000 +500 +500 1000 +0 +0 +1000 +1000 +0 +500 +0 +0 +0 +800 +2000 +2000 +800 +48Cd +100052- +531 +46Pd +51Sb +:-i. +1000 +Te +1500 +600 +1500 +1500 +1000 +600 +1000 +400 +1000 +400 +1000 +500 +500 +500 +0.0075 +500 +200 +500 +200 +500 +0: +1000 +500 +1000 +2000 +1000 +2000 +1000 +5001000 +5001000 +500 +0 +0 +0 +0 +0 + 57 La +1000 +2000 +1000 +74W +2000 +72Hf +760s +3000 +1500 +750 +1000 +1500 +0.0050 +2000 + 500 +2000 +500 +500 +1000 +1000 +！ +500 +250 +1000 +1000 + 500 +250 +500 +0 +2000 +2000 +1000 +1000 +5001000 +500 +1000 +2000 +500 +500 +1000 +2000 +0 +0 +0 +0 +600 +800 +800 +3Ho8 +81TI +600 82Pb +1500 +2000 +78pt +009 +600 +1500 +1000 +400 +1000 +400 +400 +400 +1000 +500 +200 +500 +200 +200 +200 +500 +20 +0 +250 +500 +1000 +500 +500 +1000 +500 +0 +2000 +0 +0 +0 +1000 +0 +0Figure S16. Element-wise correlation plot of the prediction results within test set using MVN. Correlation plots with +about 59 elements exist within the test set. The plotted phonon frequencies are in [cm−1]. +19/28 + +5B +9F +1H +‘C. +7N +80 +3000 +. +1500 +3000 +： +1000 +1000 +1000 +1000 +2000 +Y.. +1000 +2000 +500 +500 +500 + 500 +500 +1000 +500 +:. +1000 +2000 +500 +1000 +1000 +1000 +1000 +500 +2000 +1000 +0 +0 +Avg. Loss +0 +0 +1500 +1000 +1000 +11Na +2000 +15P +2000 +12Mg +13 Al +14Si +17 CI +1008 +1000 +750 +750 +1500 +1000 +1500 +600 +1000 +500 + 500 +1000 +1000 +400 +500 +500 + 500 +250 +500 +250 +200 +500 +: +1000 +1000 +1000 +1000 +2000 +1000 +500 +1000 +500 +1000 +500 +0 +0 +2000 +0 +0 +500 +0 +0 +1000 +20Ca + 24Cr +23V +26Fe +28 Ni +29 Cu +0.04 +400 +750 +300 +600 +200 +1000 +300 +400 +500 +500 +400 +200 +200 +100 +250 +500 +250 +200 +.. +200 +100 +100 +.. +1.0 +0 +0 +0. +70 +500 +1000 +500 +500 +500 1000 +400 +250 +200 +100 +200 +500 +200 +1000 +800 +31Ga +33 As +800 +800 +37Rb +1. +38S +Zn +1000 +1500 +750 +1500 +Sr +600 +600 +600 +400 + 500 +1000 +0.03 +1000 +400 +400 +400 +500 +250 +200 +.. +500 +200 +200 +200 +500 +. +0 +500 +500 +500 +500 +500 +1000 +1000 +500 +1000 +1000 +0 +0 +0 +0 +0 +2000 + 48Cd +46 P.d +39V +750 +3000 +1500 +1000 +1500 +1500 +009 + 500 +400 +2000 +1000 +1000 +1000 +400 + 500 +250 +1000 +500 +500 +i +200 +200： +500 +0.02 +o. +10 +0 +0 +0 +YO +500 +500 +2000 +2000 +1000 +1000 +1000 +500 +0 +0 +500 +1000 +800 + 52Te + 57 La +53 +1000 +600 +009 +600 +400 +1000 +1000 +1000 +400 +400 +400 +500 +200 +500 +500 +500 +200 +200 +200 +0 +0 +1000 +500 +1000 +1000 +500 +0 +500 +1000 +500 +0 +0 +200 +400 +0 +0 +0 +0 +0.01 +1000 +74W +77 +72Hf +200 +79 Au +600 +80Hg +75Re +2000 +78pt +1000 +: +200 +1000 +750 +200 +150 +750 +1500 +400 +500 + 500 +1000 +100 +500 +100 +200 +250 +/: +250 +500 +50 +0 +0 +70 +：0 +02 +100200 +5001000 +500 +1000 +1000 +200 +1000 +2000 +100 +200 +250 +500 +1500 +81TI +800 +83Bi +600 +1000 +009 +400 +400 +200 +500 +200 +0 +1000 +500 +0 +500 +0Figure S17. Element-wise correlation plot of the prediction results within test set of the Togo Database using MVN. +Correlation plots with about 64 elements exist within the test set of the complicated materials. The plotted phonon frequencies +are in [cm−1]. +V FULL PHONON BAND STRUCTURES PREDICTIONS +We present more results by using the k-MVN model for predicting the full phonons dispersion band within the training and +testing data from the Main Database and complex materials from the Togo database. Note that some of the phonons in DFPT +calculation have negative values, and the model is set so that it only gives positive phonon frequency. Although the phonon +bands prediction of some materials does not match up exactly with the DFPT calculation, some features of the bands, such as +an average frequency and band gaps, are well captured by the model. +Figure S18-S20 provide more results of the full phonon bands prediction using k-dependent matrix virtual nodes model. +20/28 + +3000 +3000 +3000 +3000 +3000 +3000 +2000 +2000 +2000 +2000 +2000 +2000 +2000 +2000 +! +1000 +1000 +1000 +1000 +1000 +1000 +1000 +1000 +3 Li +5B +7N +1H +4Be +O: +9F +0 +70 +C +2000 +1000 2000 +2000 +2000 +2000 +2000 +0 +2000 +0 +0 +10002000 +Avg. Loss +0 +0.035 +3000 +1500 +3000 +3000 +3000 +3000 +1500 +3000 +1000 +2000 +2000 +2000 +2000 +1000 +2000 +2000 +:..1 +500 +1000 +1000 +1000 +1000 +500 +1000 +1000 +11Na +12Mg +13 Al +17 Cl +15 p +19 K +0 +1000 +0 +1000 +0 +2000 +2000 +0 +2000 +2000 +2000 +2000 +0 +0 +0.030 +1500 +1000 +1500 +3000 +2000 +: +3000 +1500 +750 +1500 +2000 +1000 +1000 +2000 +2000 +1000 +500 +1000 +i +500 +1000 +500 +1000 +1000 +500 +250 +24Cr +20Ca +500 +22Ti +32Ge + 33 As +30Zn +31Ga +70 +0 +70 +70 +2000 +1000 +2000 +500 +1000 +0 +0 +1000 +1000 +2000 +1000 +2000 +0.025 +1500 +1000 +008 +3000 +3000 +3000 +1000 +1000 +1000 +750 +600 +2000 +2000 +2000 +500 +400 +500 + 500 +500 +1000 +1000 +1000 +40X +34Se +37Rb +250 +200 +35Br +41Nb +42 Mo +38Sr +39Y +0 +0 +0.020 +500 +1000 +500 +1000 +1000 +2000 +2000 +2000 +500 +1000 +0 +500 +0 +0 +0 +0 +0 +1500 +3000 +3000 +1000 +1000 +1000 +1000 +1000 +1000 +2000 +2000 +500 +500 + 500 +500 +500 +500 +1000 +1000 +47 Ag +49/n +50Sn +48 Cd + 51Sb + 52Te +56Ba +53 +20 +70 +0.015 +Q +500 1000 +1000 +2000 +0 +500 +1000 +500 +1000 +500 +1000 +500 +1000 +2000 +0 +0 +1500 +1500 +1500 +1500 +1500 +1000 +1000 +2000 +1000 +1000 +1000 +1000 +1000 +500 +500 +1000 +500 + 500 +500 +500 +67 Ho +500 +57 La +Jd6s +PNo9 +65Tb +66Dy +68Er +0.010 +70 +0 +o +500 +10002000 +0 +1000 +0 +1000 +0 +1000 +1000 +0 +1000 +0 +500 +1000 +0 +1000 +1000 +1500 +1000 +3000 +008 +i +1500 +750 +750 +1000 +750 +600 +1000 +2000 +1000 +500 +500 + 500 +400 +500 +500 +1000 +500 +250 +250 +71LU +250 +HzL +13Ta +200 +74W +75Re +77/r +69Tm +70yb +0.005 +O +10 +0 +1000 +1000 +1000 +500 +1000 +0 +500 +1000 +0 +0 +500 +500 +0 +2000 +1000 +800 +1500 +1000 +1500 +3000 +750 +750 +2000 +1000 +600 +1000 +1000 +2000 +500 +400 +500 +1000 +500 +500 +500 +250 +1000 +200 +250 +79 Au +78pt +82pb +90Th +92U +0 +0 +500 +500 +1000 +1000 +500 +1000 +0 +2000 +0 +1000 +5001000 +0 +2000 +0 +0 +0 +0 +0Figure S18. Direct prediction result of phonon band structures within the training set of the Main Database using +k-MVN. Each band contains 100 k-points across the Brillouin zone which varies for each material depending on its structure. +The band calculated from DFPT are labeled in black, and green, yellow, and red lines represent the predicted phonon bands in +1st-3rd tertiles. +21/28 + +HaK,Mg +Ga2S4Zn +NO.Rb2 +CCaN +Cs4O,Zr2 +As2HgK4 +400 +1500 +250- +2000 +1250 +1250 +600 +300 +200 +1000 +1000 +1500- +150 +750 +750 +400 +200- +1000- +500 +100 +1E-4. +500 +100 +200 +500 +250 +50 +250 +FO +0 +Ga2Li2Te +Ua +Cu404Pb2 +CusS.V +PtSnTi +2E-4 +600 +250 +300- +400 +300 +150 +500 +200- +400 +300 +200 +150- +200 +100 +300 +200 +100 +200 +100- +100 +100 +100 +50 +5E-4 +FO +0 +0 +0: +K,O.S2 +As,O,Sr +Al,Sb, +GePtTi +K,O.Sb, +O.Sr,Ti +300 +1250 +600 +800 +300 +AX +250 +600 +500 +1E-3 +1000 +600 +200 +400 +200 +750 +400 +150 +300 +400 - +500 +100 +200 +100 +200 +200 +250 +50 +2E-3 +100 +0- +0- +NiSnTi +PdSiTi +PdSiZr +Ba4Br: +Cs4P +F4Mg2 +300 +500 +600 +300- +250 +150 +300 +400 +500 +250 +200 +400 +200 +100 +300 +5E-3 +200 +150- +150 +300 +200 +100- +100. +50 +200 +100 +100 +50 +50 +100 +0 +0 +0 +0 +1E-2 +Cd2Te2 +Cl2Csl +FeTeTi +BaBeaN +PtSiZr +FeInK2Na +300 +1000 +500 +150- +300 +250 +300 +800 +400 +200 +300 +2E -2 +100 +200 +600 +200 +150 +200 +400 +50 +100 +100 +100 +100 +200 +50 +0 +0 +100 +CssF.TI +NY +AsF.NaRb2 +I2Li2 +BiLis +BaO +5E-2 +400 +400 +600 +500- +400- +300 +500 +400 +300 +300 +300 +400 +300 +200 +200 +300 +200 +200 +200 +100- +200 +100 +100 +LO0 +0. +100 +-100Figure S19. Direct prediction result of phonon band structures within the testing set of the Main Database using +k-MVN. Each band contains 100 k-points across the Brillouin zone which varies for each material depending on its structure. +The band calculated from DFPT are labeled in black, and green, yellow, and red lines represent the predicted phonon bands in +1st-3rd tertiles. +22/28 + +AlCdSe4 +C2CaK,O6 +CFKO Sr +Ag4K.O4 +In2OgTa2 +Cdln2Se4 +400 +1500 +1500 +800 +500 +1250 +200 +1250 +300 +400 +600 +1000 +1000 +5E-4 +150 +300 +200 +750 +750 +400 +100 +500 +200 +500 +100 +200 +50 +250 +100- +250 +0 +0 +0 +CdGa2Te4 +Cs2F.Hf +BaF.Ge +Ir4S4Sb4 +Ba2HfO4 +BaF6Pb +600 +800 +600 +500 +200 +500 +1E-3 +300 +600 +400 +150 +400 +400 +300 +200 +300 +400 +100 +200 +200 +200 +100 +200 +50 +100 +100 +0 +0 +Cs2F.InNa +CaF.Pb +Br.Cs2NaY +Br.Cs2InNa +BaBiO.Ta +F.NaRb2Rh +500 +800 +2E-3 +200 +600 +600 +200 +400 +600 +500 +150 +150 +300 +400 +400 +100 +400 +100 +300 +200 +200 +50 +200 +50 +200 +100 +100 +0 +0 +FO +0 +Ba2Cu2F2Te2 +Nb2O.SC2 +CS2F.KY +Cl.Cs2Mg2 +Cl4K2Pt +Ag2K2Se2 +500 +400 +300 +800 +150 +400 +300 +300 +600 +5E-3 +200 +300 +100 +200 +200 +200 +400 +100 +100 +50 +200 +100 +100 +0- +0 +0 +0 +0 +1E-2 +BiCs2FaK +NiPSc +CaF2 +Cs204Y2 +NaOSc +Cs2F.RbY +500 +500 +600 +400 +400 +300 +400 +400 +300 +300 +400 +300 +300 +200 +200 +200 +200 +200 +100 +200 +100 +100 +100 +100 +2E -2 +0 +100 +Rb.Te. +Na.P2 +F.Sc +In2Na2Te +AS4C04S +IRb +200 +100 +250 +400 +150 +600 +200 +80 +150 +300 +100 - +150 +400 +60 +100 +100 +200 +50 +40 +50 +200 +50 +100 +20Figure S20. Direct prediction result of phonon band structures of the Togo Database using k-MVN. Each band +contains 100 k-points across the Brillouin zone which varies for each material depending on its structure. The band calculated +from DFPT are labeled in black, and green, yellow, and red lines represent the predicted phonon bands in 1st-3rd tertiles. +23/28 + +AlCdSe4 +C2CaK,O6 +CFKO Sr +Ag4K.O4 +In2OgTa2 +Cdln2Se4 +400 +1500 +1500 +800 +500 +1250 +200 +1250 +300 +400 +600 +1000 +1000 +5E-4 +150 +300 +200 +750 +750 +400 +100 +500 +200 +500 +100 +200 +50 +250 +100- +250 +0 +0 +0 +CdGa2Te4 +Cs2F.Hf +BaF.Ge +Ir4S4Sb4 +Ba2HfO4 +BaF6Pb +600 +800 +600 +500 +200 +500 +1E-3 +300 +600 +400 +150 +400 +400 +300 +200 +300 +400 +100 +200 +200 +200 +100 +200 +50 +100 +100 +0 +0 +Cs2F.InNa +CaF.Pb +Br.Cs2NaY +Br.Cs2InNa +BaBiO.Ta +F.NaRb2Rh +500 +800 +2E-3 +200 +600 +600 +200 +400 +600 +500 +150 +150 +300 +400 +400 +100 +400 +100 +300 +200 +200 +50 +200 +50 +200 +100 +100 +0 +0 +FO +0 +Ba2Cu2F2Te2 +Nb2O.SC2 +CS2F.KY +Cl.Cs2Mg2 +Cl4K2Pt +Ag2K2Se2 +500 +400 +300 +800 +150 +400 +300 +300 +600 +5E-3 +200 +300 +100 +200 +200 +200 +400 +100 +100 +50 +200 +100 +100 +0- +0 +0 +0 +0 +1E-2 +BiCs2FaK +NiPSc +CaF2 +Cs204Y2 +NaOSc +Cs2F.RbY +500 +500 +600 +400 +400 +300 +400 +400 +300 +300 +400 +300 +300 +200 +200 +200 +200 +200 +100 +200 +100 +100 +100 +100 +2E -2 +0 +100 +Rb.Te. +Na.P2 +F.Sc +In2Na2Te +AS4C04S +IRb +200 +100 +250 +400 +150 +600 +200 +80 +150 +300 +100 - +150 +400 +60 +100 +100 +200 +50 +40 +50 +200 +50 +100 +20VI Applications +VI.1 Phonon Prediction on a Topological Weyl Semimetal +The prediction of LaAlGe Γ-phonon obtained from k-MVN model compared to the experimental results?. The phonon +dispersion of LaAlGe- a Type-II Weyl semimetal, was measured by inelastic neutron and X-ray techniques along the high +symmetric points and Weyl points. +Figure S21. Comparison of LaAlGe’s phonon spectra acquired with experiments and machine learning (Top) +LaAlGe phonon bands along the highly symmetric points. (Bottom) The phonon band structures along the x, y, z directions +within the Brillouin zone near one of the Weyl points W1. The plotted phonon frequencies are in [meV]. +VI.2 Phonon Prediction in Alloy Systems +One of the most important applications of our prediction model is phonon bands of alloy systems. Crystalline alloys have +disorders whose atomic positions can be periodic. However, the system is not periodic because atomic elements’ distribution at +one site is probabilistic. Therefore, we need to consider the atomic compositions when we embed each site’s atomic mass, as +the virtual-crystal approximation (VCA)? is based on. For example, given a binary alloy with composition VpW1–p(0 ≤ p ≤ 1), +24/28 + +25 +Intensity (arb. units) +1.0 +20 +15 +0.5 +10 +Energy (meV) +10K +0.0 +15 +10 +5 +0 +7 +Z +W. +W. +25 +25 +25 +1.0 +Intensity (arb. units) +20 +20 +20 +15 +15 +15 +0.5 +10 +10 +10 +5 +5 +5 +0 +0 +0 +0.0 +25 +25 +25 +20 +20 +20 +15 +15 +15 +10 +10 +10 +5 +5 +5 +0 +0 +0 +-102 -.052 +i-.002.048 +.098 +.403.0453 +.503.0553 +.603 +-.100 -.050 .000 .050 +.100 +k +kthe input alloy encoding vector Z can take the following form +Z = [0,..., pmV,...,(1− p)mW,...,0] +(2) +where the two-hot encoding pmV and (1− p)mW are located at the vector indices corresponding to the atomic numbers of V and +W, respectively, weighted by composition. With this definition of equation (2), the embedded feature can be directly reduced to +pure phase one-hot encoding V (or W) by simply setting p = 1 (or p = 0), and it can be generalized to more complicated alloys +directly. +We demonstrate the power of this approach with the binary alloy of SiGe. We always assume SiGe alloy has eight atoms per +cubic unit cell with a side of 5.466 ˚A, which is the intermediate of the crystalline silicon and germanium?. As the input of +kMVN model, we convert the unit cell into a primitive cell with two atoms. As the ground-truth label, we used the phonon +bands calculated with VCA approximation. +Figure.S22 shows the result of the phonon band prediction of SiGe alloy. Both ground truth (Figure.S22a) and prediction +(Figure.S22b) shows similar shapes of the band structures along the highly-symmetric points. Yellow and blue colors depict the +composition ratios of silicon and germanium, respectively. The lighter atoms generally offer phonons of higher frequencies. +Our result follows the principle, as the alloys give higher phonon frequencies as the compositions of silicon get larger. +Figure S22. Phonon spectra of SiGe alloy with various compositions. Yellow and blue indicate pure Si and Ge, +respectively, and the intermediate colors represent alloys of different composition ratios. We demonstrated phonon prediction +of SipGe1−p with p = 0.00, 0.01, 0.05, 0.10, 0.13, 0.15, 0.17, 0.20, 0.30, 0.40, 0.60, 0.70, 0.80, 0.85, 0.90, 0.96, 0.98, 0.99, +0.995, and 1.00 (from blue to orange). We show phonon along L, Γ, and X points. a. Ground truth. b. Prediction. +We next show the phonon prediction of binary alloys NiCo, NiFe, and a ternary alloy NiFeCo. For each material, we +generated the random compositions 200 times and observed how the phonon prediction went. As the ground-truth label (black +lines in each of Figure S23 ), we used the phonon bands calculated with DFPT approximation. As the input of the DFPT +calculation, we used supercells that contain 64 atoms for NiCo and NiFe and 108 atoms for NiFeCo, respectively. Figure S23 +shows the predicted phonon of the three alloys. The combination of red, blue, and yellow colors represent the compositions Ni, +Co, and Fe, respectively. The spectra were arranged from left to right in the ascending order of the prediction loss: (a) NiCo, (b) +NiFeCo, and (c) NiFe. This tendency can be explained by the atomic masses of the alloy’s components. Ni and Co have almost +the same atomic mass (58.69u and 58.93u), and Fe has a lower value (55.85 u). Compared to NiFe, NiCo and NiFeCo have +smaller differences in the atomic masses of the elements they have, which makes the atomic mass encoding more stable. +25/28 + +b +a +500 +500 +Phonon frequency (cm-1) +400 +400 +300 +300 +200 +200 +100 +100 +0 +0 +X +XFigure S23. Phonon spectra of high entropy alloys with various compositions. Red, blue and yellow indicate pure Ni, +Co, and Fe, respectively, and the intermediate colors represent alloys of different composition ratios. The black lines in each +figure are ground truth computed by DFPT. a. NiCo. b. NiFeCo. c. NiFe. +We have shown that our k-MVN model could handle phonon prediction of binary and ternary alloys. Here we demon- +strate even higher component alloys. Here, we start with binaries, moving to ternaries, then quaternaries, and finally to the +five-component quinaries. Figure.S24 shows the predicted spectra of MoTaNbWV and VWNbTaTi, as well as their lower +component alloys. +In high entropy alloy systems, both force constants and masses have fluctuated. Therefore, we need to average these properties +to represent the disordered states as primitive cells or supercells. In Figure.S24a-d, the number of MoTaNbWV’s components +increased from 2 to 5. We show a similar property for the case of VWNbTaTi, in Figure.S24e-h. Our prediction follows atomic +mass embedding of equation (2), which weights the mass of the components with the configuration ratios. Even without +incorporating the force constant averaging, our model could generate phonon spectra similar to the simulation results with +special quasirandom structures (SQS)?. We show a similar property for the case of VWNbTaTi, in Figure.S24e-h. +VII Γ-PHONON DATABASE +In previous work, the full phonon band structures and derived quantities for 1521 semiconducting inorganic crystals are +presented?. Given the high-quality phonon prediction using the machine-learning approach, particularly the MVN approach for +complex materials, here we present a new phonon database containing the phonon spectra for the entire 146,323 Materials in +Materials Project (MP) as of 2022 computed by the MVN approach. Given the importance of zone-center Γ-phonons, which +are measurable with more accessible equipment like Raman scattering, here we limit the database to Γ-phonons and will leave +the full phonon database for future works. This new Γ-phonon database offers the possibility to analyze the lattice dynamics for +many compounds. It could be used as a useful tool to be compared with Raman scattering results for crystalline materials. The +Γ-phonon database generated with the MVN method is available at https://osf.io/k5utb/. +The whole database is stored as a dictionary, whose keys are the MP ID number. The dictionary values for each material are +also dictionaries, which contain basic information such as the number of atoms per unit cell, chemical formula, and others. We +show the values of each dictionary in Table S2. We provide the test from the mass-spring model on the validity of the results, +which are summarized in Figure S25. The data displays a spread around the hyperbolic fit since the phonon frequencies are the +outcome of the interplay of the whole set of interatomic force constants and the different masses of the elements composing +the material. It can be noticed some trends can be recognized with respect to the masses of the components. Systems with +non-uniform masses (identified by the small ratio mmin/ ¯m), tend to lay on a different hyperbolic curve with respect to more +uniformly weighted systems. +26/28 + +a +b +c +NiCo +NiFeCo +NiFe +350 +400 +350 +350 +300 +300 +300 +250 +250 +250 +200 +200 +200 +150 +150 +150 +100 +100 +100 +50 +50 +50 +0 +0 +0 +X +U(K) +L +X +U(K) +L +X +U(K)Figure S24. Phonon spectra from binary to 5-component high entropy alloys Predicted phonon spectra with increasing +constituent elements: From binaries to 5-component high entropy alloys. The added element for each alloy from left to right is +shown in bold. The plotted phonon frequencies are in [THz]. a. MoTa. b. MoTaNb. c. MoTaNbW. d. MoTaNbWV. e. VW. f. +VWNb. g. VWNbTa. h. VWNbTaTi. +Table S2. The components of the database for each material +Key +Data type +Description +material id +string +MP ID number +nsites +integer +The number of atoms per unit cell +formula +string +Chemical formula +space group +integer +The index of the space group +structure +string +The material structures of the CIF format +spectra +list +The values of Γ-phonon frequencies [cm−1] +27/28 + +b +d +a +Binary +Ternary +C +Quarternary +Quinary +10 +Mo-Ta +10 +Mo-Ta-Nb +10 +Mo-Ta-Nb-W +10 +Mo-Ta-Nb-W-V +78 +8 +8 +8 +HLI +6 +6 +6 +6 +4 +4 +4 +4. +2 +2 +2 +2 +0 +0 +0 +0 +G +H +N +G +G +H +N +N +G +H +N +G +p +G +H +NGPHPN +Wave Vector +WaveVector +Wave Vector +WaveVector +e +f +h +g +10 +V-W +10 +V-W-Nb-Ta +10 +10 +V-W-Nb +V-W-Nb-Ta-Ti +[ZHI] +8 +8 +8 +8 +6 +6 +6 +4 +4 +4 +2 +2 +2 +2 +0 +0 +0 +G +H +N +G +P +H +PN +GHNGPHPN +GH +N +GPHPN +G +NGPHPN +WaveVector +Wave Vector +Wave Vector +WaveVectorFigure S25. The profile of our Γ-phonon database. a. The number of atoms per unit cell in the dataset. The dataset +contains 146323, and the number of atoms per unit cell ranges from 1 to 444. b. Average frequency of the predicted Γ-phonon +¯ω versus average atomic mass ¯m. The colors of the dots represent the magnitude of mmin/ ¯m. The black solid lines represent the +least squares that fit the hyperbolic relation ¯ω = C ¯m−1/2. The constant C is estimated from the fit as 1993. The colors of the +dots represent the ratio mmin/ ¯m. +28/28 + +a +b +mmin/m +1.0 +2000 +12000 +0.8 +10000 +1500 +8000 +(/cm) +0.6 +Count +1000 +6000 +13 +0.4 +4000- +500 +0.2 +2000 +0 +0 +100 +200 +300 +400 +0 +50 +100 +150 +200 +250 +Atoms/cell +m (amu) \ No newline at end of file diff --git a/EtE0T4oBgHgl3EQfQwBZ/content/tmp_files/load_file.txt b/EtE0T4oBgHgl3EQfQwBZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0e81813c77797d3b61c767434982c329541de35 --- /dev/null +++ b/EtE0T4oBgHgl3EQfQwBZ/content/tmp_files/load_file.txt @@ -0,0 +1,1941 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf,len=1940 +page_content='Virtual Node Graph Neural Network for Full Phonon Prediction Ryotaro Okabe1,2,†, Abhijatmedhi Chotrattanapituk1,3,†, Artittaya Boonkird1,4, Nina Andrejevic5, Xiang Fu3, Tommi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Jaakkola3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Qichen Song6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Thanh Nguyen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Nathan Drucker1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Sai Mu8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Bolin Liao9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Yongqiang Cheng10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' and Mingda Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='* 1Quantum Measurement Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 2Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 3Department of Electrical Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 4Department of Nuclear Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 5Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Lemont,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 6Department of Chemistry and Chemical Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 7Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' School of Engineering and Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 8Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' University of South Carolina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Columbia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' South Carolina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 9Department of Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 10Chemical Spectroscopy Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Spectroscopy Section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Neutron Scattering Division Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' TN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA †These authors contributed equally e-mail: mingda@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='edu ABSTRACT The structure-property relationship plays a central role in materials science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Understanding the structure-property relationship in solid-state materials is crucial for structure design with optimized properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The past few years witnessed remarkable progress in correlating structures with properties in crystalline materials, such as machine learning methods and particularly graph neural networks as a natural representation of crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' However, significant challenges remain, including predicting properties with complex unit cells input and material-dependent, variable-length output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here we present the virtual node graph neural network to address the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' By developing three types of virtual node approaches - the vector, matrix, and momentum-dependent matrix virtual nodes, we achieve direct prediction of Γ-phonon spectra and full dispersion only using atomic coordinates as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We validate the phonon bandstructures on various alloy systems, and further build a Γ-phonon database containing over 146,000 materials in the Materials Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Our work provides an avenue for rapid and high-quality prediction of phonon spectra and bandstructures in complex materials, and enables materials design with superior phonon properties for energy applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The virtual node augmentation of graph neural networks also sheds light on designing other functional properties with a new level of flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Introduction The structure-property relationship defines one of the most fundamental questions in materials science16,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The ubiquitous presence of structure-property relationships profoundly influences almost all branches of materials sciences, such as structural materials3, energy harvesting and conversion and energy storage materials5,17,19, catalysts37 and polymers13, and quantum materials15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' However, despite its central importance to materials design, building an informative structure-property relationship can be nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' On the one hand, the number of stable structures grows exponentially with unit cell size22, and the structure design efforts have been largely limited to crystalline solids with relatively small unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' On the other hand, certain material properties are challenging to acquire due to experimental or computational complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In the past few years, data-driven and machine-learning methods play an increasingly important role in materials science and significantly boost the research on building structure-property relationships6,24,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Complex structures such as porous materials1,27, nanoalloys10,36, and grain boundaries34 are becoming more feasible to handle, and properties ranging from mechanical strength to quantum ordering can be learned with increased confidence9,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' One particular powerful approach is the graph neural networks (GNNs)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' By representing atoms as graph nodes and interatomic bonds as graph edges, GNNs arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='02197v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='dis-nn] 5 Jan 2023 provide a natural representation of molecules and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For crystalline solids, crystallographic symmetry offers a further boost on the GNN performance, with a few symmetry-augmented GNNs being proposed8,30,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' A few fundamental challenges still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For one thing, many materials properties are not naturally represented as a weighted aggregation of each atom in real space, such as reciprocal and energy space properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For another thing, the output property length is usually fixed, like the heat capacity4 as a single scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In contrast, many materials’ properties have unique degrees of dimensions, such as the number of electronic and phononic bands2, frequency ranges with optical responses, and the features of magnetic structures like propagation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In this work, we propose Virtual Node Graph Neural Network (VGNN) as a generically applicable approach to augment GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In contrast to symmetry-augmented GNN which focuses on reducing the input data volume, VGNN focuses on handling the output properties with variable or even arbitrary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We study materials’ phonon spectra and dispersion relations, given that phonons bands are challenging to compute or measure with high computational cost and limited experimental resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' By using the phonon spectra as examples, we present three versions of VGNN: the vector virtual nodes (VVN), the matrix virtual nodes (MVN), and the momentum-dependent matrix virtual nodes (k-MVN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' All three VGNN models take atomic structures as input without prior knowledge of interatomic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The VVN is the simplest VGNN that takes in a crystal structure with m atoms and outputs 3m branches Γ-phonon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The MVN is a more involved VGNN that shows higher accuracy for complex materials with slightly higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Finally, the k-MVN is a VGNN that can predict full phonon band structure at arbitrary k points in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To achieve so, the crystal graphs contain "virtual-dynamical matrices", which are matrix structures that resemble phonon dynamical matrices14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Instead of performing direct ab initio calculations on each material, all matrix elements are learned from the neural network optimization process using training data comprised of all other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Our work offers an efficient technique that can compute zone-center phonon energies and full phonon band structures directly from atomic structures in complex materials and enables phonon property optimization within a larger structure design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The prediction methods has enabled us to acquire relevant information of materials such as group velocities, heat capacities, density of states as by-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Meanwhile, the virtual node structures also shed light on future flexible GNN design, that to put intermediate crucial quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' dynamical matrix) as key learning parameters without having to put target properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' phonon band structures) as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Results Virtual node augmentation for graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure 1 gives an overview of the VGNN method as a generic approach to augment GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For a crystal with m atoms per unit cell (Figure 1a), a typical GNN model converts the crystal into a crystal graph, where each graph node represents an atom, and each graph edge represents the interatomic bonding as shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The node features associated with each atomic node (gray arrays in Figure 1b) are updated by neighborhood nodes and edges connecting the nodes (gray arrows in Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After iterative layers of graph convolutions, m final-layer node features are obtained that represent the atomic features from each of the m atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The final graph output can be obtained by aggregating the final-layer node features into one fixed-sized output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figures 1c,d describe the general idea of VGNN that endows a GNN with greater flexibility for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' On top of the conventional, real-node GNN, virtual atoms are added into crystal (yellow nodes in Figure 1c), which become the virtual nodes in the corresponding GNN (yellow nodes in Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' As Figure 1d illustrates, just like the bi-directional message passing between real atomic nodes (double-arrow gray lines), the message passing (double-arrow yellow lines) between virtual nodes is also bi-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' On the other hand, to preserve the structure of the conventional GNN, the messages from real nodes to virtual (single-arrow gray-to-yellow gradient lines) are uni-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Given the flexibility of the choice of the virtual nodes, a VGNN gains huge flexibility to predict materials-dependent outputs with arbitrary lengths and in spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We will introduce three VGNN methods for phonon prediction with increased levels of predictive power and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Vector virtual nodes for Γ-phonon prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' As illustrated in Figure 1, VGNN makes it possible to adjust output dimension based on input information with flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We first introduce the vector virtual node (VVN) method, which is the simplest approach to acquire 3m phonon branches when inputting a crystal with m atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (See Methods for more detail) Figure 2 shows the VVN approach to predict Γ-phonon spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Since the virtual nodes do not pass information to real nodes, there is additional flexibility in choosing the position of the virtual node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Without loss of generality, we assign the position of the virtual nodes evenly spaced along the diagonal line of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The crystal graph is constructed with virtual and real nodes (Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After updating node features in each convolution layer, the feature vectors pass a linear layer so that virtual node features Vi,i ∈ [1,3m] are converted to 3m scalars, which represent the predicted Γ-phonon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Throughout this work, the GNN part is implemented through the Euclidean neural networks8 that are aware of the crystallographic symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Data preparation, neural network architectures, and optimizations are described in Supplementary Information I-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The main results using the VVN for Γ-phonon prediction are shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The three-row spectral comparison plots 2/11 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Overview of virtual node graph neural network (VGNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Atomic structure of a crystalline material with m atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' A GNN converts the atomic structures into a crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After layers of graph convolutions (omitted for simplicity), the final node features are aggregated into a single fixed-sized output feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' A flexible of n virtual atoms are added into the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After forming the crystal graph with both real and virtual nodes, the flexibility of virtual nodes enables the choices of output not necessarily from real-node aggregation but can have variable length and in different spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' are randomly selected samples from the test set within each error tertile (top-to-bottom rows are top-to-bottom performance tertiles, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The first four columns are taken from the same database as the training set from high-quality density- functional perturbation theory (DFPT) calculations25, and the fifth column contains additional test examples with much larger unit cells from a frozen phonon database31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' It is worthwhile mentioning that although for very complicated materials, the VNN-predicted phonons tend to have higher frequencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', third row, fourth column of Ba12I36Y4), the resulted phonon density-of-states over the entire Brillouin zone can still be largely comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' However, the prediction loss becomes larger and distributed broader as the input materials are more complicated (Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' From the correlation plot of predicted and ground-truth phonon frequencies (Figure 2d), most data points are along the diagonal line, indicating good prediction between VNN prediction and ground-truth from DFPT calculations with the number of atoms per unit cell m ≤ 24 (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For complex materials, the correlation performance could be degraded (orange dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' More test results are shown in Supplementary Information IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Matrix virtual nodes for Γ-phonon with enhanced performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In this section, we introduce another type of virtual nodes approach, the matrix virtual nodes (MVN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The MVN approach performs better Γ-phonon prediction than VVN, especially for complex materials, with a slightly higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Moreover, the structure of MVN lays the groundwork for the full phonon band structures to be discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In MVN, m copies of virtual crystals are generated for material with m atoms per unit cell, and each copy contains m virtual nodes that share the same crystal structure as the real crystal (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This results in a total of m2 virtual nodes Vi j, i, j ∈ [1,m] with more involved node connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (See Methods for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' With this graph construction scheme, after the neural network training, the virtual nodes Vij would capture the essence of the connection between Ri, and R j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Hence, after the message passes in each convolutional layer, each virtual node feature 3/11 realfeature build crystal c graph .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' R2 1 (fixed) aggregate feature R a Add n virtual 上 atoms build crystal virtual feature graph n (variable) features real-to-real real-to-virtual virtual-to-virtual message passing message passing message passingFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The vector virtual node (VVN) method to predict Γ-point phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Schematic of VVN model construction and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For material with m atoms per unit cell, 3m Virtual nodes are augmented along the diagonal vector ⃗v =⃗a+⃗b+⃗c of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We embedded the components of the crystal when building the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For instance, atomic numbers of the mth real atom (ARm) and that of the 3mth virtual atom (AV3m) are embedded as the attributes of each nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The atomic mass of mth real atom (ZRm) is set as the initial feature of that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The relative position of the node V1 with respect to Rm is⃗rV1Rm, which is used to embed the edge attribute between the two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The model predicts Γ-phonon spectra by sorting the scalar output features from virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Spectral prediction samples in the test set within each error tertile compared with ground truth (black): Test from the same database as the training set (blue), and a different database containing complex materials (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' c-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Evaluation of the test accuracy through the distribution of loss function and correlation plot between ground-truth and predicted average phonon frequencies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The heavy distribution at low loss regime of the distribution plot and the agreement along the diagonal line of the correlation plot for the test set (blue) indicates a high-quality phonon prediction at least for relatively simple materials with the number of atoms per unit cell m ≤ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The loss becomes higher with reduced performance for complex materials (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' is further converted into a three-by-three matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Each of Vij is assembled to form (i, j) block of a supermatrix ˜D of shape (3m,3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Given the structural similarity of this matrix and the dynamical matrix expressed in Equation (2) with⃗k = 0, we predict Γ-point phonon energies by solving for 3m eigenvalues of the matrix ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' It is still worthwhile mentioning that although the matrix shares a similar feature with the dynamical matrix, the matrix elements are learned from neural network training and are not necessarily the matrix elements from the real dynamical matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' An intuitive comparison is that the edge of GNN does not necessarily reflect true chemical bonding, but is more like an atomic neighbor connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The predicted phonons using MVN are summarized in Figure 3b, which shares the same structure with Figure 2b as error tertile plots from the high-quality DFPT database (blue) and database for complex materials (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MVN shows comparable performance with VVN for simple materials (blue curves in Figure 2c and Figure 3c), but shows significant performance improvement for complex materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The prediction loss distribution of MVN shows a heavier distribution toward a lower loss regime compared to VVN (orange curves in Figure 2c and Figure 3c), and the average phonon frequencies in the correlation plot align better toward ground truth (orange dots in Figure 2d and Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' More results and correlation plots are shown in Supplementary Information IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Momentum-dependent matrix virtual nodes for predicting full phonon band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The structure of MVN inspires 4/11 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='SizZr2 F202Y2 Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='KRh GaO,P Br1:OPb16Tl2 b 600 1000 600 1000 a 400 750 400- ZRm 400 500 000 500 200 200 200 250 ARm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0 Ag2Cl4CS2 Cl-F2Sr2 FePaSi4 Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Pt Lu2O3s Ta14 600- 1000 400- 200 400 300 400- 200 500 100 200 200 R: 100 XV3m R1 0: 0 0 0 口 CMgN2 Li,N HgK,O2 BraLizRb2 Ba12l36Y4 Add 3m 600- virtual atoms build crystal graph 2000 Avgm 750 300 200 1500 400 500 200 000 1000 100 TViRm 250 200 100 500 0 TV, Rm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0 C d 3500 FePSi4 40 3000 Probability Density 30 2500 sort Predicted w 2000 CMgN2 20 1500 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='SizZr2 prediction 1000 10 Lu2O3sTa14 Br1:O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Pb1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Tl2 500 Ba12l36Y4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='000 1000 2000 3000 Loss True w [cm-1]Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The matrix virtual node (MVN) method to predict Γ-point phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Augment m2 virtual nodes as m sets of virtual crystals (left) and the message passing scheme and post-processing of virtual node features (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The legends are the same as Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In contrast to VVN, where each node Vj is a scalar, here, each node Vij is a 3×3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The phonon spectra in MVN are obtained by solving the eigenproblems instead of direct output, as done in VVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Selected test examples within each error tertile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Tests from the same dataset as the training set and additional tests containing complex materials are predicted in blue and orange, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Comparison of prediction loss distribution with several examples of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The correlation plots of average phonon frequencies with the graph y = x as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Better performance for MVN is achieved than VVN for complex materials (orange color), which can be seen from both the loss distribution and the average phonon frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' us to take one step further and construct full momentum-dependent virtual dynamical matrices by taking into account the unit cell translation, termed momentum-dependent matrix virtual nodes (k-MVN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We construct virtual-dynamical matrices following Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In contrast to the MVN, which focuses on Γ-point phonons by taking⃗k = 0, here in k-MVN, we include the phase factor ei⃗k·⃗T when defining the virtual dynamical matrices, where ⃗T is the relative unit-cell translation of a neighboring unit cell origin relative to the chosen reference unit cell ⃗T0 (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' If a total number of t neighboring unit cells are included, each with translation ⃗Th,h ∈ [0,t −1] (reference cell included), then a total t copies of MVN-type virtual nodes matrices will be generated, with a total number of tm2 virtual nodes V h ij,h ∈ [0,t − 1],i, j ∈ [1,m] in k-MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To obtain the phonon band structure, each set of virtual nodes at a given ⃗Th needs to multiply by the phase factor ei⃗k·⃗Th, and all virtual nodes at each ⃗Th are summed in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Thanks to the graph connectivity within the cutoff radius (see Methods), only a small number of t is needed as long as crystal graph connectivity can be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In practice, t is materials dependent, and t = 27 (nearest neighbor unit cells) is sufficient for many materials and does not need to go beyond t = 125 (next-nearest neighbor unit cells) in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Intuitively, such a supercell approach resembles the ab initio band structure calculations with frozen phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To facilitate the training, phonons from selected high-symmetry points are included in the training data, without the need to use full phonon energies in the entire Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This significantly facilitates the training process while maintaining accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' More details are discussed in Methods and Supplementary Information V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure 4b shows the prediction results of phonon band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here 12 materials are selected from the same dataset for training (blue color) and the additional dataset for complex materials (orange color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Despite the complexity of a generic phonon band structure, the k-MVN model could predict the positions and the shapes of the phonon bands, such as gaps between different optical branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The dispersion relations of the acoustic phonons are also well generated around the Γ-points on 5/11 Ag2Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',Se4 CINSr2 Ba2Se6 Ga2HgTe4 Br18O:Pb16Tl2 b 400 400 200 三 200 400 a 200 200 100- 100 - 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Sr2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Se6 As2SrZn2 Al40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cd:O32Rb16Sis 200 750- 200 750- 500 - 100 - 500- 100- 100 250- 250 3m 0 0 0- 0 0 Add m virtual B40O84Pr16 La2N2 NNaO2 Ca2Cl4 Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2Cu4 copies 00 build crystal graph 750 200- 300- 1500 1000- 500 - 200 1000 100 500 - 100 250- 500 3m 0- 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C 100 3500 CdgO32Rb16Sig Q +0000 80 jth copy Density ←B4oOg4Pr16 60, Predicted w [ Ag2Al2Se4 2000 eigenvalues Probability 40 1500 ZONN 1000 Cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Sr2 20 500 Br1:O:Pb16Tl2 prediction 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010 2000 3000 1000 Loss True w [cm-1]the left three columns, even though we do not enforce that acoustic Γ-phonons have to be gapless with zero-energy known as acoustic sum rule28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This may enable the prediction of crystal stability for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' While there are risks that prediction performance could be degraded for the phonon bands of at higher frequencies, most of the predicted phonons follow the references, including the complex materials with more than 40 atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The momentum-dependent matrix virtual nodes (k-MVN) to predict full phonon band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (Top) Augment m2 virtual nodes for each translation vector ⃗T, and with a total t neighboring unit cells, a total tm2 virtual nodes are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (Bottom) By multiplying a phase factor by each translated unit cell, a full virtual dynamical matrix can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Selected examples in the test set within each error tertile, for the high-quality DFPT database (blue) and additional complex materials test (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Γ-point positions are labeled for each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Discussion We demonstrate the prediction of phonons directly from the materials’ atomic coordinates, using three different types of virtual nodes – the VVN, the MVN, and the k-MVN – to augment the symmetry-aware Euclidean neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The comparison between the three virtual node approaches is summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VVN directly acquires the phonon spectra from the virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The assignment of 3m virtual nodes ensures that the output phonon band number is always 3m for a crystal with a primitive unit cell containing m atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In MVN, instead of computing phonon energies directly, a virtual dynamical matrix (VDM) is constructed first, from which the phonon energies are solved as an eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This step is crucial to gain robustness for complex materials prediction since intermediate quantities like force constants and dynamical matrices are con- sidered more “fundamental” than final phonon energies to reflect the interatomic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The k-MVN goes one step further, using the unit-cell translations to generate the momentum dependence that could be used to obtain the full phonon band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Today, the ab initio calculations like frozen-phonon and DFPT remain the most accurate methods for phonon calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Even so, since the VGNN-based phonon calculation skips the direct calculation of the material-by-material dynamical matrix, it shows significantly faster computation speed while maintaining reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Additional tests on SiGe alloys, FeCoNi alloys, and other high-energy alloys are performed, which agree well with existing literature (Supplementary Information VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Finally, by using MVN, we build a database containing the Γ-phonon spectra for over 140,000 materials listed in Materials 6/11 a b Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Se In16Rb16Sb24 CaMg2Sb2 Be2Li2Sb2 175 250 500 250 150 cell 200 400 200 150 300 100 150 75 100 100 200 50 50 50 100 25 T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' := 0 cell Cs2FelnNa Ag204Y2 BrCsO3 Hg4K:P:Se24 500 500 800 F009 400 400 600 300 400 400 200 200 200 200 100 100 ild crystal graph As4OS2 Ge:O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='N2O6 AugNd16O36 1200 700 300 prediction 1000 1000 600 800 - 500 - 800 200 eigenvalues 600 400 - 600 300 400 DT D(k) 100 400 200 200 200 100 DTi(k SToProject (Supplementary Data and Supplementary Information VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' It took an eight-GPU system less than five hours to obtain all results, even though some materials contain over 400 atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Such efficiency enables the material design, searching, and optimization in a much larger design space, including alloys, interfaces, and even amorphous solids, with superior engineered phonon properties for thermal storage, energy conversion and harvesting, and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In parallel, by taking advantage of the flexibility endowed by virtual nodes, other properties that are challenging to predict for a conventional GNN can be predicted similarly, such as electronic band structures and tight-binding and k · p effective Hamiltonian with a variable number of bands, optical properties like flexible optical absorption peaks as in the Lorentz oscillator model, and magnetic properties such as the number of propagation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Comparison of how the virtual nodes contribute to phonon prediction in terms of physics and computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here m, Ntrain, t, n indicate the number of atoms per unit cell, the average of that in training data, the number of the unit cell counts, and an arbitrary, not large number respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VVN MVN k-MVN Force constants Reflected in VDM Dynamical matrices VDM VDM Phonon data Virtual nodes Eigenvalues Eigenvalues Run time O(m2) O(m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='37) O(t ×m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='37) Storage O(m) O(m2) O(t ×m2) Generalization to larger systems False True True Reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Altintas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Altundal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Keskin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Yildirim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Machine learning meets with metal organic frameworks for gas storage and separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Journal of Chemical Information and Modeling, 61(5):2131–2146, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' S.' metadata={'source': 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+page_content=' ACS Energy Letters, 6(8):2838–2843, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Gong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Xie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Gorai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Nielsch, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Grossman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Charting lattice thermal conductivity for inorganic crystals and discovering rare earth chalcogenides for thermoelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Energy & Environmental Science, 14(6):3559–3566, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Methods Phonon data preparation We trained all of our models against an ab initio DFPT computational database for phonon dispersion in harmonic model25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The data set contains material structures (the same as the primitive structure obtained from the Material Project12), second-order derivatives of energies with respect to atomic perturbations for regular points inside the irreducible zone, and phonon dispersion along highly symmetric paths of 1,521 crystalline inorganic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' These materials have 2 to 40 atoms per unit cell, with an average of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For this work, we only used the highly symmetric path phonon dispersion as our training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The dispersion is between wave vectors⃗k in the fractional reciprocal unit and response spectra in cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' All models randomly split the data into 90% training (1,365 materials), and 10% testing (156 materials) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Furthermore, we trained our models with a 5-fold cross-validation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We also got phonon dispersion of complex (more number of atoms per unit cell) materials from Atsushi Togo’s phonon database31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We used seekpath11,33 module to get the highly symmetric path of each material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Then, we fed it alongside POSCAR, FORCE_SET, and phonopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='config files from the database to Phonopy32’s python command to calculate the phonon dispersion along such path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To quality control the data, we selected materials whose lowest Γ-phonon band is higher than −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='07 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We also filtered the material to get only the ones with more than 40 atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Finally, we randomly selected 156 (the same as the number of data in the testing set for ease of comparison) out of 505 filtered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We used them as our complex material data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Computation environments We coded the models in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='13 and trained them on our GPU cluster with CUDA version 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To facilitate the model implementation, and training, we used some important python modules: Pymatgen23 and ase20 for handling material structure files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='CIF), PyTorch26 for managing model training framework, e3nn8 for implementing our neural network models in the form that is equivariant for Euclidean group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Virtual node graph neural network (VGNN) We have developed a scheme for a graph neural network (GNN) for it to be able to have variable output dimensions depending on the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For ease of understanding, we will explain the method with our work on phonon prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Considering a material with m atoms per unit cell, we add n additional virtual atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We can adjust the number n depending on the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Using both real and virtual atoms, we convert the crystal structures into periodic graphs with m real nodes for the actual atoms and n virtual nodes for the added virtual atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Then, we connect nodes with edges indicating the message-passing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To preserve the structural information of the materials and limit the computational cost, we apply the following rules for connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' First, if the distance between the two real nodes is within a specified cutoff radius rmax, the real nodes are connected through bi-directed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We also set up an edge between a real node and a virtual node according to the model description, but this edge is directed from real to virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Lastly, we embed the information of radial distance vector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',⃗rab from atom b to a, in the form of radial basis functions and spherical harmonics on the corresponding edge as edge attributes, which represent the distance and the direction of⃗rab respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Since each node represents an atom in the unit cell, we embedded the atomic numbers A information as node attributes A by passing one-hot representation vectors of length 118 through an embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' As for the model’s input, we embedded the atomic masses Z information as input node features Z by passing the product of atomic mass and one-hot representation of atomic number through an embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The constructed graph is then passed through the model message passing that operates on the features with multiple convolutions and gated activation layers18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After the final layer, which consists of only a convolution (no gated activation), each 9/11 of the n virtual node features is collected, and passed through the post-processing block, which output the 3m predicted phonon branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The post-processing block is different and will be explained in detail in the subsequent section of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The model is optimized by minimizing the mean squared error (MSE) loss function between the phonon of the training data set and the one predicted by the model after normalizing them by the maximum phonon frequency of each material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The full network structure is provided in the supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Vector virtual node method (VVN) VVN is a VGNN we designed for learning to predict Γ-phonon spectra from material structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Since, for a material with m atoms per unit cell, there are 3m phonon bands, one sensible choice of adding virtual atoms is to add 3m virtual atoms each outputs the prediction of one of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Hence, when there are m atoms in the unit cell of crystalline material, we assign the position⃗rVi of the virtual nodes Vi,i ∈ [1,3m] following equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We can set the atomic species of the virtual node as anything, and we use Fe after optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ⃗rVi = i−1 3m (⃗a+⃗b+⃗c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (1) Here⃗a,⃗b,⃗c indicates the unit cell vector of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In other words, 3m virtual atoms are placed along the diagonal line from (0,0,0) to⃗a+⃗b+⃗c with equal spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' By keeping the distances between the virtual nodes in the real space, it is possible to give position dependencies to the feature updating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In that sense, equation (1) can consistently keep virtual nodes away from each other and enables us to use the virtual 3m virtual nodes as the output nodes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To get information from the whole structure, each of the 3m virtual nodes is connected to all of the real nodes via directed edges from real to virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After each convolution layer, the virtual node features are passed to a linear layer, converted to a scalar output, and sorted based on their magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The outputted 3m scalars represent the predicted Γ-phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Matrix virtual node method (MVN) MVN is a VGNN we designed with the influence of the dynamic matrix representation of a periodic harmonic system for learning to predict Γ-phonon spectra from material structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Given the momentum vector⃗k, the dynamical matrix element ˜Dij(⃗k), which is a three-by-three matrix representing 3D harmonic interaction between atom Ri and R j, can be written as the Fourier transform of the force constant matrix Φαβ ij following equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here, ZRi is Ri atom’s atomic mass, and ⃗Tα is the αth unit cell position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Note that, for each k-vector, the system has 3m degrees of freedom and frequencies where m is the number of atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We can get the phonon dispersion relations ω(⃗k) by solving eigenvalues ω2(⃗k) of ˜D(⃗k), which is a matrix with shape (3m,3m) that composed of m2 blocks of ˜Dij(⃗k) for i, j ∈ [1,m], ˜Di j(⃗k) = ∑ α,β Φαβ i j �ZRiZR j ei⃗k·(⃗Tα−⃗Tβ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (2) In the MVN method, we generate a matrix that could work like a dynamical matrix as is written in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here, we focus on the prediction of Γ-phonon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ⃗k =⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' So, the contributions of the same atom pair, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', Ri, and R j from every unit cell separation ⃗Tα −⃗Tβ are summed without the⃗k-dependent exponential phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Hence, the model needs to predict a matrix with shape (3m,3m) representing such summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In order to do that, while preserving the relation of each matrix element, we generate m virtual crystals Cj, j ∈ [1,m] each of which has m virtual nodes Vij,i ∈ [1,m] of the same atomic species and at the same positions as the real atoms Ri,i ∈ [1,m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here, a virtual node Vij represents the interaction term from a real node Rj to another real node Ri by adding a directed edge from R j to Vij whenever there is an edge connecting Rj to Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After each convolution layer, the virtual node features are passed to a linear layer and converted to complex-valued output vectors with length 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For each output feature, we reshape the output features into three-by-three matrices and arrange them such that Vij’s matrix is the (i, j) block of ˜D supermatrix with shape (3m, 3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Finally, we solve ˜D for its 3m eigenvalues, which work as the Γ-phonon prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Momentum-dependent matrix virtual node method (k-MVN) k-MVN is a generalization of MVN model with non-zero⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Unlike the MVN case, the k-MVN model needs to predict matrices representing interactions between atoms from a unit cell, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', ⃗Tβ, to the different unit cell, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', ⃗Tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Since the phase factor depends only on the difference in unit cell positions, we can redefine ⃗T to be such a difference and simplify equation (2) into ˜Di j(⃗k) = ∑ ⃗T Φ⃗T i j �ZRiZRj ei⃗k·⃗T := ∑ ⃗T D⃗T i jei⃗k·⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (3) 10/11 With this simplification, for each ⃗T, we generate m virtual crystal C⃗T j∈[1,m] the same way as in MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' However, in this case, V⃗T i j represents the interaction term from a real node Rj to another real node Ri that is in the unit cell with unit cell position ⃗T with respect to R j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In other words, we add a directed edge from R j to V⃗T ij whenever there is an edge connecting Rj to Ri and that edge represent⃗ri −⃗r j =⃗r′ i +⃗T −⃗r′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here,⃗r′ is the atomic position relative to its unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Since GNN only considers edges with interatomic distance less than rmax, the model can generate, with this scheme, a non-zero matrix for only a finite number of ⃗T that satisfy min i, j∈[1,m]|⃗r′ i +⃗T −⃗r′ j| ≤ rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (4) Hence, before the virtual crystal generations, the model also iterates through atom pairs to find all viable ⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Similar to the MVN model, we convert virtual node features into three-by-three matrices and merge them into a matrix with shape (3m,3m) representing ˜D⃗T for each ⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Finally, we weight sum these matrices with their phase factor to get ˜D and solve for its 3m eigenvalues as phonon spectrum at wave vector⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Acknowledgements RO and AC contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' RO, AC, AB, and ML thank M Geiger, S Fang, T Smidt, and K Persson for helpful discussions, and acknowledge the support from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Department of Energy (DOE), Office of Science (SC), Basic Energy Sciences (BES), Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' DE-SC0021940, and National Science Foundation (NSF) Designing Materials to Revolutionize and Engineer our Future (DMREF) Program with Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' DMR-2118448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' BL acknowledges the support of NSF DMREF with Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' DMR-2118523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' TN, ND, and ML are partially supported by DOE BES Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' DE-SC0020148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' TN acknowledges support from Mathworks Fellowship and Sow-Hsin Chen Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ML acknowledges the support from the Class of 1947 Career Development Chair and discussions with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Yip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Competing interests The authors declare no competing interests Data Availability Statement The data that support the findings of this study are openly available in GitHub at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='com/RyotaroOKabe/ phonon_prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The Γ-phonon database generated with the MVN method is available at https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='io/ k5utb/ 11/11 Virtual Node Graph Neural Network for Full Phonon Prediction: Supplementary Information Ryotaro Okabe1,2,†, Abhijatmedhi Chotrattanapituk1,3,†, Artittaya Boonkird1,4, Nina Andrejevic5, Xiang Fu3, Tommi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Jaakkola3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Qichen Song6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Thanh Nguyen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Nathan Drucker1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Sai Mu8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Bolin Liao9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Yongqiang Cheng10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' and Mingda Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='* 1Quantum Measurement Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 2Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 3Department of Electrical Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA 10Chemical Spectroscopy Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Spectroscopy Section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Neutron Scattering Division Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' TN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' USA †These authors contributed equally e-mail: mingda@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='edu Contents I DATA PREPARATION 1 II VIRTUAL NODE GRAPH NEURAL NETWORKS 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='1 Euclidean Neural Networks and Real Node Graph Convolutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 24 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 Phonon Prediction in Alloy Systems .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 24 VII Γ-PHONON DATABASE 26 I DATA PREPARATION The data set containing full phonon bands of 1521 semiconducting inorganic materials are calculated from density functional perturbation theory (DFPT)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', which is the main dataset for training (“Main Database” for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The phonon energy values at Γ-points (for vector virtual nodes, VVN and matrix virtual nodes, MVN model), and other high symmetric points (k-MVN model) were extracted and randomly split for training (90%) and testing set (10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' High symmetric points are different for 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='02197v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='dis-nn] 5 Jan 2023 each material depending on its structure, and the reduced fractional coordinates are implemented for both real and reciprocal spaces for a primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' A consistency check has been performed for all data, ensuring the intended k-points in the fractional unit match the desired k-points in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Another set of data for testing the model was taken from the Phonon database by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Atsuhi Togo at Kyoto University (“Togo Database” for short)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', which contains the phonons of more complex materials, but meanwhile contain more phonons with imaginary phonon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We use the Main Database for training, given the more stringent convergence criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In addition, we randomly selected 156 materials in the Togo Database with the lowest Γ phonons greater than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='07 cm−1, and have at least 40 atoms per unit cell for testing the model trained from the Main Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The 156 materials from the Togo Database match the number as the testing set from the Main Database (10% of the 1521 materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Most elements appear in both the test set of the Main Database and the additional test in the Togo Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The distribution of the number of atoms per unit cell (m-value in the main text) for the Main Database and the Togo Database are shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The profiles of these data sets with respect to the elements are shown in Figures S2 - S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The distribution of atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The number of atoms per unit cell in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Training and test data set from the Main Database b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Additional test data set of complex materials from the Togo Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The total 1521 materials from the Main database contain 2 to 40 atoms per unit cell with an average of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4 atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The randomly selected 156 complex materials with good ground-truth quality in the Togo Database with 42 to 174 atoms per unit cell (an average of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='1 atoms per unit cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' II VIRTUAL NODE GRAPH NEURAL NETWORKS II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='1 Euclidean Neural Networks and Real Node Graph Convolutions Our approaches were developed within the framework of a symmetry-aware graph convolutional neural network, the Euclidean neural networks (E3NN)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. Figure S4 illustrates the overall architecture of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The model takes atomic mass and position of atoms in a unit cell as an input, the same as the previous report?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. This information is converted to a periodic graph with nodes representing atoms and edges controlling the message passing between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This input is then passed through a series of convolution layers separated by nonlinear layers, which introduce the complexity to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The convolution layer computes the tensor product of input features, and the convolution kernel is defined as: f ′ i = 1 √z ∑ j∈∂(i) f j ⊗(h(||rij||))Y(rij/||rij||) (1) where f ′ i is the output feature of atom i, and ∂(i) is the set of neighbouring atoms which rij ≤ rmax where rij is the relative position from atom j to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We sum the tensor product of the features of atom j and the convolution kernel consisting of the learned radial function (h(||ri j||)) and the spherical harmonics Y(rij/||rij||).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The normalizing factor z is the coordination number, aka the number of neighboring atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' After passing through the last convolution layer, the output features are processed through the screening virtual node and processing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The mean square error (MSE) loss is calculated and used for backpropagation to optimize the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 2/28 a 250 25 200 20 S counts count 150 15 100 10 50 5 510 20 25 40 40 60 80 100 120 140160 180 The number of atoms per unit cel The number of atoms per unit cellFigure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Training and testing data by elements in the Main Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The number of appearances by each chemical element in the training (green) and the testing (blue) data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 Virtual Nodes Augmentation and Convolutions With the advantages of parameter sharing of graph convolution and symmetry awareness of Euclidean neural networks, the augmentation of the virtual node must satisfy the basic requirements for the model to still perform correct convolution that is equivariant for Euclidean group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Virtual nodes must be embedded with the same dimensions of node attributes, and input features as the real nodes and every edge connecting virtual nodes to any other nodes must be embedded with the same dimensions of edge attributes as the edges connecting real nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In our work on phonon prediction, we decided to augment virtual nodes by adding virtual atoms that represent virtual nodes into the crystal structure and constructing the graph in the same way as when there are no virtual atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Because the graph construction treats virtual atoms and real atoms equally, the results must already satisfy all requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Although this augmentation method limits the freedom of virtual nodes to be constructed from some virtual atoms, there are still plenty of degrees of freedom for us to engineer the model construction: numbers, types, and positions of added virtual atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In fact, depending on the model, the virtual atoms do not need to be actual elements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' virtual atoms with an atomic number of 4, but with an atomic mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Another degree of freedom that the virtual node method allows is node connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' However, since we want the model to preserve the connection structure and message passing between real nodes, we decided to restrict the edges between real and virtual nodes to be directed from a real to a virtual node only, while the connections among real nodes are the same as previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Hence, with the restriction on the direction of message passing between real and virtual nodes, we can design our model with any remaining connection combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' To sum up, all of our models (VVN, MVN, and k-MVN) augment virtual nodes by adding a certain number of virtual atoms of certain types at certain locations in the original crystal cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Then, they build a graph by treating virtual atoms as real atoms to keep the functionality of convolution and symmetry awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Finally, they include all edges connecting real nodes, include only some directed edges from real to virtual nodes, and include some edges connecting virtual nodes depending on the design 3/28 400 350 300 250 200 150 100 50 H Li Be B C NOF NaMg AI Si PS CI K Ca Sc Ti Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br 400 350 300 250 200 150 100 50 Rb SrYZr Nb Mo Tc Ru Rh Pd Ag Cd In Sn SbTe Cs Ba La Hf Os Pt Au Hg TI Pb BiFigure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Additional testing data by elements in the Togo Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The number of appearances by each chemical element in testing data of complex materials is shown among the 156 randomly selected materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='3 Neural Network Architectures The generic neural network architectures is shown in Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 4/28 40 Counts 30 20 10 0 LiBeBCNO F Na Mg Al SiPS Cl Cr Fe Ni Cu Zn Ga Ge As Se H KCa Sc V 50 40 Counts 30 20 10 0 Zr Nb Ru Rh Pd Ag Cd In Sn Sb Te Br Rb SräYä Cs Ba La Pt Au Hg TI Pb Bi Hf Ta W ReFigure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Overall architecture of the equivariant neural network and convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The model consists of n layers of convolution layer separated by non-linear layers, screen virtual node layer, and post-processing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 5/28 Z A (m+n)×1 (m+n) ×1 One-hot One-hot embed onehot(Z) (m +n) x din onehot(A) (m+ n) Xdin Irl gn(rmax) × drad Embedding Embedding gn(rmax) × Imax?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (o) Z (m+n) × dem Convolution nconv × Gated block (u)Z (m+n) ×mul Convolution (m+ n) Xmul Screen n Virtual Nodes n Xmul Post Process n X dout WpredIII NEURAL NETWORK TRAINING AND OPTIMIZATION We optimize the values of the training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The set of parameters that gives the best results for VVN, MVN, k-MVN are shown in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The parameter setting of VVN Hyperparameter VVN MVN k-MVN Maximum cutoff radius (rmax[ ˚A]) 4 4 4 Multiplicity of irreducible representation (mul) 16 4 4 Number of pointwise convolution layer (nconv) 2 3 2 Number of basis for radial filters (nrad) 10 10 10 Maximum l of spherical harmonics (lmax) 2 2 2 Length of embedding feature vector (dimem) 16 32 32 Length of output feature vector (dimout) 1 18 18 AdamW optimizer learning rate (5·10−3)×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='96k (5·10−3)×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='96k (5·10−3)×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='96k AdamW optimizer weight decay coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='05 IV PERFORMANCE on Γ-PHONON PREDICTIONS IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='1 Additional Γ-Phonon Predictions We present additional Γ-phonon predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S5-S10 shows the direct prediction and correlation plots of training, testing data from the Main Database, and testing data from the Togo Database using the VVN model, MVN, and k-MVN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 6/28 Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VVN model: Direct prediction results within the training set of the Main Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Black lines indicate the Γ-phonon from DFPT calculation (ground truth) in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The colored (green, yellow, red) lines represent predicted Γ-phonon in 1st, 2nd, and 3rd tertiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (Left) Loss distribution shows that it is heavily peaked in the 1st and 2nd tertiles with lower error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The correlation plots (prediction Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ground truth) of all Γ-phonon within the training set in each tertile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 7/28 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') FaK,Zn Ag20,Sr2 Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Ca2 Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='RbzTi Cs2F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' O,RbSc CozLa2Ne Ba2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Se2 H,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='P2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='050 2000 600 750 300 3000 400 500 200 2000- 1000 1000 100 200 250 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0 0 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' CNNaO MgO BaO2 Cs202 Cs2Ga,S4 SSr 400 300- 600- 2000- 750 200- 400 500 500 200 1000 200 250 100- 三 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') 0- 0 0: 0 2500 2000 1: 3000 3000 1500 2000 2000 1000 1000 1000 t 500 0: 0 1000 2000 3000 500 1000 1500 2000 2500 2000 3000 1000Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VVN model: Direct prediction results within the testing set of the Main Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The predictions of Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd, and 3rd tertiles (green, yellow, and red) in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The loss distribution on the left shows a peak in the 1st and 2nd tertiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The correlation plots (prediction Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ground truth) of all Γ-phonon within the test set in each tertile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 8/28 Li,O,Sb Cal.' 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1st, 2nd and 3rd tertiles(green, yellow, and red) in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The loss distribution on the left shows a peak in the 1st and 2nd tertiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The correlation plots (prediction Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ground truth) of all Γ-phonon within the test set in each tertile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 11/28 a.' metadata={'source': 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+page_content='Sbz As2La2O,Zn2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='MgzRu Baals K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Ses 2000 200 400 100 200 400 100 1000 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010 50 200 100 0 0 0 Be,LizP2 BazPd,S4 CaP,Znz LizNbOs AlO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' IgSra 600 150 300 750 750 100 200 400 200 500 500 100 200 250 250 K,Li,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Znz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Si4 CuGaO2 CaCla K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='O4 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='020 600 300 600 600 750- 400 200 500 400 400 500 200 100 200 250 200 0 0 0- CuGal4 NazO Rb,Te AuBiSrz B1sCr4Y4 200 400 100- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='050 600 1000 100 400 100 200 50 500 50 200 0 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') 1500 2000 1250 3000 1500 : 1000 2000 750 1000 500 1000 500 250 Fo 0 500 1000 1500 0 500 1000 1500 2000 0 1000 2000 3000Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MVN model: Direct prediction results within test set of the Togo Database using MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The predictions of Γ-phonon comparing with DFPT calculations (black) in 1st, 2nd and 3rd tertiles(green, yellow, and red) in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The loss distribution on the left shows a peak in the 1st and 2nd tertiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The correlation plots (prediction Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' ground truth) of all Γ-phonon within the test set in each tertile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 12/28 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') FsOzoSb16 Oa2Sn4Te12 Er:OasTa14 Ca12Ga:O48Si4Sna Cl24Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Te24 Bi4eClK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='O72 750 1000 750 750 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='001 500 500 500 500 200 500 250 250 100 250- 0 BagSa2Sb16 BrsMo24Pba PbaoSsaSb32 Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='FsPb10 K,O4oP12Th4 CaBasF12024Y4 400- 400 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='002 600 1000 400- 1000 400 200 200 500 200 200 500 0 O AlaF4gPb12 I5Moz4Pba Ba24N32W: KNa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='O2:P:Zn4 112Nd,O36 Bis2OseRb16 400 750- 750 1000 600 500 500 500 400 200 500- 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='005 250 200 Ca12O3eTe12 Al20O4gY12 InieRb1Sb24 O2ReSm12 Ba17Ho1:OsPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Zng B12Ca18N24 1000 750 200 1500 500- 500 1000 500 500 100 250 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010- 01 0 FO BaoOg4Pr16 B,Ge,O4sPrs Baa2SesoSn16 B1-Pb16S40 AuaBiasBrs6 B12N12012Sr12 1500 1000 1500 400 200 1500 1000 1000 1000 200 500 100 500 005 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='020 0 0 0 Cds2l24Sb16 B12Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='O24 B24Ca12048 H1Li2020P2U2 Hz2O4oSr4 Cd12Er24Se4s 1500 400 200 3000 1000 2000 1000 2000 200 100 500 1000 1000 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') 1750 3500- 1500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1500 3000 1250 1250 2500 1000 1000 2000 750 750 1500 500 500 1000- 250 250 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 1000 1500 0 500 1000 1500 0 1000 2000 3000IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 General Validity of Prediction on Unseen Materials In Figure S11, we plot the average phonon frequency against the quadratic mean of atomic mass in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We apply the predictive model on 5000 unseen crystal structures from the Materials Project with atomic site number N in each unit cell (S11a,b) 1≤ N ≤ 20, (S11c,d) 21≤ N ≤ 40, (S11e,f) 41≤ N ≤ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For each material the average phonon frequency is related to the mass by the hyperbolic relationship ¯ω = C ¯m−1/2, where a constant C represents the rigidity of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The reasonable distribution of rigidity supports the physical validity of our model for unknown materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Moreover, we characterize the non-uniformity of atomic masses in each material by computing the ratio of the minimum mass mmin in a crystal to ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The materials with high mmin/ ¯m, containing both small and large atoms, tend to aggregate at lower ¯ω and higher ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Whereas, the materials with low mmin/ ¯m, containing atoms with similar mass, tend to aggregate at higher ¯ω and lower ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' These tendencies suggest that small atoms, such as hydrogen and oxygen, give high-frequency phonons which agree with the results in the previous study?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. Note that the VVN’s plots show higher and broader ¯ω distribution in the low ¯m region as N gets larger, while those of MVN keep narrower distribution in the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This corresponds to the results in Figure S7 that the Γ-phonon predicted by VVN gets higher than the ground truth in the case the input materials are complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 13/28 Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Evaluation of model predictions on unseen crystal structures Average frequency of the predicted Γ-phonon ¯ω versus average atomic mass ¯m using (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') VVN and (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 5000 structures with N atoms per unit cell are randomly sampled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') 1≤ N ≤ 20 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') 21 ≤ N ≤ 40 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=') 41 ≤ N ≤ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The black solid lines represent the least squares that fit the hyperbolic relation ¯ω = C ¯m−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The constant C is estimated from the fit as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C=1743 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C=1847 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C=1760 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C=2073 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C=2113 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' C=2157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The dot colors represent the magnitudes of mmin/ ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 14/28 mmin/m mmi/m a b T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 4000 1750 3500 0.' metadata={'source': 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+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 500 250 0 0 50 100 150 200 250 0 50 100 150 200 250 m [amu] m [amu] mmin/m d mmin/m C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 1200 3500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='8 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='8 1000 2500 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 4000 1200 3500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='8 1000 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='6 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='6 [cm-1 [cm 2000 600 13 13 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4 400 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 500 0 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200 m [amu] m [amu]IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='3 Element-wise Prediction of Average Phonon Energies In Figure S12-S17, we illustrate the correlation plots of the γ-phonons of all element appearing in the training and testing data sets using VVN and MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The background color represents the value of the prediction loss (yellow to blue from low to high loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We could see that it is not strict, but there are dependencies of the prediction accuracy on the periodicity for some parts of the elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' from 5B to 9F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Moreover, when compared to Figure S2-S3, it is generally found that elements corresponding to high prediction loss in Figure S12-S17 tend to appear less frequently than those with lower errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Element-wise correlation plot of the prediction results within train set using VVN Correlation plots with 63 elements existing within the train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 15/28 2000 5B 1H 2000 3Li 4Be 1500 2000 ‘C 3000 :i 9F 1500 80 0000 3000 2000 2000 1000 2000 1000 1000 1000 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1000 1000 1000 500 500 2000 2000 1000 2000 2000 1000 1000 1000 1000 2000 2000 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Loss 0 0 2000 11Na 2000 12Mg 13 Al 17CI 3000 2000 1500 3000 2000 1500 1000 2000 1000 2000 2000 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='030 1000 : 1000 500 500 1000 1000 500 1000 0 10 10002000 0 2000 1000 2000 1000 2000 10002000 2000 2000 0 1000 0 0 0 0 0 2000 1500 22Ti 25Mn 27Co 210 3000 SC 1500 1500 1000 400 1000 400 400 0000 1000 1000 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='025 500 200 1000 200 200 500 0 0 2000 1000 1000 500 2000 1000 250 500 1000 0 1500 37Rb 1000 750 1000 3000 3000 1000 500 2000 2000 500 500 500 500 500 250 1000 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='020 10 500 0 1000 5001000 1000 1000 1000 2000 2000 0 0 500 0 500 500 0 0 1000 45Rh 1500 41Nb 42Mo 43- Tc 1500 0000 600 2000 1000 1000 750 1000 2000 1000 400 500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 1000 500 1000 500 200 0os 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='015 0 01 2000 1000 1000 1000 2000 0 0 0 500 1000 0 500 2000 1500 2000 1000 800 46Pd 52- 53 750 Te 1500 1000 1500 750 1500 600 1000 500 1000 1000 500 1000 400 500 500 500 250 500 250 500 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010 O 1000 500 1000 2000 0 1000 2000 0 1000 1000 500 1000 0 500 0 0 0 57 La 72Hf 2000 2000 56Ba 1000 74W 750 3000 Ta 1500 750 1000 1500 2000 500 2000 500 1000 500 1000 500 250 1000 1000 500 500 250 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0： 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='005 2000 1000 2000 2000 1000 5001000 5001000 500 1000 0 500 2000 0 0 1500 77/r 80Hg 78pt 81TI 83Bi 600 2000 : 600 400 1000 1000 400 400 400 : 1000 500 200 200 500 200 200 200400 1000 2000 0 500 0 500 0 1000 0 500 0 1000 0 0Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Element-wise correlation plot of the prediction results within test set using VVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Correlation plots with 59 elements existing within the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 16/28 3Li 4Be 1H 9F °℃ 1000 80 3000 1500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. : 0000 1000 1000 1000 1000 2000 1000 2000 500 : 500 500 500 500 1000 1000 500 2000 1000 0 1000 1000 1000 5001000 2000 1000 0 500 0 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Loss 0 0 1000 1000 1000 2000 11Na 2000 17 CI 12Mg 19 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 14Si 15p 750 1000 750 750 1500 1500 1000 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 500 500 1000 1000 : 500 500 500 250 250 250 500 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='040 0 1000 2000 1000 2000 1000 500 1000 500 1000 0 1000 500 1000 500 0 0 1000 0 0 0 28Ni 800 29 Cu 23V 600 300 750 750 200 600 300 500 500 400 200 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='035 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=': 400 200 100 250 250 200 100 250 200 100 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. 0 500 500 200 500 400 500 500 200 100 200 0 0 0 800 800 800 33 As 37Rb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='030 307 Zn 1500 750 1500 600 600 600 400 1000 400 500 / 400 400 1000 500 /: 茶 250 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 200 200 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='025 500 500 500 1000 500 500 1000 1000 0 0 500 0 0 0 0 0 45Rh 48Cd 2000 46pd 39V 3000 600 1500 1500 600 1500 1000 2000 400 1000 1000 400 500 1000 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='020 200: 500:1 500 200 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 10 0 500 2000 1000 0 1000 0 2000 1000 0 500 0 1000 0 0 50Sn 1000 600 600 400 1000 1000 400 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='015 400 400 500 200 500 500 500 200 200 200 0 0 FC 0 1000 0 1000 500 0 500 0 200 400 1000 0 1000 500 0 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010 1000 79 Au 80Hg JHz 77/r 2000 1000 74W 78pt 600 200 1000 200 750 200 1500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 750 400 500 500 1000 100 100 500 100 200 250: 250 500 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='005 70 0 70 100 200 5001000 500 1000 1000 200 1000 2000 100 200 0 500 0 800 800 82pb 83Bi 600 600 1000 400 400 200 500 200 0 500 500 0 1000 0 0Figure S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Element-wise correlation plot of the prediction results within train set of the Togo Database using VVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Correlation plots with 64 elements existing within the test set of complicated materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 17/28 4000 20000 4000 6000 20000 20000 15000 4000 3000 : 15000 3000 15000 15000 10000 4000 10000 2000 2000 10000 10000 ii 2000 5000 2000 5000 1000 1000 5000 5000 6C 1H 3Li 4Be 5B 7N 80 9F 0 10000 20000 5000 4000 4000 5000 10000 20000 10000 20000 2500 2000 10000 2000 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Loss 3000 6000 20000 3000 20000 6000 6000 7500 4000 2000 15000 15000 4000 2000 4000 5000 10000 10000 1000 2000 2000 1000 2500 2000 11Na 5000 5000 19K 12Mg 13 Al 25005000 2000 10000 20000 5000 5000 2000 5000 1000020000 0 0 0 3000 3000 4000 1500 3000 6000 6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='8 3000 2000 2000 2000 1000 2000 4000 4000 2000 1000 1000 1000 2000 2000 500 1000 24Cr 1000 32Ge 22Ti 31Ga 33 As 70 2000 5000 1000 2000 4000 2000 2000 2000 0 5000 1500 1500 1500 : 2000 3000 4000 i 1500 10000 1000 1500 1000 1000 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='6 1000 1000 2000 5000 500 500 500 1000 500 39 500 41Nb 37Rb 42Mo 70 0 0 1000 1000 2000 1000 20004000 5000 10000 2000 1000 1000 0 0 0 0 0 0 2000 3000 3000 3000 1500 2000 15000 1000 1500 : 2000 2000 1000 2000 10000 1000 1000 500 1000 1000 500 1000 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4 500 47 Ag 48Cd 49/m 51Sb 52Te 56Ba 0 2000 2000 1000 2000 1000 2000 2000 5001000 1000 0 10000 0 0 0 0 3000 3000 3000 6000 3000 6000 1000 2000 2000 2000 2000 4000 4000 2000 1000 500 1000 1000 1000 2000 2000 57 /a 1000 PNo9 62Sm 65Tb 59Pr 67 Ho 66Dy 68Er 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 1000 5000 5000 2000 2000 2000 2000 0 0 2000 0 0 3000 4000 3000 1500 800 3000 3000 2000 3000 600 2000 2000 1500 1000 2000 2000 2000 1000 400 1000 1000 500 1000 1000 1000 70Yb 73Ta 72Hf 500 200 69Tm 71Lu 74W 75Re 2000 2000 2000 1000 4000 500 2000 0 0 2000 0 0 3000 1500 3000 3000 1000 1500 1500 4000 2000 2000 1000 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1000 1000 500 2000 1000 500 1000 500 1000 500 90Th 92V 8oHg 82pb 83Bi 78 Pt 70 0 1000 500 0 1000 2000 1000 2000 1000 2000 2500 5000 0 0 0 0Figure S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Element-wise correlation plot of the prediction results within train set using MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Correlation plots with about 63 elements existing within the train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The plotted phonon frequencies are in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 18/28 2000 2000 1H 2000 3Li 5B 2000 3000 7N 80 9F 4Be 1500 3000 1500 3000 2000 2000 1000 1000 2000 1000 1000 1000 1000 1000 1000 500 500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=': Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Loss 2000 10002000 1000 2000 1000 1000 2000 2000 2000 1000 2000 0 0 2000 12Mg 2000 13 Al 17 Cl 19 K 3000 3000 2000 1500 2000 3000 1000 1500 2000 1000 2000 2000 1000 1000 1000 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0200 500 1000 1000 1000 500 0 1000 2000 0 2000 2000 2000 2000 2000 1000 1000 5001000 1000 2000 0 600 2000 600 20Ca 23V 25Mn 28Ni 21C 3000 SC 1500 1500 1000 400 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1000 400 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0175 2000 1000 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 500 200 200 200 1000 500 500 :0 2000 1000 1000 1000 2000 1000 0 0 200 400 250 500 0 250 500 0 0 0 0 800 1500 33 As 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0150 1000 1000 1000 1000 3000 3000 600 1000 2000 2000 400 500 500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' : 500 1000 200 1000 0 C ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 500 1000 1000 5001000 0 500 1000 2000 2000 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0125 39 41Nb 42Mo 1000 600 3000 1500 1000 2000 1000 1000 750 400 2000 1000 500 500 1000 500 500 200 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0100 0 0 2000 2000 1000 1000 500 500 1000 0 0 1000 1000 0 500 0 0 0 800 2000 2000 800 48Cd 100052- 531 46Pd 51Sb :-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1000 Te 1500 600 1500 1500 1000 600 1000 400 1000 400 1000 500 500 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0075 500 200 500 200 500 0: 1000 500 1000 2000 1000 2000 1000 5001000 5001000 500 0 0 0 0 0 57 La 1000 2000 1000 74W 2000 72Hf 760s 3000 1500 750 1000 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0050 2000 500 2000 500 500 1000 1000 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 250 1000 1000 500 250 500 0 2000 2000 1000 1000 5001000 500 1000 2000 500 500 1000 2000 0 0 0 0 600 800 800 3Ho8 81TI 600 82Pb 1500 2000 78pt 009 600 1500 1000 400 1000 400 400 400 1000 500 200 500 200 200 200 500 20 0 250 500 1000 500 500 1000 500 0 2000 0 0 0 1000 0 0Figure S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Element-wise correlation plot of the prediction results within test set using MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Correlation plots with about 59 elements exist within the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The plotted phonon frequencies are in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 19/28 5B 9F 1H ‘C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 7N 80 3000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1500 3000 ： 1000 1000 1000 1000 2000 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. 1000 2000 500 500 500 500 500 1000 500 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1000 2000 500 1000 1000 1000 1000 500 2000 1000 0 0 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Loss 0 0 1500 1000 1000 11Na 2000 15P 2000 12Mg 13 Al 14Si 17 CI 1008 1000 750 750 1500 1000 1500 600 1000 500 500 1000 1000 400 500 500 500 250 500 250 200 500 : 1000 1000 1000 1000 2000 1000 500 1000 500 1000 500 0 0 2000 0 0 500 0 0 1000 20Ca 24Cr 23V 26Fe 28 Ni 29 Cu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='04 400 750 300 600 200 1000 300 400 500 500 400 200 200 100 250 500 250 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. 200 100 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 70 500 1000 500 500 500 1000 400 250 200 100 200 500 200 1000 800 31Ga 33 As 800 800 37Rb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 38S Zn 1000 1500 750 1500 Sr 600 600 600 400 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='03 1000 400 400 400 500 250 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. 500 200 200 200 500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 0 500 500 500 500 500 1000 1000 500 1000 1000 0 0 0 0 0 2000 48Cd 46 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='d 39V 750 3000 1500 1000 1500 1500 009 500 400 2000 1000 1000 1000 400 500 250 1000 500 500 i 200 200： 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='02 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 10 0 0 0 YO 500 500 2000 2000 1000 1000 1000 500 0 0 500 1000 800 52Te 57 La 53 1000 600 009 600 400 1000 1000 1000 400 400 400 500 200 500 500 500 200 200 200 0 0 1000 500 1000 1000 500 0 500 1000 500 0 0 200 400 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='01 1000 74W 77 72Hf 200 79 Au 600 80Hg 75Re 2000 78pt 1000 : 200 1000 750 200 150 750 1500 400 500 500 1000 100 500 100 200 250 /: 250 500 50 0 0 70 ：0 02 100200 5001000 500 1000 1000 200 1000 2000 100 200 250 500 1500 81TI 800 83Bi 600 1000 009 400 400 200 500 200 0 1000 500 0 500 0Figure S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Element-wise correlation plot of the prediction results within test set of the Togo Database using MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Correlation plots with about 64 elements exist within the test set of the complicated materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The plotted phonon frequencies are in [cm−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' V FULL PHONON BAND STRUCTURES PREDICTIONS We present more results by using the k-MVN model for predicting the full phonons dispersion band within the training and testing data from the Main Database and complex materials from the Togo database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Note that some of the phonons in DFPT calculation have negative values, and the model is set so that it only gives positive phonon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Although the phonon bands prediction of some materials does not match up exactly with the DFPT calculation, some features of the bands, such as an average frequency and band gaps, are well captured by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S18-S20 provide more results of the full phonon bands prediction using k-dependent matrix virtual nodes model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 20/28 3000 3000 3000 3000 3000 3000 2000 2000 2000 2000 2000 2000 2000 2000 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 1000 1000 1000 1000 1000 1000 1000 1000 3 Li 5B 7N 1H 4Be O: 9F 0 70 C 2000 1000 2000 2000 2000 2000 2000 0 2000 0 0 10002000 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Loss 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='035 3000 1500 3000 3000 3000 3000 1500 3000 1000 2000 2000 2000 2000 1000 2000 2000 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='.1 500 1000 1000 1000 1000 500 1000 1000 11Na 12Mg 13 Al 17 Cl 15 p 19 K 0 1000 0 1000 0 2000 2000 0 2000 2000 2000 2000 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='030 1500 1000 1500 3000 2000 : 3000 1500 750 1500 2000 1000 1000 2000 2000 1000 500 1000 i 500 1000 500 1000 1000 500 250 24Cr 20Ca 500 22Ti 32Ge 33 As 30Zn 31Ga 70 0 70 70 2000 1000 2000 500 1000 0 0 1000 1000 2000 1000 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='025 1500 1000 008 3000 3000 3000 1000 1000 1000 750 600 2000 2000 2000 500 400 500 500 500 1000 1000 1000 40X 34Se 37Rb 250 200 35Br 41Nb 42 Mo 38Sr 39Y 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='020 500 1000 500 1000 1000 2000 2000 2000 500 1000 0 500 0 0 0 0 0 1500 3000 3000 1000 1000 1000 1000 1000 1000 2000 2000 500 500 500 500 500 500 1000 1000 47 Ag 49/n 50Sn 48 Cd 51Sb 52Te 56Ba 53 20 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='015 Q 500 1000 1000 2000 0 500 1000 500 1000 500 1000 500 1000 2000 0 0 1500 1500 1500 1500 1500 1000 1000 2000 1000 1000 1000 1000 1000 500 500 1000 500 500 500 500 67 Ho 500 57 La Jd6s PNo9 65Tb 66Dy 68Er 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='010 70 0 o 500 10002000 0 1000 0 1000 0 1000 1000 0 1000 0 500 1000 0 1000 1000 1500 1000 3000 008 i 1500 750 750 1000 750 600 1000 2000 1000 500 500 500 400 500 500 1000 500 250 250 71LU 250 HzL 13Ta 200 74W 75Re 77/r 69Tm 70yb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='005 O 10 0 1000 1000 1000 500 1000 0 500 1000 0 0 500 500 0 2000 1000 800 1500 1000 1500 3000 750 750 2000 1000 600 1000 1000 2000 500 400 500 1000 500 500 500 250 1000 200 250 79 Au 78pt 82pb 90Th 92U 0 0 500 500 1000 1000 500 1000 0 2000 0 1000 5001000 0 2000 0 0 0 0 0Figure S18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Direct prediction result of phonon band structures within the training set of the Main Database using k-MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Each band contains 100 k-points across the Brillouin zone which varies for each material depending on its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The band calculated from DFPT are labeled in black, and green, yellow, and red lines represent the predicted phonon bands in 1st-3rd tertiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 21/28 HaK,Mg Ga2S4Zn NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Rb2 CCaN Cs4O,Zr2 As2HgK4 400 1500 250- 2000 1250 1250 600 300 200 1000 1000 1500- 150 750 750 400 200- 1000- 500 100 1E-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 500 100 200 500 250 50 250 FO 0 Ga2Li2Te Ua Cu404Pb2 CusS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='V PtSnTi 2E-4 600 250 300- 400 300 150 500 200- 400 300 200 150- 200 100 300 200 100 200 100- 100 100 100 50 5E-4 FO 0 0 0: K,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S2 As,O,Sr Al,Sb, GePtTi K,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Sb, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Sr,Ti 300 1250 600 800 300 AX 250 600 500 1E-3 1000 600 200 400 200 750 400 150 300 400 - 500 100 200 100 200 200 250 50 2E-3 100 0- 0- NiSnTi PdSiTi PdSiZr Ba4Br: Cs4P F4Mg2 300 500 600 300- 250 150 300 400 500 250 200 400 200 100 300 5E-3 200 150- 150 300 200 100- 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 50 200 100 100 50 50 100 0 0 0 0 1E-2 Cd2Te2 Cl2Csl FeTeTi BaBeaN PtSiZr FeInK2Na 300 1000 500 150- 300 250 300 800 400 200 300 2E -2 100 200 600 200 150 200 400 50 100 100 100 100 200 50 0 0 100 CssF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='TI NY AsF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='NaRb2 I2Li2 BiLis BaO 5E-2 400 400 600 500- 400- 300 500 400 300 300 300 400 300 200 200 300 200 200 200 100- 200 100 100 LO0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 100 100Figure S19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Direct prediction result of phonon band structures within the testing set of the Main Database using k-MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Each band contains 100 k-points across the Brillouin zone which varies for each material depending on its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The band calculated from DFPT are labeled in black, and green, yellow, and red lines represent the predicted phonon bands in 1st-3rd tertiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 22/28 AlCdSe4 C2CaK,O6 CFKO Sr Ag4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='O4 In2OgTa2 Cdln2Se4 400 1500 1500 800 500 1250 200 1250 300 400 600 1000 1000 5E-4 150 300 200 750 750 400 100 500 200 500 100 200 50 250 100- 250 0 0 0 CdGa2Te4 Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Hf BaF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Ge Ir4S4Sb4 Ba2HfO4 BaF6Pb 600 800 600 500 200 500 1E-3 300 600 400 150 400 400 300 200 300 400 100 200 200 200 100 200 50 100 100 0 0 Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='InNa CaF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Pb Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2NaY Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2InNa BaBiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Ta F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='NaRb2Rh 500 800 2E-3 200 600 600 200 400 600 500 150 150 300 400 400 100 400 100 300 200 200 50 200 50 200 100 100 0 0 FO 0 Ba2Cu2F2Te2 Nb2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='SC2 CS2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='KY Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2Mg2 Cl4K2Pt Ag2K2Se2 500 400 300 800 150 400 300 300 600 5E-3 200 300 100 200 200 200 400 100 100 50 200 100 100 0- 0 0 0 0 1E-2 BiCs2FaK NiPSc CaF2 Cs204Y2 NaOSc Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='RbY 500 500 600 400 400 300 400 400 300 300 400 300 300 200 200 200 200 200 100 200 100 100 100 100 2E -2 0 100 Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='P2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Sc In2Na2Te AS4C04S IRb 200 100 250 400 150 600 200 80 150 300 100 - 150 400 60 100 100 200 50 40 50 200 50 100 20Figure S20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Direct prediction result of phonon band structures of the Togo Database using k-MVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Each band contains 100 k-points across the Brillouin zone which varies for each material depending on its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The band calculated from DFPT are labeled in black, and green, yellow, and red lines represent the predicted phonon bands in 1st-3rd tertiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 23/28 AlCdSe4 C2CaK,O6 CFKO Sr Ag4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='O4 In2OgTa2 Cdln2Se4 400 1500 1500 800 500 1250 200 1250 300 400 600 1000 1000 5E-4 150 300 200 750 750 400 100 500 200 500 100 200 50 250 100- 250 0 0 0 CdGa2Te4 Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Hf BaF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Ge Ir4S4Sb4 Ba2HfO4 BaF6Pb 600 800 600 500 200 500 1E-3 300 600 400 150 400 400 300 200 300 400 100 200 200 200 100 200 50 100 100 0 0 Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='InNa CaF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Pb Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2NaY Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2InNa BaBiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Ta F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='NaRb2Rh 500 800 2E-3 200 600 600 200 400 600 500 150 150 300 400 400 100 400 100 300 200 200 50 200 50 200 100 100 0 0 FO 0 Ba2Cu2F2Te2 Nb2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='SC2 CS2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='KY Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Cs2Mg2 Cl4K2Pt Ag2K2Se2 500 400 300 800 150 400 300 300 600 5E-3 200 300 100 200 200 200 400 100 100 50 200 100 100 0- 0 0 0 0 1E-2 BiCs2FaK NiPSc CaF2 Cs204Y2 NaOSc Cs2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='RbY 500 500 600 400 400 300 400 400 300 300 400 300 300 200 200 200 200 200 100 200 100 100 100 100 2E -2 0 100 Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='P2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Sc In2Na2Te AS4C04S IRb 200 100 250 400 150 600 200 80 150 300 100 - 150 400 60 100 100 200 50 40 50 200 50 100 20VI Applications VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='1 Phonon Prediction on a Topological Weyl Semimetal The prediction of LaAlGe Γ-phonon obtained from k-MVN model compared to the experimental results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. The phonon dispersion of LaAlGe- a Type-II Weyl semimetal, was measured by inelastic neutron and X-ray techniques along the high symmetric points and Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Comparison of LaAlGe’s phonon spectra acquired with experiments and machine learning (Top) LaAlGe phonon bands along the highly symmetric points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' (Bottom) The phonon band structures along the x, y, z directions within the Brillouin zone near one of the Weyl points W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The plotted phonon frequencies are in [meV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 Phonon Prediction in Alloy Systems One of the most important applications of our prediction model is phonon bands of alloy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Crystalline alloys have disorders whose atomic positions can be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' However, the system is not periodic because atomic elements’ distribution at one site is probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Therefore, we need to consider the atomic compositions when we embed each site’s atomic mass, as the virtual-crystal approximation (VCA)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' is based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For example, given a binary alloy with composition VpW1–p(0 ≤ p ≤ 1), 24/28 25 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' units) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 20 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='5 10 Energy (meV) 10K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 15 10 5 0 7 Z W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 25 25 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' units) 20 20 20 15 15 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='5 10 10 10 5 5 5 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 25 25 25 20 20 20 15 15 15 10 10 10 5 5 5 0 0 0 102 -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='052 i-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='048 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='098 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0453 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0553 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='603 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='100 -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='050 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='050 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='100 k kthe input alloy encoding vector Z can take the following form Z = [0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=', pmV,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',(1− p)mW,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=',0] (2) where the two-hot encoding pmV and (1− p)mW are located at the vector indices corresponding to the atomic numbers of V and W, respectively, weighted by composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' With this definition of equation (2), the embedded feature can be directly reduced to pure phase one-hot encoding V (or W) by simply setting p = 1 (or p = 0), and it can be generalized to more complicated alloys directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We demonstrate the power of this approach with the binary alloy of SiGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We always assume SiGe alloy has eight atoms per cubic unit cell with a side of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='466 ˚A, which is the intermediate of the crystalline silicon and germanium?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. As the input of kMVN model, we convert the unit cell into a primitive cell with two atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' As the ground-truth label, we used the phonon bands calculated with VCA approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S22 shows the result of the phonon band prediction of SiGe alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Both ground truth (Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S22a) and prediction (Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S22b) shows similar shapes of the band structures along the highly-symmetric points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Yellow and blue colors depict the composition ratios of silicon and germanium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The lighter atoms generally offer phonons of higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Our result follows the principle, as the alloys give higher phonon frequencies as the compositions of silicon get larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Phonon spectra of SiGe alloy with various compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Yellow and blue indicate pure Si and Ge, respectively, and the intermediate colors represent alloys of different composition ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We demonstrated phonon prediction of SipGe1−p with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='13, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='20, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='80, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='96, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='995, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='00 (from blue to orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We show phonon along L, Γ, and X points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We next show the phonon prediction of binary alloys NiCo, NiFe, and a ternary alloy NiFeCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' For each material, we generated the random compositions 200 times and observed how the phonon prediction went.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' As the ground-truth label (black lines in each of Figure S23 ), we used the phonon bands calculated with DFPT approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' As the input of the DFPT calculation, we used supercells that contain 64 atoms for NiCo and NiFe and 108 atoms for NiFeCo, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure S23 shows the predicted phonon of the three alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The combination of red, blue, and yellow colors represent the compositions Ni, Co, and Fe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The spectra were arranged from left to right in the ascending order of the prediction loss: (a) NiCo, (b) NiFeCo, and (c) NiFe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This tendency can be explained by the atomic masses of the alloy’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Ni and Co have almost the same atomic mass (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='69u and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='93u), and Fe has a lower value (55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='85 u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Compared to NiFe, NiCo and NiFeCo have smaller differences in the atomic masses of the elements they have, which makes the atomic mass encoding more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 25/28 b a 500 500 Phonon frequency (cm-1) 400 400 300 300 200 200 100 100 0 0 X XFigure S23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Phonon spectra of high entropy alloys with various compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Red, blue and yellow indicate pure Ni, Co, and Fe, respectively, and the intermediate colors represent alloys of different composition ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The black lines in each figure are ground truth computed by DFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' NiCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' NiFeCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' NiFe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We have shown that our k-MVN model could handle phonon prediction of binary and ternary alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here we demon- strate even higher component alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Here, we start with binaries, moving to ternaries, then quaternaries, and finally to the five-component quinaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S24 shows the predicted spectra of MoTaNbWV and VWNbTaTi, as well as their lower component alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In high entropy alloy systems, both force constants and masses have fluctuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Therefore, we need to average these properties to represent the disordered states as primitive cells or supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' In Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S24a-d, the number of MoTaNbWV’s components increased from 2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We show a similar property for the case of VWNbTaTi, in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S24e-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Our prediction follows atomic mass embedding of equation (2), which weights the mass of the components with the configuration ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Even without incorporating the force constant averaging, our model could generate phonon spectra similar to the simulation results with special quasirandom structures (SQS)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. We show a similar property for the case of VWNbTaTi, in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='S24e-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VII Γ-PHONON DATABASE In previous work, the full phonon band structures and derived quantities for 1521 semiconducting inorganic crystals are presented?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='. Given the high-quality phonon prediction using the machine-learning approach, particularly the MVN approach for complex materials, here we present a new phonon database containing the phonon spectra for the entire 146,323 Materials in Materials Project (MP) as of 2022 computed by the MVN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Given the importance of zone-center Γ-phonons, which are measurable with more accessible equipment like Raman scattering, here we limit the database to Γ-phonons and will leave the full phonon database for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' This new Γ-phonon database offers the possibility to analyze the lattice dynamics for many compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' It could be used as a useful tool to be compared with Raman scattering results for crystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The Γ-phonon database generated with the MVN method is available at https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='io/k5utb/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The whole database is stored as a dictionary, whose keys are the MP ID number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The dictionary values for each material are also dictionaries, which contain basic information such as the number of atoms per unit cell, chemical formula, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We show the values of each dictionary in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' We provide the test from the mass-spring model on the validity of the results, which are summarized in Figure S25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The data displays a spread around the hyperbolic fit since the phonon frequencies are the outcome of the interplay of the whole set of interatomic force constants and the different masses of the elements composing the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' It can be noticed some trends can be recognized with respect to the masses of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Systems with non-uniform masses (identified by the small ratio mmin/ ¯m), tend to lay on a different hyperbolic curve with respect to more uniformly weighted systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 26/28 a b c NiCo NiFeCo NiFe 350 400 350 350 300 300 300 250 250 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 X U(K) L X U(K) L X U(K)Figure S24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Phonon spectra from binary to 5-component high entropy alloys Predicted phonon spectra with increasing constituent elements: From binaries to 5-component high entropy alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The added element for each alloy from left to right is shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The plotted phonon frequencies are in [THz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MoTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MoTaNb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MoTaNbW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' MoTaNbWV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VWNb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VWNbTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' VWNbTaTi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The components of the database for each material ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Data type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='material id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='MP ID number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='nsites ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='The number of atoms per unit cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='formula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Chemical formula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='space group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='The index of the space group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='The material structures of the CIF format ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='spectra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='list ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='The values of Γ-phonon frequencies [cm−1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='27/28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='a ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Mo-Ta-Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Mo-Ta-Nb-W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='Mo-Ta-Nb-W-V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='78 ' 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GHNGPHPN GH N GPHPN G NGPHPN WaveVector Wave Vector Wave Vector WaveVectorFigure S25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The profile of our Γ-phonon database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The number of atoms per unit cell in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The dataset contains 146323, and the number of atoms per unit cell ranges from 1 to 444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' Average frequency of the predicted Γ-phonon ¯ω versus average atomic mass ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The colors of the dots represent the magnitude of mmin/ ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The black solid lines represent the least squares that fit the hyperbolic relation ¯ω = C ¯m−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The constant C is estimated from the fit as 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' The colors of the dots represent the ratio mmin/ ¯m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content=' 28/28 a b mmin/m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='0 2000 12000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='8 10000 1500 8000 (/cm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='6 Count 1000 6000 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='4 4000- 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} +page_content='2 2000 0 0 100 200 300 400 0 50 100 150 200 250 Atoms/cell m (amu)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfQwBZ/content/2301.02197v1.pdf'} diff --git a/FNAyT4oBgHgl3EQf4_q8/content/tmp_files/2301.00797v1.pdf.txt b/FNAyT4oBgHgl3EQf4_q8/content/tmp_files/2301.00797v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c9a25fd1111760434058396ed25d0710e466eb0 --- /dev/null +++ b/FNAyT4oBgHgl3EQf4_q8/content/tmp_files/2301.00797v1.pdf.txt @@ -0,0 +1,1681 @@ +Parameterized Lower Bounds for Problems in P +via Fine-Grained Cross-Compositions +Klaus Heeger � +Technische Universität Berlin, Algorithmics and Computational Complexity, Germany +André Nichterlein � +Technische Universität Berlin, Algorithmics and Computational Complexity, Germany +Rolf Niedermeier +Technische Universität Berlin, Algorithmics and Computational Complexity, Germany +Abstract +We provide a general framework to exclude parameterized running times of the form O(ℓβ + nγ) +for problems that have polynomial running time lower bounds under hypotheses from fine-grained +complexity. Our framework is based on cross-compositions from parameterized complexity. We +(conditionally) exclude running times of the form O(ℓγ/(γ−1)−ε + nγ) for any 1 < γ < 2 and ε > 0 +for the following problems: +Longest Common (Increasing) Subsequence: Given two length-n strings over an alphabet Σ +(over N) and ℓ ∈ N, is there a common (increasing) subsequence of length ℓ in both strings? +Discrete Fréchet Distance: Given two lists of n points each and k ∈ N, is the Fréchet +distance of the lists at most k? Here ℓ is the maximum number of points which one list is ahead +of the other list in an optimum traversal. +Planar Motion Planning: Given a set of n non-intersecting axis-parallel line segment obstacles +in the plane and a line segment robot (called rod), can the rod be moved to a specified target +without touching any obstacles? Here ℓ is the maximum number of segments any segment has in +its vicinity. +Moreover, we exclude running times O(ℓ2γ/(γ−1)−ε + nγ) for any 1 < γ < 3 and ε > 0 for: +Negative Triangle: Given an edge-weighted graph with n vertices, is there a triangle whose +sum of edge-weights is negative? Here ℓ is the order of a maximum connected component. +Triangle Collection: Given a vertex-colored graph with n vertices, is there for each triple +of colors a triangle whose vertices have these three colors? Here ℓ is the order of a maximum +connected component. +2nd Shortest Path: Given an n-vertex edge-weighted directed graph, two vertices s and t, +and k ∈ N, has the second longest s-t-path length at most k? Here ℓ is the directed feedback +vertex set. +Except for 2nd Shortest Path all these running time bounds are tight, that is, algorithms +with running time O(ℓγ/(γ−1) + nγ) for any 1 < γ < 2 and O(ℓ2γ/(γ−1) + nγ) for any 1 < γ < 3, +respectively, are known. Our running time lower bounds also imply lower bounds on kernelization +algorithms for these problems. +2012 ACM Subject Classification Theory of computation → Graph algorithms analysis; Theory of +computation → Parameterized complexity and exact algorithms +Keywords and phrases FPT in P, Kernelization, Decomposition +Funding Klaus Heeger: Supported by DFG project NI 369/16 “FPTinP”. +Acknowledgements In memory of Rolf Niedermeier, our colleague, friend, and mentor, who sadly +passed away before this paper was finished. +We thank the anonymous reviewers for their thoughtful and constructive feedback. +arXiv:2301.00797v1 [cs.DS] 2 Jan 2023 + +2 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +1 +Introduction +In recent years, many results in Fine-Grained Complexity showed that many decade-old +textbook algorithms for polynomial-time solvable problems are essentially optimal: Consider +as an example Longest Common Subsequence (LCS) where, given two input strings +with n characters each, the task is to find a longest string that appears as subsequence +in both input strings. The classic O(n2)-time algorithm is often taught in introductory +courses to dynamic programming [16]. Bringmann and Künnemann [13] and Abboud et al. +[1] independently showed that an algorithm solving LCS in O(n2−ε) time for any ε > 0 would +refute the Strong Exponential Time Hypothesis (SETH). Such conditional lower bounds have +been shown for many polynomial-time solvable problems in the recent years [50]. +One approach to circumvent such lower bounds is “FPT in P” [2, 32]. For Longest +Common Subsequence there is a (quite old) parameterized algorithm running in O(kn + +n log n) time, where k is the length of the longest common subsequence [37]. Thus, if k is small +(e. g. O(n0.99)), then the O(n2) barrier can be broken (without refuting the SETH). A natural +question is whether we can do better. As k ≤ n, an algorithm running in O(k1−εn) time +for any ε > 0 would break the SETH. However, there are no obvious arguments excluding a +running time of O(k2+n). In fact, such additive running times are not only desirable (as again, +for small k this would be faster than even O(kn)) but also quite common in parameterized +algorithmics by employing kernelization: For Longest Common Subsequence the question +would be whether there are linear-time applicable data reduction rules that shrink the input +to size O(k). Then we could simply apply the textbook algorithm to solve Longest Common +Subsequence in overall O(k2 + n) time. Kernelization is well-studied in the parameterized +community [5, 29] and also effective in practice for polynomial-time solvable problems such +as Maximum Matching [40] or Minimum Cut [36]. +In this work, we prove that Longest Common Subsequence does not admit an +O(k2 + n)-time algorithm assuming the SETH. This also implies that no such kernelization +algorithm as mentioned above is likely to exist. +More precisely, we provide a general +framework to (conditionally) exclude algorithms with running time O(kβ + nγ) for problems +admitting conditional running time lower bounds. We apply our framework to various string +and graph problems as well as problems from computational geometry. As a result, we +get tight trade-offs between β and γ showing that the trivial trade-offs are often the best +one can hope for. For example, the algorithm of Hirschberg [37] for Longest Common +Subsequence implies an O(k3 + n1.5)-time algorithm. We show that any algorithm running +in O(k3−ε + n1.5) time for ε > 0 refutes the SETH (see Section 1.2 for a more detailed +overview). +1.1 +Related work +Fine-grained complexity is an active field of research with hundreds of papers. We refer to the +survey of Vassilevska Williams [50] for an overview of the results and employed hypotheses. +Over the last couple of years there has been a lot of work in the direction of “FPT +in P” for various problems such as Maximum Matching [17, 20, 28, 35, 38, 40, 42, 46], +Hyperbolicity [17, 27], and Diameter [1, 17]. Parameterized lower bounds are rare in this +line of work. Certain linear-time reductions can be used to exclude any kind of meaningful +FPT-running times; this is also known as General-Problem-Hardness [8]. Closer to our work +are Fluschnik et al. [26]. They provide lower bounds for strict kernelization (i. e. kernels +where the parameter is not allowed to increase) for subgraph detection problems such as +Negative Weight Triangle and Triangle Collection. Conceptually, they use the + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +3 +diminisher-framework [15, 25] which was originally developed to exclude polynomial-size +strict kernels under the assumption P ̸= NP. The basic idea is to iteratively apply a diminisher +(an algorithm that reduces the parameter at a cost of increasing the instance size) and an +(assumed) strict kernel (to shrink and control the instance size) to an instance I of an NP-hard +problem. After a polynomial number of rounds, this overall polynomial-time algorithm will +return a constant size instance which is equivalent to I, thus arriving at P = NP. Fluschnik +et al. [26] applied the same idea to polynomial-time solvable problems. In contrast, we rely +and adjust the composition-framework by Bodlaender et al. [10] which was developed to +exclude (general) polynomial-size kernels under the stronger assumption NP ̸⊆ coNP / poly. +The composition framework works as follows. Consider the example of the NP-hard +problem Negative-Weight Clique: Given an edge-weighted graph G and an integer k, +does G contain a negative-weight k-clique, that is, a clique on k vertices where the sum of +the edge-weights of the edges within the clique is negative. +Let (G1, k), (G2, k), . . . , (Gt, k) be several instances of Negative-Weight Clique with +the same k. Clearly, the graph G obtained by taking the disjoint union of all Gi contains a +negative-weight k-clique if and only if some Gi contains a negative-weight k-clique. Moreover, +the largest connected component of G has order maxi∈[t]{|V (Gi)|}. +Now assume that +Negative-Weight Clique has a kernel of size O(ℓc) for some constant c where ℓ is the +order of a largest connected component. By choosing t = kc+1, it follows that kernelizing +the instance (G, k) yields an instance of size less than ℓ, that is, less bits than the number +of instances encoded in G. Given the NP-hardness of Negative-Weight Clique such a +compression seems challenging; indeed it would imply NP ⊆ coNP / poly [30], which in turn +results in a collapse of the polynomial hierarchy. +Compositions and their extension cross-composition [11] are extensively employed in the +parameterized complexity literature. Moreover, to exclude kernels whose size is bounded +by polynomials of a specific degree adjustments have been made to the composition frame- +work [18]. +Parameter trade-offs. +For several of our running time lower bounds we have tight upper +bounds that are derived from a simple case distinction argument. +▶ Observation 1.1 (folklore). If a problem P admits an �O(ℓβnγ)-time algorithm1, then it +admits for every λ > 0 an �O(ℓβ+ γ·β +λ + nγ+λ)-time algorithm. +Proof. If ℓ ≤ n +λ +β , then the �O(ℓβnγ)-time algorithm runs in �O(nγ+λ) time. Otherwise n ≤ ℓ +β +λ . +Then the �O(ℓβnγ)-algorithm then in �O(ℓβ+ γ·β +λ ) time. +◀ +1.2 +Our Results & Technique +We provide a composition-based framework to establish parameterized running time lower +bounds and apply the framework to Longest Common Subsequence, Longest Common +(Weakly) Increasing Subsequence, Discrete Fréchet Distance, Planar Motion +Planning, Negative Triangle, and 2nd Shortest Path (see Section 1.3 for the +problem definitions). Using similar ideas we obtain running time lower bounds for Triangle +Collection. For all these problems except 2nd Shortest Path parameterized by the +directed feedback vertex set there exist matching running time upper bounds. We refer +to Table 1 for an overview on the specific results and the parameterization. Moreover, we +1 The � +O hides polylogarithmic factors. + +4 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Table 1 Overview of achievable running times. The upper part of the table lists the results +for four problems that can be solved in O(n2) time but under SETH or 3SUM-hypothesis not +in O(n2−ε) time for any ε > 0. The lower part lists results for three graph problems that, based on +the APSP-hypothesis, do not admit O(n3−ε)-time algorithms. The parameterized upper and lower +bounds are visualized in Figure 1. +problems +Longest Common Subsequence +ℓˆ=solution size +Longest Common (Weakly) Increasing Subsequence +ℓˆ=solution size +Discrete Fréchet Distance +ℓˆ=maximum shift +Planar Motion Planning +ℓˆ=max. number of segments in the vincinty of any segment +results +upper bounds +lower bounds +o(n2) [6, 7, 33, 52] +no O(n2−ε) assuming SETH / 3SUM [1, 12, 31] +� +O(ℓn) [37, 43, 49] +� +O(ℓγ/(γ−1) + nγ) for each γ > 1 +no O(ℓγ/(γ−1)−ε + nγ) for any γ < 2 +(Observation 1.1) +(Corollaries 4.6, 4.11, and 4.13) +problems +Negative Triangle +ℓˆ=size of maximum component +Triangle Collection +ℓˆ=size of maximum component +2nd Shortest Path (only lower bounds) +ℓˆ=directed feedback vertex set size +results +upper bounds +lower bounds +O(n3/2Θ(log0.5 n)) [14, 53] +no O(n3−ε) assuming APSP [51] +� +O(ℓ2n) (folklore) +� +O(ℓ2γ/(γ−1) + nγ) for each γ > 1 +no O(ℓ2γ/(γ−1)−ε + nγ) for any γ < 3 +(Observation 1.1) +(Corollaries 4.15 and 4.17 and Proposition 4.18) +visualize in Figure 1 the trade-offs in the running times that are (im-)possible. +Framework. +We adjust the cross-composition framework to obtain lower bounds for poly- +nomial time solvable problems. As an example, consider Negative-Weight Triangle, +that is, Negative-Weight Clique with k fixed to three. Assuming the APSP-hypothesis, +Negative-Weight Clique cannot be solved in O(n3−ε) time [51]. The first difference to +the cross-composition framework is that we start with one instance G of Negative-Weight +Triangle which we then decompose into many small instances as follows: Partition the +vertices V (G) of G into z many sets V1, . . . , Vz of size n/z, where z is chosen depending +on the running time we want to exclude (see the proof of Lemma 3.3 in Section 3 for the +actual formula specifying z). Then, we create z3 instances of Negative-Weight Clique: +for each (i, j, k) ∈ [z]3 take the graph G[Vi ∪ Vj ∪ Vk]. Clearly, we have that G contains +a negative-weight triangle if and only if at least one of the created instances contains a +negative-weight triangle. +Next, we apply the composition as explained above (the disjoint union) for Negative- +Weight Clique to obtain an instance G′ with n′ = z3 · n/z = z2n vertices. Note that the +size ℓ of a largest connected component in G′ is at most 3n/z. Hence, an algorithm running +in time O(nγ + ℓβ) for Negative-Weight Triangle solves G′ in time O(z2γnγ + (3n/z)β). +By carefully choosing z as a function in n, β, and γ, we get that this is in O(n3−ε) for various +combinations of γ and β. +The property that Negative-Weight Triangle can be decomposed as above is not +unique to the problem. In fact, this has been observed already: “Many problems, like SAT, + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +5 +1 +2 +3 +4 +1 +2 +3 +4 +5 +γ +β +O(n2)-solvable +1 +2 +3 +4 +1 +2 +3 +4 +5 +γ +β +O(n3)-solvable +Figure 1 Overview on the possible (in green) and unlikely (in red) trade-offs in running times of +the form O(nγ + ℓβ). Left: First category for O(n2)-time solvable problems (upper part in Table 1); +right: second category for O(n3)-time solvable problems (lower part in Table 1). +have a simple self-reduction proving that the “Direct-OR” version is hard, assuming the prob- +lem itself is hard” [4]. Our framework formalizes this notion of decomposition (see Section 2 +for a definition) and adjusts the cross-composition definition. We furthermore show that +commonly used “hard” problems such as Orthogonal Vectors, 3-Sum, and Negative- +Weight k-Clique are decomposable. Thus, it remains to show cross-compositions in order +to apply our framework and obtain lower bounds. +1.3 +Preliminaries and Notation. +Problem definitions. +For ℓ ∈ N we set [ℓ] := {1, 2, . . . , ℓ}. +Orthogonal Vectors +Input: Two size-n sets A, B ⊆ {0, 1}d for some d ∈ N. +Question: Are there a ∈ A and b ∈ B so that a and b are orthogonal, i.e., for each i ∈ [d], +the i-th coordinate of a or the i-th coordinate of b is zero. +We denote the restriction of Orthogonal Vectors to instances with d ≤ O(log n) as +Orthogonal Vectors with logarithmic dimension. +3-Sum +Input: An array A of n integers. +Question: Are there i, j, h ∈ [n] such that A[i] + A[j] + A[h] = 0? +Negative-Weight k-Clique +Input: An edge-weighted graph G on n vertices. +Question: Does G contain a k-clique of negative weight? +Longest Common Subsequence +Input: Two strings x1 and x2 of length n over an alphabet Σ and k ∈ N. +Question: Decide whether there is a common subsequence of length k of x1 and x2. +Longest Common (Weakly) Increasing Subsequence +Input: Two strings x1 and x2 of length n over N and k ∈ N. +Question: Decide whether there is a common subsequence y of length k of x1 and x2 +with y[i] < y[i + 1] (y[i] ≤ y[i + 1]) for all i ∈ [k − 1]. + +6 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Discrete Fréchet Distance +Input: Two lists P = (p1, . . . , pn), Q = (q1, . . . , qn) of points in the plane and k ∈ N. +Question: Is the Fréchet distance of P and Q at most k, that is, there are two surjective, +non-decreasing functions fP , fQ : [2n] → [n] with fP (1) = 1 = fQ(1), fP (2n) = n = +fQ(2n) and maxi∈[2n] dist(pfP (i), qfQ(i)) ≤ k? +Planar Motion Planning +Input: A set of n non-intersecting, non-touching, axis-parallel line segment obstacles in +the plane and a line segment robot (a rod or ladder), a given source, and a given goal. +Question: Can the rod be moved (allowing both translation and rotation) from the +source to the goal without colliding with the obstacles? +2nd Shortest Path +Input: An n-vertex edge-weighted directed graph G, vertices s and t, and k ∈ N. +Question: Has the 2nd-shortest s-t-path length at most k? +Triangle Collection +Input: A vertex-colored graph G on n vertices. +Question: For each combination of three colors, does G contain a triangle whose vertices +are colored with three colors? +Hypotheses. +The conditional lower bounds in this work are based on SETH, 3-Sum-, and +the APSP-Hypothesis (see Vassilevska Williams [50] for more details). +▶ Hypothesis 1 (SETH). For every ε > 0 there exists a k ∈ N such that k-SAT cannot be +solved in O(2(1−ε)n) time, where n is the number of variables. +▶ Hypothesis 2 (3SUM). 3-Sum on n integers in {−n4, . . . , n4} cannot be solved in O(n2−ε) +time for any ε > 0. +▶ Hypothesis 3 (APSP). All Pairs Shortest Path on n-vertex graphs with polynomial +edge weights cannot be solved in O(n3−ε) time for any ε > 0. +Parameterized Complexity. +In many of the above problems, n is not the input size but +a parameter and the input size is bounded by nO(1). A parameterized problem is a set +of instances (I, p) ∈ Σ∗ × Σ∗, where Σ denotes a finite alphabet, I denotes the classical +instance and p the parameter. A kernelization is a polynomial-time algorithm that maps any +instance (I, p) to an equivalent instance (I′, p′) (the kernel) such that |I′| + p′ ≤ f(p) for +some computable function f. If f is a polynomial, then (I′, p′) is a polynomial-size kernel. +In this work we restrict ourselves to the following: Either p = n (p is a single parameter) +or p = (n, ℓ) (p is a combined parameter). Moreover, both n and ℓ are always nonnegative +integers, n is related to the input size but ℓ is not (ℓ can be seen as “classical” parameter). +2 +Framework +Our framework has the following three steps (see Section 1.2 for a high-level description). +1. Start with an instance (I, nP) of a “hard” problem P and decompose it into the disjunction +of t instances (I1, n1), . . . , (It, nt) of P. In Section 3, we provide such decompositions +for the frequently used hard problems 3-Sum, Orthogonal Vectors, and Negative +Weight k-Clique. + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +7 +2. Compose (I1, n1), . . . , (It, nt) into one instance (J, (n, ℓ)) of the “target” problem using +an OR-cross-composition. This step has to be done for the application at hand. +3. Apply the assumed �O(nγ + ℓβ)-time algorithm to J. If the combination of γ and β is +small enough, then the resulting algorithm will be faster than the lower bound for P. +To give a more formal description of our framework, we first define decompositions and +cross-compositions. Note that all mentioned problems are parameterized problems. +▶ Definition 2.1 (OR-decomposition). For α > 1 an α-OR-decomposition for a problem P +is an algorithm that, given λ > 0 and an instance (I, n) of P, computes for some α′ < α in +�O(nα′) time t ∈ �O(nαλ/(α+λ)) many instances (I1, n1), . . . , (It, nt) of P such that +(I, n) ∈ P if and only if (Ii, ni) ∈ P for some i ∈ [t], and +ni ∈ �O(nα/(α+λ)) for all i ∈ [t]. +We say a problem P is α-OR-decomposable if there exists an α-OR-decomposition for +it. For some problems it is easier to show OR-decomposability than others. Thus, using +appropriate reductions to transfer OR-decomposability can be desirable (we do so in Section 3 +when showing that 3-Sum is 2-OR-decomposable). Quasi-linear time reductions that do +not increase the parameter to much are one option. To this end, we say a reduction that +given an instance (IP, nP) of P produces an instance (IQ, nQ) of Q is parameter-preserving +if nQ ∈ �O(nP). +▶ Proposition 2.2. Let P and Q be two problems such that there are quasi-linear-time +parameter-preserving reductions from P to Q and from Q to P. Then for any α > 1, P is +α-OR-decomposable if and only if Q is. +Proof. Assume that P is α-OR-decomposable (the case that Q is α-OR-decomposable is +symmetric). We now give an α-OR-decomposition for Q. Given an instance (IQ, nQ) and +λ > 0, we first reduce (IQ, nQ) to an instance (IP, nP). Afterwards, we apply the α-OR- +decomposition from P, resulting in instances (IP +1 , nP +1 ), . . . , (IP +t , nP +t ) of P. Finally, we reduce +each instance (IP +i , nP +i ) to an instance (IP +i , nP +i ). This clearly is an α-OR-decomposition +for Q. +◀ +For the second step of our framework, we introduce fine-grained OR-cross-compositions: +▶ Definition 2.3 (fine-grained OR-cross-composition). For ν ≥ 1, µ ≥ 0 an (ν, µ)-OR- +cross-composition from a problem P to a problem Q is an algorithm A which takes as +an input t instances (IP +1 , nP +1 ), . . . , (IP +t , nP +t ) of P, runs in �O(t · nν +max + �t +i=1 |IP +i |) time +with nmax := maxi∈[t] nP +i , and computes an instance (IQ, (nQ, ℓQ)) of Q such that +1. (IQ, (nQ, ℓQ)) ∈ Q if and only if (IP +i , nP +i ) ∈ P for some i ∈ [t], and +2. nQ ∈ �O(t · nν +max) and ℓQ ∈ �O(nµ +max). +We say a problem P (ν, µ)-OR-cross-composes into a problem Q if there exists an +(ν, µ)-OR-cross-composition from P to Q. +▶ Theorem 2.4. Let α > ν ≥ 1, γ > 1, and µ > 0 with α > ν · γ. Let P be an α-OR- +decomposable problem with parameter nP that (ν, µ)-OR-cross-composes into a problem Q +with parameters nQ and ℓQ. If there is an �O(nγ +Q + ℓβ +Q)-time algorithm for Q and +0 < β < γ · (α − ν) +(γ − 1) · µ , +then P can be solved in O(nα−ε +P +) time for some ε > 0 time. + +8 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Proof. Let (IP, nP) be an instance of P. +Our algorithm to solve (IP, nP) runs in the +following steps: +1. Apply the α-OR-decomposition (with λ specified below) to obtain the instances (I1, n1), +. . . , (It, nt) with maxi∈[t] ni ≤ q := nα/(α+λ) +P +and t := nαλ/(α+λ) +P += qλ. Note that, by +definition, this step runs in �O(nα′ +P ) time for some α′ < α. +2. Apply the (ν, µ)-OR-cross-composition to compute the instance (IQ, (nQ, ℓQ)) for Q +from (I1, n1), . . . , (It, nt). Note that the running time and nQ is in �O(t·qν +�t +i=1 |IP +i |) = +�O(qλ+ν + nα′ +P ). Moreover, ℓQ ∈ �O(qµ). +3. Apply the algorithm with running time �O(nγ +Q + ℓβ +Q). This requires �O(q(λ+ν)γ + qµ·β) +time. +It remains to show that all three steps run in O(nα−ε +P +) time for some ε > 0. To this end, we +now specify λ = βµ/γ−ν. Note that λ+ν = βµ/γ < µ·β and thus it suffices to show that the +third step runs in �O(nα−ϵ +P +) for some ϵ > 0. The last step runs in �O(qµ·β) = �O(nαµ·β/(α+λ) +P +) +time. The exponent is +αµβ +α + λ = +αµβ +α + βµ/γ − ν = +αβγµ +γ(α − ν) + βµ = +αγµ +γ(α − ν)/β + µ . +By assumption we have β < γ·(α−ν) +(γ−1)·µ and thus the exponent is +αγµ +γ(α − ν)/β + µ < +αγµ +γ(α − ν)/ +� +γ·(α−ν) +(γ−1)·µ +� ++ µ += +αγµ +(γ − 1) · µ + µ = α. +There is still one thing left to do: We must ensure that λ > 0. This will not always be the +case as when β → 0, λ gets negative. However, an �O(nγ +Q + ℓβ +Q)-time algorithm also implies +for any β′ > β an �O(nγ +Q + ℓβ′ +Q )-time algorithm. Thus, we can simply pick some larger β′ such +that the corresponding λ′ is larger than 0. To do so, let βmax := γ·(α−ν) +(γ−1)·µ the upper bound +for β. Note that α > ν · γ implies that +λmax := βmaxµ +γ +− ν = +γ·(α−ν) +(γ−1)·µ µ +γ +− ν = α − ν +γ − 1 − ν = α − ν − ν · (γ − 1) +γ − 1 += α − ν · γ +γ − 1 +> 0 +Thus, we can pick β < β′ < βmax such that λ′ := β′µ +γ − ν > 0. +◀ +Note that if for P there is a (conditional) running time lower bound of Ω(nα), then +Theorem 2.4 excludes (conditionally) running times of the form �O(nP + ℓβ) for any β ∈ R as +limγ→1 +γ·(α−ν) +(γ−1)·µ = ∞. This running time also excludes linear-time computable polynomial-size +kernels. More precisely, we get the following. +▶ Corollary 2.5. Let α > ν ≥ 1, γ > 1, and µ > 0 with α > ν · γ. Let P be an α-OR- +decomposable problem with parameter nP that (ν, µ)-OR-cross-composes into a problem Q +with parameters nQ and ℓQ. Assume that there is an �O(nξ +Q) algorithm for deciding Q and +that nQ is upper bounded by the input size. If there exists an �O(ℓβ +Q)-size �O(nγ +Q)-time kernel +for Q for some γ > 1, β ∈ R, and +0 < β < +γ · (α − ν) +(γ − 1) · µ · ξ , +then P can be solved in O(nα−ε +P +) time for some ε > 0. +Proof. An �O(ℓβ +Q)-size �O(nγ +Q)-time kernel together with an �O(nξ +Q)-time algorithm solving Q +yields an �O(nγ +Q + ℓβξ +Q ) algorithm for Q. The corollary now follows from Theorem 2.4. +◀ + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +9 +Our general approach to apply our framework is follows. Start with a problem P that +(under some hypothesis) cannot be solved in O(nα−ε) time for ε > 0. Then construct an +α-decomposition for P followed by a (1,1)-OR-cross-composition into the target problem. +3 +OR-decomposable problems +In order to apply our framework, we first need some OR-decomposable problems. We will +observe that three fundamental problems from fine-grained complexity, namely Orthogonal +Vectors, 3-SUM and Negative-Weight k-Clique, are OR-decomposable. These prob- +lems are also our source for running time lower bounds: Note that the former two problems +cannot be solved in O(n2−ε) time unless the SETH respectively 3-Sum-hypothesis fail [50]. +We moreover use that Negative-Weight 3-Clique (= Negative Triangle) cannot be +solved in O(n3−ε) time unless APSP-hypothesis fails [51]. +We will use that “Many problems, like SAT, have a simple self-reduction proving that the +“Direct-OR” version is hard, assuming the problem itself is hard” [4]. This self-reduction is +based on partitioning the instance into many small ones, with at least one of them containing +the small desired structure (i.e., a pair of orthogonal vectors, three numbers summing to 0, +or a negative triangle). +Orthogonal Vectors. +We now show that Orthogonal Vectors is 2-OR-decomposable. +▶ Lemma 3.1. Orthogonal Vectors parameterized by the number of vectors is 2-OR- +decomposable. +Proof. Let (I, n) be an instance of Orthogonal Vectors and λ > 0. Set ϵ := +2 +2+λ. +Partition A into z := ⌈n1−ϵ⌉ many sets A1, . . . , Az of at most ⌈nϵ⌉ vectors each. Sym- +metrically, partition B into B1, . . . , Bz of at most ⌈nϵ⌉ vectors each. We assume without +loss of generality that |Ai| = |Bj| = ⌈nϵ⌉ =: n′ (this can be achieved e.g. by adding the +all-one vector). For each pair (i, j) ∈ [z]2, create an instance (I(i,j), n′) = ((Ai, Bj), n′) of +Orthogonal Vectors. We claim that this constitutes a 2-OR-decomposition. +The number of vectors n′ of each instance I(i,j) is n′ = O(n +2/(2+λ)). The number of +created instances is z2 = O(n2·(1−ϵ)) = O(n2·(1−2/(2+λ))) = O(n +(4+2λ−4)/(2+λ)) = O(n +2λ/(2+λ)). +The running time to compute the decomposition is �O(z2 · n′) = �O(n2−ϵ). +It remains to show that (I, n) is a “Yes”-instance if and only if (I(i,j), n′) is a “Yes”- +instance for some (i, j) ∈ [z]2. First assume that (I, n) is a “Yes”-instance. Then there exists +some a ∈ A and b ∈ B such that a and b are orthogonal. Let i ∈ [z] such that a ∈ Ai and +j ∈ [z] such that b ∈ Bj. Then a and b are orthogonal vectors in (I(i,j), n′), showing that +(I(i,j), n′) is a “Yes”-instance. +Finally, assume that there exists (i∗, j∗) ∈ [z]2 such that (I(i∗,j∗), n′) is a “Yes”-instance. +Then there exists a ∈ Ai∗ and b ∈ Bj∗ which are orthogonal. Consequently, a and b are +orthogonal vectors in I, implying that (I, n) is a “Yes”-instance. +◀ +▶ Remark 3.2. Note that the above decomposition does not change the dimension d. Thus, +even restricted versions of Orthogonal Vectors with d ∈ O(log n) are 2-OR-decomposable. +Further, we can assume that all constructed instances of Orthogonal Vectors have the +same number of vectors and the same dimension. +Negative-Weight k-Clique. +We now show that Negative-Weight k-Clique is k-OR- +decomposable. + +10 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +▶ Lemma 3.3. For any k ≥ 3, Negative-Weight k-Clique parameterized by the number +of vertices is k-OR-decomposable. +Proof. The proof follows the ideas from Lemma 3.1. Let (I = (G, w), n) be an instance +of Negative-Weight k-Clique and λ > 0. Set ϵ := +k +k+λ. Partition the set V (G) of +vertices into z := ⌈n1−ϵ⌉ many sets V1, . . . , Vz of size at most ⌈nϵ⌉. We assume without loss +of generality that |Vi| = ⌈nϵ⌉ for all i ∈ [z] (this can be achieved e.g. by adding isolated +vertices). Let n(i1,...,ik) := |{i1, . . . , ik}| · |V1|. For each tuple (i1, . . . , ik) ∈ [z]k, create an +instance (I(i1,...,ik) = G[Ai1 ∪ · · · ∪ Aik], n(i1,...,ik)) of Negative-Weight k-Clique. We +claim that this constitutes a k-OR-decomposition. +Each instance (I(i,j), n(i1,...,ik)) has at most O(k ·nϵ) = O(n +k/(k+λ)) vertices (note that k is +a constant here). The number of created instances is zk = O(nk·(1−ϵ)) = O(nk·(1−k/(k+λ))) = +O(n +(k2+kλ−k2)/(k+λ)) = O(n +kλ/(k+λ)). Further, the summed size of all created instances and +therefore also the running time is �O(zk · k · n2ϵ) = �O(nk−kϵ+2ϵ) = �O(nk−(k−2)ϵ) = �O(nk′) for +some k′ < k. +It remains to show that (I, n) is a “Yes”-instance if and only if (I(i1,...,ik), n(i1,...,ik)) is a +“Yes”-instance for some (i1, . . . , ik) ∈ [z]k. First assume that (I, n) is a “Yes”-instance. Thus, +G contains a negative-weight clique C = {v1, . . . , vk}. Let ij such that vj ∈ Vij. Then C is a +negative-weight k-clique in (I(i1,...,ik), n(i1,...,ik)). +Finally, assume that there exists (i1, . . . , ik) ∈ [n]k such that (I(i1,...,ik), n(i1,...,ik)) is a +“Yes”-instance. Then there exists a clique in G[Vi1∪· · ·∪Vik]. Since G contains G[Vi1∪· · ·∪Vik], +also G contains a negative-weight k-clique. +◀ +3-Sum. +Showing that 3-Sum is 2-OR-decomposable requires some more work. +▶ Lemma 3.4. 3-Sum parameterized by the number of numbers is 2-OR-decomposable. +Instead of directly considering 3-Sum, we consider a variation called Convolution +3-Sum of 3-Sum which was shown to be equivalent (under quasi-linear time Las-Vegas +reductions) to 3-Sum [41]. +Convolution 3-Sum +Input: An array A of n integers. +Question: Are there i, j ∈ [n] such that A[i] + A[j] = A[i + j]? +For simplicity, we will not directly work with Convolution 3-Sum, but with a “multicolored” +version of it: +Multicolored Convolution 3-Sum +Input: Three arrays A, B, and C, each of n integers. +Question: Are there i, j ∈ [n] such that A[i] + B[j] = C[i + j]? +First, we show how to reduce Convolution 3-Sum to Multicolored Convolution +3-Sum and vice versa. +▶ Lemma 3.5. There is a linear-time parameter-preserving reduction from Convolution +3-Sum parameterized by the number of numbers to Multicolored Convolution 3-Sum +parameterized by the number of numbers and vice versa. +Proof. Let (A, n) be an instance of Convolution 3-Sum. We create an instance ((A, B, C), n) +of Multicolored Convolution 3-Sum by setting B[i] := A[i] and C[i] := A[i]. Note that +for i, j ∈ [n], we have A[i] + A[j] = A[i + j] if and only if A[i] + B[i] = A[i] + A[j] = C[i + j], +implying that the two instances are equivalent. + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +11 +Now consider an instance (I = (A, B, C), n) of Multicolored Convolution 3-Sum. +Let zmax := maxi∈[n] max{A[i], B[i], C[i]} + 1. +We create an instance (A′, n′ = 4n) of +Convolution 3-Sum as follows. The array A′ has length 4n with +A′[i] := +� +� +� +� +� +� +� +� +� +� +� +−5zmax +1 ≤ i ≤ n +A[i − n] + zmax +n + 1 ≤ i ≤ 2n, +B[i − 2n] + 3zmax +2n + 1 ≤ i ≤ 3n, +C[i − 3n] + 4zmax +3n + 1 ≤ i ≤ 4n . +First, note that a solution A[i] + B[j] = C[i + j] to (I, n) implies a solution to (A′, n′) as +A′[i+n]+A′[j +2n] = A[i]+zmax +B[j]+3zmax = C[i+j]+4zmax = A′[3n+i+j]. Second, +consider a solution A′[i′]+A′[j′] = A′[i′+j′]. Then i′, j′ > n as otherwise A′[i′]+A′[j′] < 0 and +A′[i′] + A′[j′] ̸= −5zmax. Further, i′, j′ ≤ 3n as otherwise A′[i′] + A′[j′] > 5zmax > A′[i′ + j′]. +Further, we have without loss of generality n + 1 ≤ i′ ≤ 2n and 2n + 1 ≤ j′ ≤ 3n. Thus, we +have A[i′−n]+B[j′−2n] = A′[i′]−zmax+A′[j′]−3zmax = A[i′+j′]−4zmax = C[i′+j′−3n]. +◀ +Next, we show that Multicolored Convolution 3-Sum is 2-OR-decomposable. +▶ Lemma 3.6. Multicolored Convolution 3-Sum parameterized by the number of +numbers is 2-OR-decomposable. +Proof. The idea behind the proof is essentially the same as for Lemma 3.1. For an array A +we denote the array consisting of the entries (A[i], A[i + 1], . . . , A[j]) by A[i, j]. Let (I = +(A, B, C), n) be an instance of Multicolored Convolution 3-Sum and λ > 0. Set +ϵ := +2 +2+λ and q := ⌈nϵ⌉. Partition A into z := ⌈n1−ϵ⌉ many arrays A1 = A[1, q], A2 = +A[q+1, 2q], . . . , Az = A[(z−1)q+1, n]. For simplicity, we assume that n = z·q. Symmetrically, +partition B into z many arrays B1 = B[1, q], B2 = B[q + 1, 2q], . . . , Bz = B[(z − 1)q + 1, n]. +For (ℓ, p) ∈ [z]2 we denote the interval C[(ℓ + p − 2)q + 2, (ℓ + p)q] by C(ℓ,p). For each pair +(ℓ, p) ∈ [z]2, create an instance (I(ℓ,p) = (Aℓ, Bp, C(ℓ,p)), q) of Convolution 3-Sum. We +claim that this constitutes a 2-OR-decomposition. +Each instance (I(ℓ,p), q) has at most O(q) = O(nϵ) = O(n +2/(2+λ)). +The number of +created instances is z2 = O(n2·(1−ϵ)) = O(n2·(1−2/(2+λ))) = O(n +(4+2λ−4)/(2+λ)) = O(n +2λ/(2+λ)). +Computing the decomposition can be done in �O(z2 · q) = �O(n2−ϵ) time. +It remains to show that (I, n) is a “Yes”-instance if and only if (I(ℓ,p), q) is a “Yes”- +instance for some (ℓ, p) ∈ [z]2. First assume that (I, n) is a “Yes”-instance. Then there +exists some i = (ℓ − 1)q + i∗, j = (p − 1)q + j∗ such that A[i] + B[j] = C[i + j]. Note that +A[i] = Aℓ[i∗], B[j] = Bp[j∗], and C[i+j] = C(ℓ,p)[i∗ +j∗]. Thus, (I(ℓ,p), q) is a “Yes”-instance. +Finally, assume that there exists (ℓ, p) ∈ [z]2 such that (I(ℓ,p), q) is a “Yes”-instance. +Then Aℓ[i∗] + Bp[j∗] = C(ℓ,p)[i∗ + j∗] for some i∗, j∗ ∈ [q]. Consequently, A[(ℓ − 1)q + i∗] + +B[(p − 1)q + j∗] = C(ℓ,p)[i∗ + j∗] = C[(ℓ + p − 2)q + i∗ + j∗], implying that (I, n) is a +“Yes”-instance. +◀ +The 2-OR-decomposability of 3-Sum now follows from the quasi-linear time parameter- +preserving reductions between 3-Sum and Convolution 3-Sum [41] as well as between +Convolution 3-Sum and Multicolored Convolution 3-Sum Lemma 3.5. +Proof of Lemma 3.4. Follows from Lemma 3.6, the equivalence of 3-Sum and Convolu- +tion 3-Sum [41], the equivalence of Convolution 3-Sum and Multicolored Convolu- +tion 3-Sum, and Proposition 2.2. +◀ + +12 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Applying the framework. +The above results make our framework easier to apply. To apply +Theorem 2.4 we only need to provide a suitable OR-cross composition from one of the three +problems discussed above. We thus arrive at the following. +▶ Proposition 3.7. Let Q be a problem with parameters nQ and ℓQ. If Orthogonal +Vectors resp. 3-Sum parameterized by n (1, 1)-OR-cross-composes into Q, then an O(ℓβ +Q + +nγ +Q)-time algorithm for Q for any 2 > γ > 1 and β < γ/γ−1 refutes the SETH respectively +the 3-Sum-Hypothesis. +If Negative Triangle parameterized by the number of vertices (1, 1)-OR-cross-composes +into Q, then an O(ℓβ +Q + nγ +Q)-time algorithm for Q for any 3 > γ > 1 and β < 2γ/γ−1 refutes +the APSP-Hypothesis. +4 +Applications +We now apply our framework from Theorem 2.4 to several problems from different areas +such as string problems (Section 4.1), computational geometry (Section 4.2), and subgraph +isomorphism (Section 4.3). +4.1 +String problems +We start with Longest Common Subsequence that can be solved in O(n2) time al- +gorithm [16]. Assuming SETH, there is no algorithm solving Longest Common Sub- +sequence in O(n2−ε) time for any ε > 0 [1, 13]. However, Longest Common Sub- +sequence can be solved in O(kn+n log n) time, where k is the length of the longest common +subsequence [37]. +A string over an alphabet Σ is an element from Σ∗. We access the i-th element of a +string x via x[i]. A subsequence of a string x is a string y such that there is an injective, +strictly increasing function f with y[i] = x[f(i)] for all i. A common subsequence of two +strings x and x′ is a string which is a subsequence of both x and x′. For two strings x and y, +we denote their concatenation (i.e. the string starting with x and ending with y) by x ◦ y. +▶ Lemma 4.1. Orthogonal Vectors with logarithmic dimension parameterized by the +number of vectors (1, 1)-OR-cross-composes into Longest Common Subsequence paramet- +erized by the length of the input strings and the length k of the longest common subsequence. +Proof. Let (IOV +1 +, n1), . . . , (IOV +t +, nt) be instances of Orthogonal Vectors with logarithmic +dimension. We denote by di the dimension of IOV +i +. Abboud, Backurs, and Williams [1] gave +a reduction from Orthogonal Vectors to Longest Common Subsequence which, +given an instance of Orthogonal Vectors with n vectors of dimension d, constructs +in n · dO(1) time an equivalent instance of Longest Common Subsequence with strings +of length n · dO(1). We apply this reduction to each instance (IOV +i +, ni) of OV, giving us +an instance ILCS +i += ((x1 +i , x2 +i ), (nLCS +i +, ki)) with ki = O(ni · dO(1) +i +) and nLCS +i += ni · dO(1) +i +. We +assume that ki = kj for every i, j (this can be achieved by appending identical sequences +of appropriate length to strings x1 +i and x2 +i ), and set k := ki. Further, we assume that the +alphabets used for ILCS +i +and ILCS +j +are disjoint for i ̸= j. We define x1 := x1 +1 ◦ x1 +2 ◦ · · · ◦ x1 +t +and x2 := x2 +t ◦ x2 +t−1 ◦ · · · ◦ x2 +1. The OR-cross-composition constructs the instance (x1, x2, k) +(the parameter is (nLCS, k) with nLCS = �t +i=1 nLCS +i +). +We now show correctness of the reduction. First assume that (IOV +i +, ni) is a “Yes”-instance +for some i ∈ [t]. Then x1 +i and x2 +i contain a subsequence y of length ki = k. This subsequence y +is also a subsequence of x1 and x2, so (x1, x2, k) is a “Yes”-instance. Vice versa, assume that + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +13 +(x1, x2, k) is a “Yes”-instance, i.e., x1 and x2 contain a subsequence y of length k. Let i ∈ [t] +such that the first letter of y is contained in x1 +i . We claim that all letters from y are contained +in x1 +i . Since the first letter y[1] of y is contained in x1 +i , no letter of y can be contained +in x1 +j for j < i. For j > i, note that (as any letter from x2 +j only appears in x2 +j and x1 +j) +any letter from x1 +j appears only before x2 +i in x2 and thus would have to appear before y[1] +in y, a contradiction to y[1] being the first letter of y. Thus, y is contained in x1 +i and x2 +i . +Consequently, (x1 +i , x2 +i , k) is a “Yes”-instance, implying that (Ii, ni) is a “Yes”-instance. +Computing the OR-cross-composition can be done in t · maxi∈t ni · (maxi∈[t] di)O(1) = +�O(t maxi∈[t] ni) time and k = O(maxi∈t ni · (maxi∈t di)O(1)) = �O(maxi=1 ni) (we use here +that di ∈ O(log ni)). Further, we have nLCS = �t +i=1 nLCS +i += O(�t +i=1 ni · dO(1) +i +) = �O(�t +i=1). +Thus, it is an (1, 1)-OR-cross-composition. +◀ +We remark that the use of alphabets of non-constant size in the above composition is +probably necessary as it excludes a linear-time computable kernel of polynomial size. In +contrast, Longest Common Subsequence with constant alphabet size parameterized +by solution size admits a linear-time computable polynomial-size kernel (note that the +SETH-based O(n2−ϵ) lower bound also holds for alphabets of size seven [1]): +▶ Proposition 4.2. Longest Common Subsequence with constant alphabet size paramet- +erized by solution size k admits a linear-time computable polynomial-size kernel. +Proof. Consider the following reduction rule which directly leads a polynomial kernel. A +substring of a string x is a string y such that y = (x[i], x[i + 1], . . . , x[j]) for some i < j. +▶ Reduction Rule 4.3. If, for some t ∈ N and i ∈ {1, 2}, a string xi contains a substring x′ +of length at least (k + 1)t over only t different letters and not containing a substring x′′ of +length (k+1)t−1 over at most t−1 different letters, then we can delete all but the first (k+1)t +letters of x′. +First, we argue that the reduction rule is safe. Let x′ be such a substring and t the number +of different letters it contains. Assume that there is a substring x′ of xi of length (k + 1)t +containing only t different letters. By assumption x′ contains no substring of length (k+1)t−1 +which contains at most t − 1 different letters. Thus, in the first (k + 1)t−1 − 1 letters +of x′, each of the t letters appears at least once. More generally, for any i ∈ [k], each +letter appears at least once among x′[(i − 1) · (k + 1)t−1 + 1], . . . , x′[i · (k + 1)t−1]. We +claim that we can delete all letters appearing after position k · (k + 1)t−1 ≤ (k + 1)t: Let +I = (x1, x2, k) be an instance of Longest Common Subsequence. We assume without +loss of generality that the reduction rule can be applied to x1, i.e., x1 contains a substring x′ +of length (k + 1)t which contains only t different letters, and all substrings of x′ containing +only t′ < t different letters have length at most (k + 1)t′. Let Ired = (x1 +red, x2, k) be the +instance arising from applying Reduction Rule 4.3. Clearly, if I is a “No”-instance, then also +Ired is also a “No”-instance. Now assume that I is a “Yes”-instance, i.e., x1 and x2 contain +a common subsequence z of length k. We claim that z is also a subsequence of x1 +red (and +thus a common subsequence of x1 +red and x2), showing that Ired is also a “Yes”-instance. Let +z = (x1[i1], x1[i2], . . . , x1[ik]). Let x1[ir], x1[ir+1], . . . , x1[is] be the intersection of z with x′. +By our above observation, z[r − 1 + i] appears at least once among x′[(i − 1) · (k + 1)t−1 + +1], . . . , x′[i(k+1)t−1] for each i ∈ [s−r], say z[r−1+i] = x′[(i−1)·(k+1)t−1 +ℓi]. Then z = +(x1[i1], . . . , x1[ir−1], x1 +ℓ1, x1 +(k+1)t−1+ℓ2], . . . , x1 +(s−r−1)·(k+1)t−1+ℓs−r], x1[is+1], x1[is+2], . . . , x1[ik]) +is also a subsequence of x1 +red, showing that Ired is a “Yes”-instance. +Next, we analyze the time needed to exhaustively apply Reduction Rule 4.3. + +14 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +▷ Claim 4.4. +For constant alphabet size, Reduction Rule 4.3 can be exhaustively applied in +linear time. +Proof. We process x1 and x2 from left to right. For each subset S of the alphabet, we have a +variable storing the number of consecutive letters from S are before the current letter. If this +number exceeds (k + 1)|S| for some letter, then we delete this letter. Note that whenever we +found a string of length (k + 1)|S| containing only letters from S, then this string does not +contain a substring of length (k + 1)|S|−1 containing only letters from S \ {ℓ} for some ℓ ∈ S +as in this case, the number stored for S \ {ℓ} would have exceeded (k + 1)|S|−1. +The above procedure clearly runs in linear time as the alphabet size is constant. +◁ +After the exhaustive application of Reduction Rule 4.3, the size of the instance is clearly +O(k|Σ|). As |Σ| is constant, this is a polynomial-sized kernel. +◀ +Analogously to Longest Common Subsequence, we derive similar hardness results +for the related problems Longest Common Weakly Increasing Subsequence and +Longest Common Increasing Subsequence. Both problems can be solved in slightly +subquadratic time [7, 21]. +▶ Lemma 4.5. Orthogonal Vectors with logarithmic dimension (1, 1)-OR-cross-composes +into Longest Common (Weakly) Increasing Subsequence parameterized by the +length n of the strings and the length k of the longest common (weakly) increasing sub- +sequence. +Proof. The proof is analogous to the proof of Lemma 4.1. Let (IOV +1 +, n1), . . . , (IOV +t +, nt) be +instances of Orthogonal Vector with logarithmic dimension. We denote by di the +dimension of IOV +i +. Polak [47] and Duraj, Künnemann, and Polak [22] gave a reduction from +Orthogonal Vectors to Longest Common (Weakly) Increasing Subsequence +which, given an instance of Orthogonal Vectors with n vectors of dimension d, constructs +in �O(n·dO(1)) time an instance of Longest Common (Weakly) Increasing Subsequence +of length �O(n·dO(1)). We apply this reduction to each instance (IOV +i +, ni) of OV, giving us an +instance ILCIS +i += (x1 +i , x2 +i , ki) with ki = ni · dO(1) +i +and whose strings have length �O(n · dO(1)). +We assume that ki = kj for every i, j (this can be achieved by appending identical sequences +of increasing numbers of appropriate length to strings x1 +i and x2 +i ), and set k := ki. Let Cmax +be the largest number appearing in any of the instances ILCIS +1 +, . . . , ILCIS +t +. We modify +instance ILCIS +i +by increasing each number by i · Cmax. We define x1 := x1 +1 ◦ x1 +2 ◦ · · · ◦ x1 +t +and x2 := x2 +t ◦ x2 +t−1 ◦ · · · ◦ x2 +1. The OR-cross-composition constructs the instance (x1, x2, k). +The correctness of the reduction can be proven in analogy to Lemma 4.1. +The OR-cross-composition can be computed in �O(t · maxi∈t ni · (maxi∈[t] di)O(1)) = +�O(t maxi∈[t] ni) time and k = maxi∈t ni · (maxi∈t di)O(1) = �O(maxi=1 ni) (we use here that +di ∈ O(log ni)). Thus, it is an (1, 1)-OR-cross-composition. +◀ +Combining Proposition 3.7 and Lemmas 4.1 and 4.5, we get the following result: +▶ Corollary 4.6. Unless the SETH fails, there is no �O(nγ + kβ)-time algorithm for Longest +Common Subsequence or Longest Common (Weakly) Increasing Subsequence +parameterized by solution size k for 1 < γ < 2 and β < +γ +γ−1. +Unless the SETH fails, there is no �O(nγ)-time, �O(kβ)-size kernel for Longest Common +Subsequence or Longest Common (Weakly) Increasing Subsequence parameterized +by solution size for 1 < γ < 2 and β < +γ +2·(γ−1). + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +15 +We remark that the running time lower bounds are tight; the tight upper bound follows +from the known O(kn+n log n) time algorithm for Longest Common Subsequence [37] or +the O(nk log log n + n log n) time algorithm for Longest Common (Weakly) Increasing +Subsequence [43] and Observation 1.1. +4.2 +Computational Geometry +We now turn to problems from computational geometry. We will denote the Euclidean +distance between two points p and q in the plane by dist(p, q). We start by defining the +discrete Fréchet distance. +For a list of points P, a tour through P is a surjective, non-increasing function fP : [2n] → +[n] such that fP (1) = 1 and fP (2n) = n. The input of Discrete Fréchet Distance +contains two lists of points P = (p1, . . . , pn) and Q = (q1, . . . , qn), and a pair (fP , fQ) of +tours through P and Q is called a traversal. We call maxi∈[2n] dist(pfP (i), qfQ(i)) the width +of the traversal. We say that a traversal (fP , fQ) traverses a sublist pi, pi+1, . . . , pj of P (or +qi, qi+1, . . . , qj of Q) if, for some ℓ ∈ [2n] and i∗ ∈ [n]), we have fP (ℓ) = pi and fQ(ℓ) = i∗, +and for any ℓ′ ∈ [j − i], we have fP (ℓ + ℓ′) = i + ℓ′ and fQ(ℓ + ℓ′) = i∗ (or fP (ℓ) = i∗ and +fQ(ℓ) = i, and for any ℓ′ ∈ [j − i], we have fP (ℓ + ℓ′) = i∗ and fQ(ℓ + ℓ′) = i + ℓ′). +Discrete Fréchet Distance can be solved in O(n2) time via dynamic program- +ming [23]. Bringmann [12] gave a linear-time reduction from Orthogonal Vectors2 +to Discrete Fréchet Distance, showing that assuming SETH, Discrete Fréchet +Distance cannot be solved in O(n2−ϵ) time for any ϵ > 0. We use this reduction to get a +(1, 1)-OR-composition for the parameter maxi∈[2n] |fP (i) − fQ(i)|, i.e., the maximum number +of time steps which P may be ahead of Q (or vice versa) in the optimal solution. We will +call the parameter maxi∈[2n] |fP (i) − fQ(i)| the maximum shift of curves fP and fQ. +▶ Lemma 4.7. Orthogonal Vectors admits a (1, 1)-OR-cross-composition into Discrete +Fréchet Distance parameterized by the length of the input lists of points and the maximum +shift in a solution. +Our OR-cross-composition is based on the reduction from Orthogonal Vectors to +Discrete Fréchet Distance by Bringmann [12] which has the following properties. +▶ Proposition 4.8 (Bringmann [12, Section III.A]). Given an instance ((A, B), n) of Or- +thogonal Vectors one can compute in �O(n) time an instance (PA, PB, 1) of Discrete +Fréchet Distance such that +1. the Fréchet distance of PA and PB is 1 if and only if ((A, B), n) is a “Yes”-instance, +2. PA starts at sA := (− 1 +3, 1 +5) and sA appears only at the start of PA, +3. PB starts at sB := (− 1 +3, 0), +4. each point from PB has distance at most 1 from cA := (0, 1 +3), +5. each point from PA has distance at most 1 from cB := sB, +6. ℓA := (− 4 +3, 0) has distance larger than 1 to all points from Q \ {sB}, and +7. ℓB := (− 4 +3, 1 +5) has distance larger than 1 to all points of PA \ {sA}. +8. one point in PB has distance more than 1 from sA. +2 Bringmann [12] reduces from Satisfiability, but his reduction implicitly reduces Satisfiability to +Orthogonal Vectors and then Orthogonal Vectors to Discrete Fréchet Distance. + +16 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Proof of Lemma 4.7. Let (IOV +1 += (A1, B1), n1), . . . , (IOV +t += (At, Bt), nt) be instances of +Orthogonal Vectors. To simplify notation, we assume that IOV +1 +, . . . , IOV +t +all are of the +same dimension d and that there is some n ∈ N such that ni = n for all i ∈ [t]. Further, +we assume that no two vectors of different instances of IOV +t +are orthogonal. This can be +achieved by increasing the dimension by 2 log t and using the additional dimensions to add +to each a ∈ Ai the binary encoding of i and its “inverse” (i.e., replacing 0’s by 1’s and vice +versa). To each b ∈ Bi, we append the “inverse” of the binary encoding of i and the binary +encoding of i. +We apply the reduction from Proposition 4.8 to each instance (IOV +i +, n), resulting in +an instance IDFD +i += (P i +A, P i +B, 1) of Discrete Fréchet Distance. We set nDFD +i +:= |P i +A| +to be the length of P i +A. For i ∈ [t], let si +A := sA, si +B := sB, ci +A := cA, ℓi +A := ℓA, ℓi +B := ℓB +(where sA, sB, cA, cB, ℓA, and ℓB are be defined as in Proposition 4.8). Further, let IDFD +t+1 +be an instance arising from applying Proposition 4.8 to an arbitrary “No”-instance of +Orthogonal Vectors. From this, we construct an instance (IDFD = (PA, PB, 1), (2t + +�t+1 +i=1 nDFD +i +, 2 maxi∈[t+1] nDFD +i +)) of Discrete Fréchet Distance parameterized by the +length of the point lists and the maximum shift as follows. We set PA := ⃝i∈[t](P i +A ◦ ℓi +A ◦ +ci +A) ◦ P t+1 +A +and PB := ⃝i∈[t+1](P i +B ◦ ℓi +B) ◦ cB. To ensure that PA and PB have the same +number of points, we append an appropriate number of copies of cB at the end of PB. +On an intuitive level, the idea of the construction is as follows: The asymmetry in the +construction allows to traverse an arbitrary part of PA while being in si +B = cB in PB for +some i ∈ [t + 1]. However, the converse is not true and, thus, traversing in PB as far as +possible is desirable. In order to traverse ℓi +B we have to be in sj +A or ℓj +A for some i ∈ [t + 1]. +We cannot stay in sj +A since by Property 8 at least one point in PBi has distance larger than +one from si +A. Hence, there are two options: The traversal contains (sj +A, ℓi +B) for i, j ∈ [t + 1]. +As Property 8 forces us out of the sA-points, we will have j > i when following this option. +Thus, only using this option results in getting stuck at the end as the last ℓBt+1 cannot +be traversed. To prevent this and “get ahead” in PB, we have to use the second option at +some point: The traversal contains (ℓi +A, ℓi +B) for some i ∈ [t]. We prove that in this case PAi +and PBi have to be traversed simultaneously, that is, IDFD +i +is a yes-instance. The benefit of +the second option is that we can follow (ℓi +A, ℓi +B) by (ci +A, si+1 +B ) and then traverse PBi+1. Then, +being “ahead” in PB, we can use the first option for the rest of the traversal. +We now show the correctness of the OR-cross-composition. +▷ Claim 4.9. +If there is some i∗ ∈ [t] such that there are a ∈ Ai∗ and b ∈ Bi∗ which +are orthogonal, then there is a traversal T of (PA, PB) of width at most 1 and maximum +shift 2 maxi∈[t+1] nDFD +i +. +Proof. We define a traversal as follows: We start with (s1 +A, s1 +B). For each i < i∗, we define the +following traversal Ti (which first traverses P i +A ◦ℓi +A ◦ci +A and afterwards P i +B ◦ℓi +B): Traverse PA +until the end of P i +A. By Property 5, this traversal so far has width 1. Afterwards, go +to (ℓi +A, si +B), followed by (ci +A, si +B). Then traverse P i +B. By Property 4, this traversal still has +width 1. We continue with (si+1 +A , ℓi +B) and (si+1 +A , si+1 +B ). +By Property 1, there is a traversal of (P i∗ +A , P i∗ +B ) of width 1. We continue with this +traversal. We continue with (ℓi∗ +A, ℓi∗ +B) and (ci∗ +A, si∗+1 +B +). Then, we traverse P i∗+1 +B +. This is +followed by (si∗+1 +A +, ℓi∗+1 +B +) and (si∗+1 +A +, si∗+2 +B +). Thus, we are now at the start of P i∗+1 +A +in PA +and P i∗+2 +B +in PB. +Afterwards, for t ≥ i > i∗, we traverse P i +A and P i+1 +B +as in the first step: First, traverse P i +A +until the end of P i +A. By Property 5, this traversal so far has width 1. Afterwards, go +to (ℓi +A, si+1 +B ), followed by (ci +A, si+1 +B ). Then traverse P i+1 +B +. By Property 4, this traversal still +has width 1. We continue with (si+1 +A , ℓi+1 +B ) and (si+1 +A , si+2 +B ). + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +17 +Finally, we are at st+1 +A +and the end of P t+1 +B +. We go to (st+1 +A , ℓt+1 +B ), followed by (st+1 +A , cB) +and afterwards traverse P t+1 +A +. By Property 5, the constructed traversal still has width 1. +Thus, we found a traversal of (PA, PB) of width at most 1. It is easy to verify that this +traversal has a maximum shift of 2 maxi∈[t+1] nDFD +i +. +◁ +▷ Claim 4.10. +If there is no i∗ ∈ [t] such that there are a ∈ Ai∗ and b ∈ Bi∗ which are +orthogonal, then the Fréchet distance of PA and PB is larger than 1. +Proof. Assume towards a contradiction that there is a traversal T of width 1. First, we show +that T does not reach the first point si +B of P i +B for each i ∈ [t + 1] before it reaches the first +point si +A of P i +A. +Assume towards a contradiction that si +B is reached before si +A and that i is minimal. +Clearly i > 1 as T starts with (s1 +A, s1 +B). By choice of i we have that si−1 +A +is reached before or +at the same time as si−1 +B . Note that before si +B is reached (which is before si +A is reached) ℓi−1 +B +has to be reached. By Property 7, the only points from PA at distance at most one to ℓi−1 +B +are ℓj +A and sj +A for j ∈ [t]. By choice of i it follows that (si−1 +A , ℓi−1 +B ) or (ℓi−1 +A , ℓi−1 +B ) is part of T. +We show contradictions in both cases: Since si−1 +A +is reached before or simultaneously to si−1 +B , +it follows from Property 8 that (si−1 +A , ℓi−1 +B ) is not part of T. Note that (ℓi−1 +A , ℓi−1 +B ) cannot be +part of T as the immediate predecessor consists of the respective last points in PAi and PBi: +As si−1 +A +is reached before si−1 +B , this would imply PAi and PBi having a Fréchet distance of at +most 1, contradicting Property 1 and PAi and PBi containing no orthogonal vector. +Thus, for each j ∈ [t + 1] we have that T reaches sj +A before sj +B. Property 1 implies that +there is no traversal of width 1 that reaches the last point of P t+1 +B +as well as the last point +of P t+1 +A +. Thus, T must reach the last point of P t+1 +B +while only partially traversing P t+1 +A +. +Afterwards, T must move to ℓt+1 +B , implying that T is at st+1 +A +at this point (as all other points +have distance larger than 1 to ℓt+1 +B ). However, by Property 8, there is some point in P t+1 +B +of +distance larger than 1 to st+1 +A , a contradiction to T traversing through P t+1 +B +while staying +at st+1 +A . +◁ +The OR-cross-composition can clearly be computed in �O(t maxi∈[t] ni) time. The correct- +ness follows from Claims 4.9 and 4.10. The maximum shift of the constructed instance is +O(n) by Claim 4.9. Consequently, it is an (1, 1)-OR-composition. +◀ +Combining Proposition 3.7 and Lemma 4.7, yields the following: +▶ Corollary 4.11. Unless the SETH fails, there is no �O(nγ + ℓβ)-time algorithm Discrete +Fréchet Distance parameterized by maximum shift ℓ for 1 < γ < 2 and β < +γ +γ−1. +Unless the SETH fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for Discrete Fréchet +Distance parameterized by maximum shift for 1 < γ < 2 and β < +γ +2·(γ−1). +It is easy to extend the O(n2)-algorithm for Discrete Fréchet Distance [23] to run +in O(nℓ) time, where ℓ is the minimal (over all solutions) maximum shift of an optimal +solution. This together with Observation 1.1 shows that the running time lower bound from +Corollary 4.11 is tight. +We now give lower bounds for another problem from computational geometry, Planar +Motion Planning, based not on the hardness of Orthogonal Vectors but on the +hardness of 3-Sum. Planar Motion Planning can be solved in �O(n2) time [45]. Assum- +ing the 3-Sum conjecture, Planar Motion Planning cannot be solved in O(n2−ϵ) for +any ϵ > 0 [31]. We say a segment is in the vicinity of another segment if they have distance +at most the length of the rod. + +18 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Figure 2 Left: Exemplary illustration of an instance of Planar Motion Planning constructed +by the reduction from 3-Sum from Gajentaan and Overmars [31]. Right: An example for the +OR-cross-composition of three instances of 3-Sum into Planar Motion Planning. The goal +position of the rod is surrounded by red. +▶ Lemma 4.12. 3-Sum (1, 1)-OR-cross-composes into Planar Motion Planning para- +meterized by the maximum number of segments any segment has in its vicinity. +Proof sketch. Let I3-Sum +1 +, . . . , I3-Sum +t +be instances of 3-Sum. We denote by ni the number of +numbers of I3-Sum +1 +. Gajentaan and Overmars [31] gave a reduction from 3-Sum to Planar +Motion Planning which, given an instance of 3-Sum with n numbers, constructs in �O(n) +time an instance of Planar Motion Planning where the rod initially is in a large “upper” +rectangle and has to reach a “lower” rectangle through a narrow passage (see Figure 2 for a +proof by picture). We apply this reduction to each instance I3-Sum +i +of 3-Sum, giving us an +instance IPMP +i +with �O(ni) many segments. From these instances, we construct an instance +of Planar Motion Planning as follows: We identify the large rectangles in which the +rod starts from each IPMP +i +. The narrow passages are copied next to each other (if the large +starting rectangle is not wide enough, then we make it wider. +The correctness of the reduction is obvious. +Any segment from an instance I3-Sum +i +has distance at most the length of the rod except +for the bounding boxes. By splitting the segments of the bounding box into many small +segments not longer than the rod, we get that for each segment s there are at most O(n) +segments whose distance to s is at most the length of the rod. +◀ +Combining Proposition 3.7 and Lemma 4.12 yields the following: +▶ Corollary 4.13. Unless the 3SUM-hypothesis fails, there is no �O(nγ + ℓβ)-time algorithm +Planar Motion Planning parameterized by the maximum number ℓ of segments any +segment has in its vicinity for β < +γ +γ−1. +Unless the 3SUM-hypothesis fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for Planar +Motion Planning parameterized by the maximum number ℓ of segments any segment has +in its vicinity for β < +γ +2·(γ−1). +The lower bound for the running time is tight by the O(K2 log n) time algorithm [49] +(where K is the number of segment pairs whose distance is less than the length of the rod), +the observation that K2 ≤ n · ℓ, and Observation 1.1. +4.3 +Graph Problems +We now turn to graph problems. First, we consider Minimum Weight k-Clique paramet- +erized by the maximum size of a connected component. + +Klaus Heeger, André Nichterlein, Rolf Niedermeier +19 +▶ Proposition 4.14. Minimum Weight k-Clique (1, 1)-OR-cross-composes into Minimum +Weight k-Clique parameterized by the maximum size of a connected component. +Proof. Let G1, . . . , Gt be instances of Minimum Weight k-Clique. The cross-composition +just computes the disjoint union G of G1, . . . , Gt. +Clearly, the cross-composition is correct, runs in linear time, and the maximum size of a +connected component is bounded by the maximum size of G1, . . . , Gs. +◀ +As Negative Triangle is the special case of Minimum Weight k-Clique, combining +Proposition 3.7 and Proposition 4.14 yields the following lower bound: +▶ Corollary 4.15. Unless the APSP-hypothesis fails, there is no �O(nγ + ℓβ)-time algorithm +for Negative Triangle parameterized by the maximum size ℓ of a connected component +for 1 < γ < 3 and β < +2·γ +γ−1. +Unless the APSP-hypothesis fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for Negative +Triangle parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 +and β < +2·γ +3·(γ−1). +An algorithm running in O(ℓ2n) where ℓ is the maximum size of a connected component is +trivial. This together with Observation 1.1 implies that the running time lower bound is +tight. +Next, we turn to 2nd Shortest Path which can be solved in �O(mn) [44] or in O(Mnω) +time (where M is the largest edge weight) [34]. If the graph is undirected [39] or one aims to +find a 2nd shortest walk [24], then there is a quasi-linear-time algorithm. For unweighted +directed graphs, the problem can be solved in �O(m√n) time [48]. An ϵ-approximation can +be computed in �O( m +ϵ ) time [9]. +▶ Lemma 4.16. Negative Triangle (1, 1)-OR-cross-composes into 2nd Shortest Path +parameterized by directed feedback vertex number. +Proof. Let (INT +1 +, n1), . . . , (INT +t +, nt) be instances of Negative Triangle. +We assume +without loss of generality that each instance of Negative Triangle has the same num- +ber n of vertices, i.e., ni = n for all i ∈ [t], and that the largest absolute value of an +edge weight is the same for all instances. Vassilevska Williams and Williams [51] gave a +linear-time reduction from Negative Triangle to 2nd Shortest Path which, given +an instance (INT +i += (Gi, wi), n) of Negative Triangle, creates an instance I2SP +i +of 2nd +Shortest Path as follows: For each vertex v ∈ V (Gi), add three vertices av, bv, and +cv. Further, add two vertices si and ti as well as a path si, pi +1, pi +2, . . . , pi +n, ti, all of whose +edges have weight 0. Further, there are edges (pi, av), (av, bu), (bu, cv), and (cv, pi) for +every i ∈ [n], u, v ∈ V (Gi). All these edges have positive weight (depending on the edge- +weights in E(Gi)). Identifying si with sj, ti with tj, and pi +ℓ with pj +ℓ for all i, j ∈ [t] and j ∈ [ℓ] +then results in one instance of 2nd Shortest Path with a directed feedback vertex set +of size n, namely {p1, . . . , pn}. As the constructed instance is equivalent to the disjunction +of I2SP +1 +, . . . , I2SP +t +(it is never beneficial to leave s, p1, . . . , pn, t more than once as all edges +leaving this path have positive weight), we have a (1, 1)-OR-cross-composition. +◀ +Combining Proposition 3.7 with Lemma 4.16 yields the following: +▶ Corollary 4.17. Unless the APSP-hypothesis fails, there is no �O(nγ + ℓβ)-time algorithm +for 2nd Shortest Path parameterized by directed feedback vertex set ℓ parameterized by +the maximum size ℓ of a connected component for 1 < γ < 3 and β < +2γ +γ−1. + +20 +Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions +Unless the APSP-hypothesis fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for 2nd +Shortest Path parameterized by directed feedback vertex set parameterized by the maximum +size ℓ of a connected component for 1 < γ < 3 and β < +2γ +3(γ−1). +In contrast to the other problems studied in this paper, we do not know whether the +running time lower bounds are tight. +4.4 +Triangle Collection +Last, we consider the Triangle Collection problem. Triangle Collection can trivially +be solved in O(n3) time, but does not admit an O(n3−ϵ) time algorithm assuming SETH, +the 3-Sum conjecture, or the APSP conjecture [3]. For this problem, we were unable to +apply our framework directly, i.e., find an OR-cross-composition from an OR-decomposable +problem to Triangle Collection. However, we can still get a lower bound in a very +similar fashion by combining decomposition and composition into one step. The difference is +that the decomposition of Triangle Collection that we use in the proof of the following +result does not admit the “OR”-property. +▶ Proposition 4.18. Unless Orthogonal Vectors, 3-Sum, or APSP-hypothesis fails, +there is no �O(nγ+ℓβ)-time algorithm for Negative Triangle parameterized by the maximum +size ℓ of a connected component for 1 < γ < 3 and β < +2·γ +γ−1. +Unless Orthogonal Vectors, 3-Sum, or APSP-hypothesis fails, there is no �O(nγ)- +time, �O(ℓβ)-size kernel for Triangle Collection parameterized by the maximum size ℓ of +a connected component for 1 < γ < 3 and β < +2·γ +3·(γ−1). +Proof. Fix 1 < γ < 3. Let G be in an instance of Triangle Collection. Partition V (G) +into z := n +λ/3+λ sets V1, . . . , Vz of size q := n +3/(3+λ), where λ := β/γ−1. For each (i, j, k) ∈ [z]3, +let G(i,j,k) be the graph induced by Vi ∪ Vj ∪ Vk. Finally, let H be the union of all G(i,j,k). +Note that H corresponds to the output of the OR-decomposition in our framework. Clearly +G has a triangle collection if and only if H has. Further, H can be computed in �O(q · z3) = +�O(n +3(λ+1)/(3+λ)) time. +Now assume that there is a �O(nγ + ℓβ)-algorithm for Triangle Collection with +β < +2·γ +γ−1 and apply this algorithm to H. This clearly solves H. The running time for this +step is �O((z3 · q)γ + qβ) = �O(nγ·(3·(λ+1)/(3+λ)) + nβ·3/(3+λ)) time. Note that γ·3·(λ+1)/(3+λ) = +3 · +β/γ +2+β/γ < 3. Further, it holds that +β · 3/(3+λ) = 3· +β +2 + β/γ = 3· +βγ +2γ + β = 3· +γ +2γ/β + 1 < 3· +γ +2γ/ +� +2γ +(γ−1) +� ++ 1 = 3· +γ +(γ − 1) + 1 = 3 +where we used the assumption β < +2·γ +γ−1 for the inequality. Consequently, we can solve +Triangle Collection in O(n3−ϵ) time for some ϵ > 0. +The statement for kernels follows from the observation that an O(ℓβ)-size, O(nγ)-time +kernel for Triangle Collection directly implies an O(nγ + ℓ3·β)-time algorithm for +Triangle Collection. +◀ +As an O(nℓ2) time algorithm for Triangle Collection is trivial, it follows from +Observation 1.1 that the running time lower bound is tight. +5 +Conclusion +We introduced a framework for extending conditional running time lower bounds to para- +meterized running time lower bounds and applied it to various problems. Beyond the clear + +REFERENCES +21 +task to apply the framework to further problems there are further challenges for future work. +For example, can we get “AND-hard” problems so that we can use AND-cross compositions +similar to the ones used to exclude compression [19]? 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URL https://doi.org/10.1137/15M1024524. + diff --git a/FNAyT4oBgHgl3EQf4_q8/content/tmp_files/load_file.txt b/FNAyT4oBgHgl3EQf4_q8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54949076a5ac592e302665f0ed1df775a3023f1d --- /dev/null +++ b/FNAyT4oBgHgl3EQf4_q8/content/tmp_files/load_file.txt @@ -0,0 +1,1343 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf,len=1342 +page_content='Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Klaus Heeger � Technische Universität Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Algorithmics and Computational Complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Germany André Nichterlein � Technische Universität Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Algorithmics and Computational Complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Germany Rolf Niedermeier Technische Universität Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Algorithmics and Computational Complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Germany Abstract We provide a general framework to exclude parameterized running times of the form O(ℓβ + nγ) for problems that have polynomial running time lower bounds under hypotheses from fine-grained complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Our framework is based on cross-compositions from parameterized complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We (conditionally) exclude running times of the form O(ℓγ/(γ−1)−ε + nγ) for any 1 < γ < 2 and ε > 0 for the following problems: Longest Common (Increasing) Subsequence: Given two length-n strings over an alphabet Σ (over N) and ℓ ∈ N, is there a common (increasing) subsequence of length ℓ in both strings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Discrete Fréchet Distance: Given two lists of n points each and k ∈ N, is the Fréchet distance of the lists at most k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Here ℓ is the maximum number of points which one list is ahead of the other list in an optimum traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Planar Motion Planning: Given a set of n non-intersecting axis-parallel line segment obstacles in the plane and a line segment robot (called rod), can the rod be moved to a specified target without touching any obstacles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Here ℓ is the maximum number of segments any segment has in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, we exclude running times O(ℓ2γ/(γ−1)−ε + nγ) for any 1 < γ < 3 and ε > 0 for: Negative Triangle: Given an edge-weighted graph with n vertices, is there a triangle whose sum of edge-weights is negative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Here ℓ is the order of a maximum connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Triangle Collection: Given a vertex-colored graph with n vertices, is there for each triple of colors a triangle whose vertices have these three colors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Here ℓ is the order of a maximum connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 2nd Shortest Path: Given an n-vertex edge-weighted directed graph, two vertices s and t, and k ∈ N, has the second longest s-t-path length at most k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Here ℓ is the directed feedback vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Except for 2nd Shortest Path all these running time bounds are tight, that is, algorithms with running time O(ℓγ/(γ−1) + nγ) for any 1 < γ < 2 and O(ℓ2γ/(γ−1) + nγ) for any 1 < γ < 3, respectively, are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Our running time lower bounds also imply lower bounds on kernelization algorithms for these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Graph algorithms analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Theory of computation → Parameterized complexity and exact algorithms Keywords and phrases FPT in P, Kernelization, Decomposition Funding Klaus Heeger: Supported by DFG project NI 369/16 “FPTinP”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Acknowledgements In memory of Rolf Niedermeier, our colleague, friend, and mentor, who sadly passed away before this paper was finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We thank the anonymous reviewers for their thoughtful and constructive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='00797v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='DS] 2 Jan 2023 2 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions 1 Introduction In recent years, many results in Fine-Grained Complexity showed that many decade-old textbook algorithms for polynomial-time solvable problems are essentially optimal: Consider as an example Longest Common Subsequence (LCS) where, given two input strings with n characters each, the task is to find a longest string that appears as subsequence in both input strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The classic O(n2)-time algorithm is often taught in introductory courses to dynamic programming [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Bringmann and Künnemann [13] and Abboud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' [1] independently showed that an algorithm solving LCS in O(n2−ε) time for any ε > 0 would refute the Strong Exponential Time Hypothesis (SETH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Such conditional lower bounds have been shown for many polynomial-time solvable problems in the recent years [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' One approach to circumvent such lower bounds is “FPT in P” [2, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For Longest Common Subsequence there is a (quite old) parameterized algorithm running in O(kn + n log n) time, where k is the length of the longest common subsequence [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, if k is small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' O(n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='99)), then the O(n2) barrier can be broken (without refuting the SETH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A natural question is whether we can do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' As k ≤ n, an algorithm running in O(k1−εn) time for any ε > 0 would break the SETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' However, there are no obvious arguments excluding a running time of O(k2+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In fact, such additive running times are not only desirable (as again, for small k this would be faster than even O(kn)) but also quite common in parameterized algorithmics by employing kernelization: For Longest Common Subsequence the question would be whether there are linear-time applicable data reduction rules that shrink the input to size O(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then we could simply apply the textbook algorithm to solve Longest Common Subsequence in overall O(k2 + n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Kernelization is well-studied in the parameterized community [5, 29] and also effective in practice for polynomial-time solvable problems such as Maximum Matching [40] or Minimum Cut [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In this work, we prove that Longest Common Subsequence does not admit an O(k2 + n)-time algorithm assuming the SETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This also implies that no such kernelization algorithm as mentioned above is likely to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' More precisely, we provide a general framework to (conditionally) exclude algorithms with running time O(kβ + nγ) for problems admitting conditional running time lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We apply our framework to various string and graph problems as well as problems from computational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' As a result, we get tight trade-offs between β and γ showing that the trivial trade-offs are often the best one can hope for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For example, the algorithm of Hirschberg [37] for Longest Common Subsequence implies an O(k3 + n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5)-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We show that any algorithm running in O(k3−ε + n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5) time for ε > 0 refutes the SETH (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2 for a more detailed overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 Related work Fine-grained complexity is an active field of research with hundreds of papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We refer to the survey of Vassilevska Williams [50] for an overview of the results and employed hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Over the last couple of years there has been a lot of work in the direction of “FPT in P” for various problems such as Maximum Matching [17, 20, 28, 35, 38, 40, 42, 46], Hyperbolicity [17, 27], and Diameter [1, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Parameterized lower bounds are rare in this line of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Certain linear-time reductions can be used to exclude any kind of meaningful FPT-running times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' this is also known as General-Problem-Hardness [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Closer to our work are Fluschnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' They provide lower bounds for strict kernelization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' kernels where the parameter is not allowed to increase) for subgraph detection problems such as Negative Weight Triangle and Triangle Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Conceptually, they use the Klaus Heeger, André Nichterlein, Rolf Niedermeier 3 diminisher-framework [15, 25] which was originally developed to exclude polynomial-size strict kernels under the assumption P ̸= NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The basic idea is to iteratively apply a diminisher (an algorithm that reduces the parameter at a cost of increasing the instance size) and an (assumed) strict kernel (to shrink and control the instance size) to an instance I of an NP-hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' After a polynomial number of rounds, this overall polynomial-time algorithm will return a constant size instance which is equivalent to I, thus arriving at P = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Fluschnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' [26] applied the same idea to polynomial-time solvable problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In contrast, we rely and adjust the composition-framework by Bodlaender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' [10] which was developed to exclude (general) polynomial-size kernels under the stronger assumption NP ̸⊆ coNP / poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The composition framework works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consider the example of the NP-hard problem Negative-Weight Clique: Given an edge-weighted graph G and an integer k, does G contain a negative-weight k-clique, that is, a clique on k vertices where the sum of the edge-weights of the edges within the clique is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (G1, k), (G2, k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (Gt, k) be several instances of Negative-Weight Clique with the same k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Clearly, the graph G obtained by taking the disjoint union of all Gi contains a negative-weight k-clique if and only if some Gi contains a negative-weight k-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, the largest connected component of G has order maxi∈[t]{|V (Gi)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Now assume that Negative-Weight Clique has a kernel of size O(ℓc) for some constant c where ℓ is the order of a largest connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By choosing t = kc+1, it follows that kernelizing the instance (G, k) yields an instance of size less than ℓ, that is, less bits than the number of instances encoded in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Given the NP-hardness of Negative-Weight Clique such a compression seems challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' indeed it would imply NP ⊆ coNP / poly [30], which in turn results in a collapse of the polynomial hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Compositions and their extension cross-composition [11] are extensively employed in the parameterized complexity literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, to exclude kernels whose size is bounded by polynomials of a specific degree adjustments have been made to the composition frame- work [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Parameter trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For several of our running time lower bounds we have tight upper bounds that are derived from a simple case distinction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 (folklore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If a problem P admits an �O(ℓβnγ)-time algorithm1, then it admits for every λ > 0 an �O(ℓβ+ γ·β λ + nγ+λ)-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If ℓ ≤ n λ β , then the �O(ℓβnγ)-time algorithm runs in �O(nγ+λ) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Otherwise n ≤ ℓ β λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then the �O(ℓβnγ)-algorithm then in �O(ℓβ+ γ·β λ ) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2 Our Results & Technique We provide a composition-based framework to establish parameterized running time lower bounds and apply the framework to Longest Common Subsequence, Longest Common (Weakly) Increasing Subsequence, Discrete Fréchet Distance, Planar Motion Planning, Negative Triangle, and 2nd Shortest Path (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3 for the problem definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Using similar ideas we obtain running time lower bounds for Triangle Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For all these problems except 2nd Shortest Path parameterized by the directed feedback vertex set there exist matching running time upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We refer to Table 1 for an overview on the specific results and the parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, we 1 The � O hides polylogarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 4 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Table 1 Overview of achievable running times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The upper part of the table lists the results for four problems that can be solved in O(n2) time but under SETH or 3SUM-hypothesis not in O(n2−ε) time for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The lower part lists results for three graph problems that, based on the APSP-hypothesis, do not admit O(n3−ε)-time algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The parameterized upper and lower bounds are visualized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' problems Longest Common Subsequence ℓˆ=solution size Longest Common (Weakly) Increasing Subsequence ℓˆ=solution size Discrete Fréchet Distance ℓˆ=maximum shift Planar Motion Planning ℓˆ=max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' number of segments in the vincinty of any segment results upper bounds lower bounds o(n2) [6, 7, 33, 52] no O(n2−ε) assuming SETH / 3SUM [1, 12, 31] � O(ℓn) [37, 43, 49] � O(ℓγ/(γ−1) + nγ) for each γ > 1 no O(ℓγ/(γ−1)−ε + nγ) for any γ < 2 (Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1) (Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='11, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='13) problems Negative Triangle ℓˆ=size of maximum component Triangle Collection ℓˆ=size of maximum component 2nd Shortest Path (only lower bounds) ℓˆ=directed feedback vertex set size results upper bounds lower bounds O(n3/2Θ(log0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5 n)) [14, 53] no O(n3−ε) assuming APSP [51] � O(ℓ2n) (folklore) � O(ℓ2γ/(γ−1) + nγ) for each γ > 1 no O(ℓ2γ/(γ−1)−ε + nγ) for any γ < 3 (Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1) (Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='17 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='18) visualize in Figure 1 the trade-offs in the running times that are (im-)possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We adjust the cross-composition framework to obtain lower bounds for poly- nomial time solvable problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' As an example, consider Negative-Weight Triangle, that is, Negative-Weight Clique with k fixed to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assuming the APSP-hypothesis, Negative-Weight Clique cannot be solved in O(n3−ε) time [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The first difference to the cross-composition framework is that we start with one instance G of Negative-Weight Triangle which we then decompose into many small instances as follows: Partition the vertices V (G) of G into z many sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Vz of size n/z, where z is chosen depending on the running time we want to exclude (see the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3 in Section 3 for the actual formula specifying z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then, we create z3 instances of Negative-Weight Clique: for each (i, j, k) ∈ [z]3 take the graph G[Vi ∪ Vj ∪ Vk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Clearly, we have that G contains a negative-weight triangle if and only if at least one of the created instances contains a negative-weight triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Next, we apply the composition as explained above (the disjoint union) for Negative- Weight Clique to obtain an instance G′ with n′ = z3 · n/z = z2n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that the size ℓ of a largest connected component in G′ is at most 3n/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Hence, an algorithm running in time O(nγ + ℓβ) for Negative-Weight Triangle solves G′ in time O(z2γnγ + (3n/z)β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By carefully choosing z as a function in n, β, and γ, we get that this is in O(n3−ε) for various combinations of γ and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The property that Negative-Weight Triangle can be decomposed as above is not unique to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In fact, this has been observed already: “Many problems, like SAT, Klaus Heeger, André Nichterlein, Rolf Niedermeier 5 1 2 3 4 1 2 3 4 5 γ β O(n2)-solvable 1 2 3 4 1 2 3 4 5 γ β O(n3)-solvable Figure 1 Overview on the possible (in green) and unlikely (in red) trade-offs in running times of the form O(nγ + ℓβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Left: First category for O(n2)-time solvable problems (upper part in Table 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' right: second category for O(n3)-time solvable problems (lower part in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' have a simple self-reduction proving that the “Direct-OR” version is hard, assuming the prob- lem itself is hard” [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Our framework formalizes this notion of decomposition (see Section 2 for a definition) and adjusts the cross-composition definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We furthermore show that commonly used “hard” problems such as Orthogonal Vectors, 3-Sum, and Negative- Weight k-Clique are decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, it remains to show cross-compositions in order to apply our framework and obtain lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3 Preliminaries and Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Problem definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For ℓ ∈ N we set [ℓ] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Orthogonal Vectors Input: Two size-n sets A, B ⊆ {0, 1}d for some d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Are there a ∈ A and b ∈ B so that a and b are orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', for each i ∈ [d], the i-th coordinate of a or the i-th coordinate of b is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We denote the restriction of Orthogonal Vectors to instances with d ≤ O(log n) as Orthogonal Vectors with logarithmic dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3-Sum Input: An array A of n integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Are there i, j, h ∈ [n] such that A[i] + A[j] + A[h] = 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Negative-Weight k-Clique Input: An edge-weighted graph G on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Does G contain a k-clique of negative weight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Longest Common Subsequence Input: Two strings x1 and x2 of length n over an alphabet Σ and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Decide whether there is a common subsequence of length k of x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Longest Common (Weakly) Increasing Subsequence Input: Two strings x1 and x2 of length n over N and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Decide whether there is a common subsequence y of length k of x1 and x2 with y[i] < y[i + 1] (y[i] ≤ y[i + 1]) for all i ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 6 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Discrete Fréchet Distance Input: Two lists P = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , pn), Q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , qn) of points in the plane and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Is the Fréchet distance of P and Q at most k, that is, there are two surjective, non-decreasing functions fP , fQ : [2n] → [n] with fP (1) = 1 = fQ(1), fP (2n) = n = fQ(2n) and maxi∈[2n] dist(pfP (i), qfQ(i)) ≤ k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Planar Motion Planning Input: A set of n non-intersecting, non-touching, axis-parallel line segment obstacles in the plane and a line segment robot (a rod or ladder), a given source, and a given goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Can the rod be moved (allowing both translation and rotation) from the source to the goal without colliding with the obstacles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 2nd Shortest Path Input: An n-vertex edge-weighted directed graph G, vertices s and t, and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Has the 2nd-shortest s-t-path length at most k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Triangle Collection Input: A vertex-colored graph G on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: For each combination of three colors, does G contain a triangle whose vertices are colored with three colors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The conditional lower bounds in this work are based on SETH, 3-Sum-, and the APSP-Hypothesis (see Vassilevska Williams [50] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Hypothesis 1 (SETH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For every ε > 0 there exists a k ∈ N such that k-SAT cannot be solved in O(2(1−ε)n) time, where n is the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Hypothesis 2 (3SUM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3-Sum on n integers in {−n4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , n4} cannot be solved in O(n2−ε) time for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Hypothesis 3 (APSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' All Pairs Shortest Path on n-vertex graphs with polynomial edge weights cannot be solved in O(n3−ε) time for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Parameterized Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In many of the above problems, n is not the input size but a parameter and the input size is bounded by nO(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A parameterized problem is a set of instances (I, p) ∈ Σ∗ × Σ∗, where Σ denotes a finite alphabet, I denotes the classical instance and p the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A kernelization is a polynomial-time algorithm that maps any instance (I, p) to an equivalent instance (I′, p′) (the kernel) such that |I′| + p′ ≤ f(p) for some computable function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If f is a polynomial, then (I′, p′) is a polynomial-size kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In this work we restrict ourselves to the following: Either p = n (p is a single parameter) or p = (n, ℓ) (p is a combined parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, both n and ℓ are always nonnegative integers, n is related to the input size but ℓ is not (ℓ can be seen as “classical” parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 2 Framework Our framework has the following three steps (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2 for a high-level description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Start with an instance (I, nP) of a “hard” problem P and decompose it into the disjunction of t instances (I1, n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (It, nt) of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In Section 3, we provide such decompositions for the frequently used hard problems 3-Sum, Orthogonal Vectors, and Negative Weight k-Clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Klaus Heeger, André Nichterlein, Rolf Niedermeier 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Compose (I1, n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (It, nt) into one instance (J, (n, ℓ)) of the “target” problem using an OR-cross-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This step has to be done for the application at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Apply the assumed �O(nγ + ℓβ)-time algorithm to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If the combination of γ and β is small enough, then the resulting algorithm will be faster than the lower bound for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To give a more formal description of our framework, we first define decompositions and cross-compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that all mentioned problems are parameterized problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 (OR-decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For α > 1 an α-OR-decomposition for a problem P is an algorithm that, given λ > 0 and an instance (I, n) of P, computes for some α′ < α in �O(nα′) time t ∈ �O(nαλ/(α+λ)) many instances (I1, n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (It, nt) of P such that (I, n) ∈ P if and only if (Ii, ni) ∈ P for some i ∈ [t], and ni ∈ �O(nα/(α+λ)) for all i ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We say a problem P is α-OR-decomposable if there exists an α-OR-decomposition for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For some problems it is easier to show OR-decomposability than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, using appropriate reductions to transfer OR-decomposability can be desirable (we do so in Section 3 when showing that 3-Sum is 2-OR-decomposable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Quasi-linear time reductions that do not increase the parameter to much are one option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To this end, we say a reduction that given an instance (IP, nP) of P produces an instance (IQ, nQ) of Q is parameter-preserving if nQ ∈ �O(nP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let P and Q be two problems such that there are quasi-linear-time parameter-preserving reductions from P to Q and from Q to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then for any α > 1, P is α-OR-decomposable if and only if Q is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assume that P is α-OR-decomposable (the case that Q is α-OR-decomposable is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We now give an α-OR-decomposition for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Given an instance (IQ, nQ) and λ > 0, we first reduce (IQ, nQ) to an instance (IP, nP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Afterwards, we apply the α-OR- decomposition from P, resulting in instances (IP 1 , nP 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (IP t , nP t ) of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Finally, we reduce each instance (IP i , nP i ) to an instance (IP i , nP i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This clearly is an α-OR-decomposition for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ For the second step of our framework, we introduce fine-grained OR-cross-compositions: ▶ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3 (fine-grained OR-cross-composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For ν ≥ 1, µ ≥ 0 an (ν, µ)-OR- cross-composition from a problem P to a problem Q is an algorithm A which takes as an input t instances (IP 1 , nP 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (IP t , nP t ) of P, runs in �O(t · nν max + �t i=1 |IP i |) time with nmax := maxi∈[t] nP i , and computes an instance (IQ, (nQ, ℓQ)) of Q such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' (IQ, (nQ, ℓQ)) ∈ Q if and only if (IP i , nP i ) ∈ P for some i ∈ [t], and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' nQ ∈ �O(t · nν max) and ℓQ ∈ �O(nµ max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We say a problem P (ν, µ)-OR-cross-composes into a problem Q if there exists an (ν, µ)-OR-cross-composition from P to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let α > ν ≥ 1, γ > 1, and µ > 0 with α > ν · γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let P be an α-OR- decomposable problem with parameter nP that (ν, µ)-OR-cross-composes into a problem Q with parameters nQ and ℓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If there is an �O(nγ Q + ℓβ Q)-time algorithm for Q and 0 < β < γ · (α − ν) (γ − 1) · µ , then P can be solved in O(nα−ε P ) time for some ε > 0 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 8 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (IP, nP) be an instance of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Our algorithm to solve (IP, nP) runs in the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Apply the α-OR-decomposition (with λ specified below) to obtain the instances (I1, n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (It, nt) with maxi∈[t] ni ≤ q := nα/(α+λ) P and t := nαλ/(α+λ) P = qλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that, by definition, this step runs in �O(nα′ P ) time for some α′ < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Apply the (ν, µ)-OR-cross-composition to compute the instance (IQ, (nQ, ℓQ)) for Q from (I1, n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (It, nt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that the running time and nQ is in �O(t·qν +�t i=1 |IP i |) = �O(qλ+ν + nα′ P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, ℓQ ∈ �O(qµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Apply the algorithm with running time �O(nγ Q + ℓβ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This requires �O(q(λ+ν)γ + qµ·β) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' It remains to show that all three steps run in O(nα−ε P ) time for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To this end, we now specify λ = βµ/γ−ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that λ+ν = βµ/γ < µ·β and thus it suffices to show that the third step runs in �O(nα−ϵ P ) for some ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The last step runs in �O(qµ·β) = �O(nαµ·β/(α+λ) P ) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The exponent is αµβ α + λ = αµβ α + βµ/γ − ν = αβγµ γ(α − ν) + βµ = αγµ γ(α − ν)/β + µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By assumption we have β < γ·(α−ν) (γ−1)·µ and thus the exponent is αγµ γ(α − ν)/β + µ < αγµ γ(α − ν)/ � γ·(α−ν) (γ−1)·µ � + µ = αγµ (γ − 1) · µ + µ = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' There is still one thing left to do: We must ensure that λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This will not always be the case as when β → 0, λ gets negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' However, an �O(nγ Q + ℓβ Q)-time algorithm also implies for any β′ > β an �O(nγ Q + ℓβ′ Q )-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, we can simply pick some larger β′ such that the corresponding λ′ is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To do so, let βmax := γ·(α−ν) (γ−1)·µ the upper bound for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that α > ν · γ implies that λmax := βmaxµ γ − ν = γ·(α−ν) (γ−1)·µ µ γ − ν = α − ν γ − 1 − ν = α − ν − ν · (γ − 1) γ − 1 = α − ν · γ γ − 1 > 0 Thus, we can pick β < β′ < βmax such that λ′ := β′µ γ − ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Note that if for P there is a (conditional) running time lower bound of Ω(nα), then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4 excludes (conditionally) running times of the form �O(nP + ℓβ) for any β ∈ R as limγ→1 γ·(α−ν) (γ−1)·µ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This running time also excludes linear-time computable polynomial-size kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' More precisely, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let α > ν ≥ 1, γ > 1, and µ > 0 with α > ν · γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let P be an α-OR- decomposable problem with parameter nP that (ν, µ)-OR-cross-composes into a problem Q with parameters nQ and ℓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assume that there is an �O(nξ Q) algorithm for deciding Q and that nQ is upper bounded by the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If there exists an �O(ℓβ Q)-size �O(nγ Q)-time kernel for Q for some γ > 1, β ∈ R, and 0 < β < γ · (α − ν) (γ − 1) · µ · ξ , then P can be solved in O(nα−ε P ) time for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' An �O(ℓβ Q)-size �O(nγ Q)-time kernel together with an �O(nξ Q)-time algorithm solving Q yields an �O(nγ Q + ℓβξ Q ) algorithm for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The corollary now follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Klaus Heeger, André Nichterlein, Rolf Niedermeier 9 Our general approach to apply our framework is follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Start with a problem P that (under some hypothesis) cannot be solved in O(nα−ε) time for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then construct an α-decomposition for P followed by a (1,1)-OR-cross-composition into the target problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3 OR-decomposable problems In order to apply our framework, we first need some OR-decomposable problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We will observe that three fundamental problems from fine-grained complexity, namely Orthogonal Vectors, 3-SUM and Negative-Weight k-Clique, are OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' These prob- lems are also our source for running time lower bounds: Note that the former two problems cannot be solved in O(n2−ε) time unless the SETH respectively 3-Sum-hypothesis fail [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We moreover use that Negative-Weight 3-Clique (= Negative Triangle) cannot be solved in O(n3−ε) time unless APSP-hypothesis fails [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We will use that “Many problems, like SAT, have a simple self-reduction proving that the “Direct-OR” version is hard, assuming the problem itself is hard” [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This self-reduction is based on partitioning the instance into many small ones, with at least one of them containing the small desired structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', a pair of orthogonal vectors, three numbers summing to 0, or a negative triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Orthogonal Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We now show that Orthogonal Vectors is 2-OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Orthogonal Vectors parameterized by the number of vectors is 2-OR- decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (I, n) be an instance of Orthogonal Vectors and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Set ϵ := 2 2+λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Partition A into z := ⌈n1−ϵ⌉ many sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Az of at most ⌈nϵ⌉ vectors each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Sym- metrically, partition B into B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Bz of at most ⌈nϵ⌉ vectors each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We assume without loss of generality that |Ai| = |Bj| = ⌈nϵ⌉ =: n′ (this can be achieved e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' by adding the all-one vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For each pair (i, j) ∈ [z]2, create an instance (I(i,j), n′) = ((Ai, Bj), n′) of Orthogonal Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We claim that this constitutes a 2-OR-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The number of vectors n′ of each instance I(i,j) is n′ = O(n 2/(2+λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The number of created instances is z2 = O(n2·(1−ϵ)) = O(n2·(1−2/(2+λ))) = O(n (4+2λ−4)/(2+λ)) = O(n 2λ/(2+λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The running time to compute the decomposition is �O(z2 · n′) = �O(n2−ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' It remains to show that (I, n) is a “Yes”-instance if and only if (I(i,j), n′) is a “Yes”- instance for some (i, j) ∈ [z]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First assume that (I, n) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then there exists some a ∈ A and b ∈ B such that a and b are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let i ∈ [z] such that a ∈ Ai and j ∈ [z] such that b ∈ Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then a and b are orthogonal vectors in (I(i,j), n′), showing that (I(i,j), n′) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Finally, assume that there exists (i∗, j∗) ∈ [z]2 such that (I(i∗,j∗), n′) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then there exists a ∈ Ai∗ and b ∈ Bj∗ which are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consequently, a and b are orthogonal vectors in I, implying that (I, n) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ ▶ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that the above decomposition does not change the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, even restricted versions of Orthogonal Vectors with d ∈ O(log n) are 2-OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, we can assume that all constructed instances of Orthogonal Vectors have the same number of vectors and the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Negative-Weight k-Clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We now show that Negative-Weight k-Clique is k-OR- decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 10 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For any k ≥ 3, Negative-Weight k-Clique parameterized by the number of vertices is k-OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The proof follows the ideas from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (I = (G, w), n) be an instance of Negative-Weight k-Clique and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Set ϵ := k k+λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Partition the set V (G) of vertices into z := ⌈n1−ϵ⌉ many sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Vz of size at most ⌈nϵ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We assume without loss of generality that |Vi| = ⌈nϵ⌉ for all i ∈ [z] (this can be achieved e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' by adding isolated vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let n(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik) := |{i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , ik}| · |V1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For each tuple (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , ik) ∈ [z]k, create an instance (I(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik) = G[Ai1 ∪ · · · ∪ Aik], n(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik)) of Negative-Weight k-Clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We claim that this constitutes a k-OR-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Each instance (I(i,j), n(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik)) has at most O(k ·nϵ) = O(n k/(k+λ)) vertices (note that k is a constant here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The number of created instances is zk = O(nk·(1−ϵ)) = O(nk·(1−k/(k+λ))) = O(n (k2+kλ−k2)/(k+λ)) = O(n kλ/(k+λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, the summed size of all created instances and therefore also the running time is �O(zk · k · n2ϵ) = �O(nk−kϵ+2ϵ) = �O(nk−(k−2)ϵ) = �O(nk′) for some k′ < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' It remains to show that (I, n) is a “Yes”-instance if and only if (I(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik), n(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik)) is a “Yes”-instance for some (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , ik) ∈ [z]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First assume that (I, n) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, G contains a negative-weight clique C = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let ij such that vj ∈ Vij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then C is a negative-weight k-clique in (I(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik), n(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Finally, assume that there exists (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , ik) ∈ [n]k such that (I(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik), n(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=',ik)) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then there exists a clique in G[Vi1∪· · ·∪Vik].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Since G contains G[Vi1∪· · ·∪Vik], also G contains a negative-weight k-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ 3-Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Showing that 3-Sum is 2-OR-decomposable requires some more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3-Sum parameterized by the number of numbers is 2-OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Instead of directly considering 3-Sum, we consider a variation called Convolution 3-Sum of 3-Sum which was shown to be equivalent (under quasi-linear time Las-Vegas reductions) to 3-Sum [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Convolution 3-Sum Input: An array A of n integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Are there i, j ∈ [n] such that A[i] + A[j] = A[i + j]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For simplicity, we will not directly work with Convolution 3-Sum, but with a “multicolored” version of it: Multicolored Convolution 3-Sum Input: Three arrays A, B, and C, each of n integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Question: Are there i, j ∈ [n] such that A[i] + B[j] = C[i + j]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First, we show how to reduce Convolution 3-Sum to Multicolored Convolution 3-Sum and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' There is a linear-time parameter-preserving reduction from Convolution 3-Sum parameterized by the number of numbers to Multicolored Convolution 3-Sum parameterized by the number of numbers and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (A, n) be an instance of Convolution 3-Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We create an instance ((A, B, C), n) of Multicolored Convolution 3-Sum by setting B[i] := A[i] and C[i] := A[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that for i, j ∈ [n], we have A[i] + A[j] = A[i + j] if and only if A[i] + B[i] = A[i] + A[j] = C[i + j], implying that the two instances are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Klaus Heeger, André Nichterlein, Rolf Niedermeier 11 Now consider an instance (I = (A, B, C), n) of Multicolored Convolution 3-Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let zmax := maxi∈[n] max{A[i], B[i], C[i]} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We create an instance (A′, n′ = 4n) of Convolution 3-Sum as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The array A′ has length 4n with A′[i] := � � � � � � � � � � � −5zmax 1 ≤ i ≤ n A[i − n] + zmax n + 1 ≤ i ≤ 2n, B[i − 2n] + 3zmax 2n + 1 ≤ i ≤ 3n, C[i − 3n] + 4zmax 3n + 1 ≤ i ≤ 4n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First, note that a solution A[i] + B[j] = C[i + j] to (I, n) implies a solution to (A′, n′) as A′[i+n]+A′[j +2n] = A[i]+zmax +B[j]+3zmax = C[i+j]+4zmax = A′[3n+i+j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Second, consider a solution A′[i′]+A′[j′] = A′[i′+j′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then i′, j′ > n as otherwise A′[i′]+A′[j′] < 0 and A′[i′] + A′[j′] ̸= −5zmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, i′, j′ ≤ 3n as otherwise A′[i′] + A′[j′] > 5zmax > A′[i′ + j′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, we have without loss of generality n + 1 ≤ i′ ≤ 2n and 2n + 1 ≤ j′ ≤ 3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, we have A[i′−n]+B[j′−2n] = A′[i′]−zmax+A′[j′]−3zmax = A[i′+j′]−4zmax = C[i′+j′−3n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Next, we show that Multicolored Convolution 3-Sum is 2-OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Multicolored Convolution 3-Sum parameterized by the number of numbers is 2-OR-decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The idea behind the proof is essentially the same as for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For an array A we denote the array consisting of the entries (A[i], A[i + 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , A[j]) by A[i, j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (I = (A, B, C), n) be an instance of Multicolored Convolution 3-Sum and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Set ϵ := 2 2+λ and q := ⌈nϵ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Partition A into z := ⌈n1−ϵ⌉ many arrays A1 = A[1, q], A2 = A[q+1, 2q], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Az = A[(z−1)q+1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For simplicity, we assume that n = z·q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Symmetrically, partition B into z many arrays B1 = B[1, q], B2 = B[q + 1, 2q], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Bz = B[(z − 1)q + 1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For (ℓ, p) ∈ [z]2 we denote the interval C[(ℓ + p − 2)q + 2, (ℓ + p)q] by C(ℓ,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For each pair (ℓ, p) ∈ [z]2, create an instance (I(ℓ,p) = (Aℓ, Bp, C(ℓ,p)), q) of Convolution 3-Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We claim that this constitutes a 2-OR-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Each instance (I(ℓ,p), q) has at most O(q) = O(nϵ) = O(n 2/(2+λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The number of created instances is z2 = O(n2·(1−ϵ)) = O(n2·(1−2/(2+λ))) = O(n (4+2λ−4)/(2+λ)) = O(n 2λ/(2+λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Computing the decomposition can be done in �O(z2 · q) = �O(n2−ϵ) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' It remains to show that (I, n) is a “Yes”-instance if and only if (I(ℓ,p), q) is a “Yes”- instance for some (ℓ, p) ∈ [z]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First assume that (I, n) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then there exists some i = (ℓ − 1)q + i∗, j = (p − 1)q + j∗ such that A[i] + B[j] = C[i + j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that A[i] = Aℓ[i∗], B[j] = Bp[j∗], and C[i+j] = C(ℓ,p)[i∗ +j∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, (I(ℓ,p), q) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Finally, assume that there exists (ℓ, p) ∈ [z]2 such that (I(ℓ,p), q) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then Aℓ[i∗] + Bp[j∗] = C(ℓ,p)[i∗ + j∗] for some i∗, j∗ ∈ [q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consequently, A[(ℓ − 1)q + i∗] + B[(p − 1)q + j∗] = C(ℓ,p)[i∗ + j∗] = C[(ℓ + p − 2)q + i∗ + j∗], implying that (I, n) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ The 2-OR-decomposability of 3-Sum now follows from the quasi-linear time parameter- preserving reductions between 3-Sum and Convolution 3-Sum [41] as well as between Convolution 3-Sum and Multicolored Convolution 3-Sum Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='6, the equivalence of 3-Sum and Convolu- tion 3-Sum [41], the equivalence of Convolution 3-Sum and Multicolored Convolu- tion 3-Sum, and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ 12 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Applying the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The above results make our framework easier to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4 we only need to provide a suitable OR-cross composition from one of the three problems discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We thus arrive at the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let Q be a problem with parameters nQ and ℓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If Orthogonal Vectors resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3-Sum parameterized by n (1, 1)-OR-cross-composes into Q, then an O(ℓβ Q + nγ Q)-time algorithm for Q for any 2 > γ > 1 and β < γ/γ−1 refutes the SETH respectively the 3-Sum-Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If Negative Triangle parameterized by the number of vertices (1, 1)-OR-cross-composes into Q, then an O(ℓβ Q + nγ Q)-time algorithm for Q for any 3 > γ > 1 and β < 2γ/γ−1 refutes the APSP-Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 4 Applications We now apply our framework from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4 to several problems from different areas such as string problems (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1), computational geometry (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2), and subgraph isomorphism (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 String problems We start with Longest Common Subsequence that can be solved in O(n2) time al- gorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assuming SETH, there is no algorithm solving Longest Common Sub- sequence in O(n2−ε) time for any ε > 0 [1, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' However, Longest Common Sub- sequence can be solved in O(kn+n log n) time, where k is the length of the longest common subsequence [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A string over an alphabet Σ is an element from Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We access the i-th element of a string x via x[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A subsequence of a string x is a string y such that there is an injective, strictly increasing function f with y[i] = x[f(i)] for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A common subsequence of two strings x and x′ is a string which is a subsequence of both x and x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For two strings x and y, we denote their concatenation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' the string starting with x and ending with y) by x ◦ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Orthogonal Vectors with logarithmic dimension parameterized by the number of vectors (1, 1)-OR-cross-composes into Longest Common Subsequence paramet- erized by the length of the input strings and the length k of the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (IOV 1 , n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (IOV t , nt) be instances of Orthogonal Vectors with logarithmic dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We denote by di the dimension of IOV i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Abboud, Backurs, and Williams [1] gave a reduction from Orthogonal Vectors to Longest Common Subsequence which, given an instance of Orthogonal Vectors with n vectors of dimension d, constructs in n · dO(1) time an equivalent instance of Longest Common Subsequence with strings of length n · dO(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We apply this reduction to each instance (IOV i , ni) of OV, giving us an instance ILCS i = ((x1 i , x2 i ), (nLCS i , ki)) with ki = O(ni · dO(1) i ) and nLCS i = ni · dO(1) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We assume that ki = kj for every i, j (this can be achieved by appending identical sequences of appropriate length to strings x1 i and x2 i ), and set k := ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, we assume that the alphabets used for ILCS i and ILCS j are disjoint for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We define x1 := x1 1 ◦ x1 2 ◦ · · · ◦ x1 t and x2 := x2 t ◦ x2 t−1 ◦ · · · ◦ x2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The OR-cross-composition constructs the instance (x1, x2, k) (the parameter is (nLCS, k) with nLCS = �t i=1 nLCS i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We now show correctness of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First assume that (IOV i , ni) is a “Yes”-instance for some i ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then x1 i and x2 i contain a subsequence y of length ki = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This subsequence y is also a subsequence of x1 and x2, so (x1, x2, k) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Vice versa, assume that Klaus Heeger, André Nichterlein, Rolf Niedermeier 13 (x1, x2, k) is a “Yes”-instance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', x1 and x2 contain a subsequence y of length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let i ∈ [t] such that the first letter of y is contained in x1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We claim that all letters from y are contained in x1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Since the first letter y[1] of y is contained in x1 i , no letter of y can be contained in x1 j for j < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For j > i, note that (as any letter from x2 j only appears in x2 j and x1 j) any letter from x1 j appears only before x2 i in x2 and thus would have to appear before y[1] in y, a contradiction to y[1] being the first letter of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, y is contained in x1 i and x2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consequently, (x1 i , x2 i , k) is a “Yes”-instance, implying that (Ii, ni) is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Computing the OR-cross-composition can be done in t · maxi∈t ni · (maxi∈[t] di)O(1) = �O(t maxi∈[t] ni) time and k = O(maxi∈t ni · (maxi∈t di)O(1)) = �O(maxi=1 ni) (we use here that di ∈ O(log ni)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, we have nLCS = �t i=1 nLCS i = O(�t i=1 ni · dO(1) i ) = �O(�t i=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, it is an (1, 1)-OR-cross-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ We remark that the use of alphabets of non-constant size in the above composition is probably necessary as it excludes a linear-time computable kernel of polynomial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In contrast, Longest Common Subsequence with constant alphabet size parameterized by solution size admits a linear-time computable polynomial-size kernel (note that the SETH-based O(n2−ϵ) lower bound also holds for alphabets of size seven [1]): ▶ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Longest Common Subsequence with constant alphabet size paramet- erized by solution size k admits a linear-time computable polynomial-size kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consider the following reduction rule which directly leads a polynomial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A substring of a string x is a string y such that y = (x[i], x[i + 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x[j]) for some i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Reduction Rule 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If, for some t ∈ N and i ∈ {1, 2}, a string xi contains a substring x′ of length at least (k + 1)t over only t different letters and not containing a substring x′′ of length (k+1)t−1 over at most t−1 different letters, then we can delete all but the first (k+1)t letters of x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First, we argue that the reduction rule is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let x′ be such a substring and t the number of different letters it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assume that there is a substring x′ of xi of length (k + 1)t containing only t different letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By assumption x′ contains no substring of length (k+1)t−1 which contains at most t − 1 different letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, in the first (k + 1)t−1 − 1 letters of x′, each of the t letters appears at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' More generally, for any i ∈ [k], each letter appears at least once among x′[(i − 1) · (k + 1)t−1 + 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x′[i · (k + 1)t−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We claim that we can delete all letters appearing after position k · (k + 1)t−1 ≤ (k + 1)t: Let I = (x1, x2, k) be an instance of Longest Common Subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We assume without loss of generality that the reduction rule can be applied to x1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', x1 contains a substring x′ of length (k + 1)t which contains only t different letters, and all substrings of x′ containing only t′ < t different letters have length at most (k + 1)t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let Ired = (x1 red, x2, k) be the instance arising from applying Reduction Rule 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Clearly, if I is a “No”-instance, then also Ired is also a “No”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Now assume that I is a “Yes”-instance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', x1 and x2 contain a common subsequence z of length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We claim that z is also a subsequence of x1 red (and thus a common subsequence of x1 red and x2), showing that Ired is also a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let z = (x1[i1], x1[i2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x1[ik]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let x1[ir], x1[ir+1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x1[is] be the intersection of z with x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By our above observation, z[r − 1 + i] appears at least once among x′[(i − 1) · (k + 1)t−1 + 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x′[i(k+1)t−1] for each i ∈ [s−r], say z[r−1+i] = x′[(i−1)·(k+1)t−1 +ℓi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then z = (x1[i1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x1[ir−1], x1 ℓ1, x1 (k+1)t−1+ℓ2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x1 (s−r−1)·(k+1)t−1+ℓs−r], x1[is+1], x1[is+2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , x1[ik]) is also a subsequence of x1 red, showing that Ired is a “Yes”-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Next, we analyze the time needed to exhaustively apply Reduction Rule 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 14 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions ▷ Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For constant alphabet size, Reduction Rule 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3 can be exhaustively applied in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We process x1 and x2 from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For each subset S of the alphabet, we have a variable storing the number of consecutive letters from S are before the current letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If this number exceeds (k + 1)|S| for some letter, then we delete this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that whenever we found a string of length (k + 1)|S| containing only letters from S, then this string does not contain a substring of length (k + 1)|S|−1 containing only letters from S \\ {ℓ} for some ℓ ∈ S as in this case, the number stored for S \\ {ℓ} would have exceeded (k + 1)|S|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The above procedure clearly runs in linear time as the alphabet size is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◁ After the exhaustive application of Reduction Rule 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3, the size of the instance is clearly O(k|Σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' As |Σ| is constant, this is a polynomial-sized kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Analogously to Longest Common Subsequence, we derive similar hardness results for the related problems Longest Common Weakly Increasing Subsequence and Longest Common Increasing Subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Both problems can be solved in slightly subquadratic time [7, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Orthogonal Vectors with logarithmic dimension (1, 1)-OR-cross-composes into Longest Common (Weakly) Increasing Subsequence parameterized by the length n of the strings and the length k of the longest common (weakly) increasing sub- sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The proof is analogous to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (IOV 1 , n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (IOV t , nt) be instances of Orthogonal Vector with logarithmic dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We denote by di the dimension of IOV i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Polak [47] and Duraj, Künnemann, and Polak [22] gave a reduction from Orthogonal Vectors to Longest Common (Weakly) Increasing Subsequence which, given an instance of Orthogonal Vectors with n vectors of dimension d, constructs in �O(n·dO(1)) time an instance of Longest Common (Weakly) Increasing Subsequence of length �O(n·dO(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We apply this reduction to each instance (IOV i , ni) of OV, giving us an instance ILCIS i = (x1 i , x2 i , ki) with ki = ni · dO(1) i and whose strings have length �O(n · dO(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We assume that ki = kj for every i, j (this can be achieved by appending identical sequences of increasing numbers of appropriate length to strings x1 i and x2 i ), and set k := ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let Cmax be the largest number appearing in any of the instances ILCIS 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , ILCIS t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We modify instance ILCIS i by increasing each number by i · Cmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We define x1 := x1 1 ◦ x1 2 ◦ · · · ◦ x1 t and x2 := x2 t ◦ x2 t−1 ◦ · · · ◦ x2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The OR-cross-composition constructs the instance (x1, x2, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The correctness of the reduction can be proven in analogy to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The OR-cross-composition can be computed in �O(t · maxi∈t ni · (maxi∈[t] di)O(1)) = �O(t maxi∈[t] ni) time and k = maxi∈t ni · (maxi∈t di)O(1) = �O(maxi=1 ni) (we use here that di ∈ O(log ni)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, it is an (1, 1)-OR-cross-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7 and Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='5, we get the following result: ▶ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the SETH fails, there is no �O(nγ + kβ)-time algorithm for Longest Common Subsequence or Longest Common (Weakly) Increasing Subsequence parameterized by solution size k for 1 < γ < 2 and β < γ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the SETH fails, there is no �O(nγ)-time, �O(kβ)-size kernel for Longest Common Subsequence or Longest Common (Weakly) Increasing Subsequence parameterized by solution size for 1 < γ < 2 and β < γ 2·(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Klaus Heeger, André Nichterlein, Rolf Niedermeier 15 We remark that the running time lower bounds are tight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' the tight upper bound follows from the known O(kn+n log n) time algorithm for Longest Common Subsequence [37] or the O(nk log log n + n log n) time algorithm for Longest Common (Weakly) Increasing Subsequence [43] and Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='2 Computational Geometry We now turn to problems from computational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We will denote the Euclidean distance between two points p and q in the plane by dist(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We start by defining the discrete Fréchet distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For a list of points P, a tour through P is a surjective, non-increasing function fP : [2n] → [n] such that fP (1) = 1 and fP (2n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The input of Discrete Fréchet Distance contains two lists of points P = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , pn) and Q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , qn), and a pair (fP , fQ) of tours through P and Q is called a traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We call maxi∈[2n] dist(pfP (i), qfQ(i)) the width of the traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We say that a traversal (fP , fQ) traverses a sublist pi, pi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , pj of P (or qi, qi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , qj of Q) if, for some ℓ ∈ [2n] and i∗ ∈ [n]), we have fP (ℓ) = pi and fQ(ℓ) = i∗, and for any ℓ′ ∈ [j − i], we have fP (ℓ + ℓ′) = i + ℓ′ and fQ(ℓ + ℓ′) = i∗ (or fP (ℓ) = i∗ and fQ(ℓ) = i, and for any ℓ′ ∈ [j − i], we have fP (ℓ + ℓ′) = i∗ and fQ(ℓ + ℓ′) = i + ℓ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Discrete Fréchet Distance can be solved in O(n2) time via dynamic program- ming [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Bringmann [12] gave a linear-time reduction from Orthogonal Vectors2 to Discrete Fréchet Distance, showing that assuming SETH, Discrete Fréchet Distance cannot be solved in O(n2−ϵ) time for any ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We use this reduction to get a (1, 1)-OR-composition for the parameter maxi∈[2n] |fP (i) − fQ(i)|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', the maximum number of time steps which P may be ahead of Q (or vice versa) in the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We will call the parameter maxi∈[2n] |fP (i) − fQ(i)| the maximum shift of curves fP and fQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Orthogonal Vectors admits a (1, 1)-OR-cross-composition into Discrete Fréchet Distance parameterized by the length of the input lists of points and the maximum shift in a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Our OR-cross-composition is based on the reduction from Orthogonal Vectors to Discrete Fréchet Distance by Bringmann [12] which has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='8 (Bringmann [12, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Given an instance ((A, B), n) of Or- thogonal Vectors one can compute in �O(n) time an instance (PA, PB, 1) of Discrete Fréchet Distance such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' the Fréchet distance of PA and PB is 1 if and only if ((A, B), n) is a “Yes”-instance, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' PA starts at sA := (− 1 3, 1 5) and sA appears only at the start of PA, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' PB starts at sB := (− 1 3, 0), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' each point from PB has distance at most 1 from cA := (0, 1 3), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' each point from PA has distance at most 1 from cB := sB, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ℓA := (− 4 3, 0) has distance larger than 1 to all points from Q \\ {sB}, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ℓB := (− 4 3, 1 5) has distance larger than 1 to all points of PA \\ {sA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' one point in PB has distance more than 1 from sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 2 Bringmann [12] reduces from Satisfiability, but his reduction implicitly reduces Satisfiability to Orthogonal Vectors and then Orthogonal Vectors to Discrete Fréchet Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 16 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (IOV 1 = (A1, B1), n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (IOV t = (At, Bt), nt) be instances of Orthogonal Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To simplify notation, we assume that IOV 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , IOV t all are of the same dimension d and that there is some n ∈ N such that ni = n for all i ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, we assume that no two vectors of different instances of IOV t are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This can be achieved by increasing the dimension by 2 log t and using the additional dimensions to add to each a ∈ Ai the binary encoding of i and its “inverse” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', replacing 0’s by 1’s and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To each b ∈ Bi, we append the “inverse” of the binary encoding of i and the binary encoding of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We apply the reduction from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='8 to each instance (IOV i , n), resulting in an instance IDFD i = (P i A, P i B, 1) of Discrete Fréchet Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We set nDFD i := |P i A| to be the length of P i A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For i ∈ [t], let si A := sA, si B := sB, ci A := cA, ℓi A := ℓA, ℓi B := ℓB (where sA, sB, cA, cB, ℓA, and ℓB are be defined as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, let IDFD t+1 be an instance arising from applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='8 to an arbitrary “No”-instance of Orthogonal Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' From this, we construct an instance (IDFD = (PA, PB, 1), (2t + �t+1 i=1 nDFD i , 2 maxi∈[t+1] nDFD i )) of Discrete Fréchet Distance parameterized by the length of the point lists and the maximum shift as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We set PA := ⃝i∈[t](P i A ◦ ℓi A ◦ ci A) ◦ P t+1 A and PB := ⃝i∈[t+1](P i B ◦ ℓi B) ◦ cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To ensure that PA and PB have the same number of points, we append an appropriate number of copies of cB at the end of PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' On an intuitive level, the idea of the construction is as follows: The asymmetry in the construction allows to traverse an arbitrary part of PA while being in si B = cB in PB for some i ∈ [t + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' However, the converse is not true and, thus, traversing in PB as far as possible is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In order to traverse ℓi B we have to be in sj A or ℓj A for some i ∈ [t + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We cannot stay in sj A since by Property 8 at least one point in PBi has distance larger than one from si A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Hence, there are two options: The traversal contains (sj A, ℓi B) for i, j ∈ [t + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' As Property 8 forces us out of the sA-points, we will have j > i when following this option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, only using this option results in getting stuck at the end as the last ℓBt+1 cannot be traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' To prevent this and “get ahead” in PB, we have to use the second option at some point: The traversal contains (ℓi A, ℓi B) for some i ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We prove that in this case PAi and PBi have to be traversed simultaneously, that is, IDFD i is a yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The benefit of the second option is that we can follow (ℓi A, ℓi B) by (ci A, si+1 B ) and then traverse PBi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then, being “ahead” in PB, we can use the first option for the rest of the traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We now show the correctness of the OR-cross-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▷ Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If there is some i∗ ∈ [t] such that there are a ∈ Ai∗ and b ∈ Bi∗ which are orthogonal, then there is a traversal T of (PA, PB) of width at most 1 and maximum shift 2 maxi∈[t+1] nDFD i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We define a traversal as follows: We start with (s1 A, s1 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For each i < i∗, we define the following traversal Ti (which first traverses P i A ◦ℓi A ◦ci A and afterwards P i B ◦ℓi B): Traverse PA until the end of P i A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 5, this traversal so far has width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Afterwards, go to (ℓi A, si B), followed by (ci A, si B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then traverse P i B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 4, this traversal still has width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We continue with (si+1 A , ℓi B) and (si+1 A , si+1 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 1, there is a traversal of (P i∗ A , P i∗ B ) of width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We continue with this traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We continue with (ℓi∗ A, ℓi∗ B) and (ci∗ A, si∗+1 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then, we traverse P i∗+1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This is followed by (si∗+1 A , ℓi∗+1 B ) and (si∗+1 A , si∗+2 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, we are now at the start of P i∗+1 A in PA and P i∗+2 B in PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Afterwards, for t ≥ i > i∗, we traverse P i A and P i+1 B as in the first step: First, traverse P i A until the end of P i A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 5, this traversal so far has width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Afterwards, go to (ℓi A, si+1 B ), followed by (ci A, si+1 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Then traverse P i+1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 4, this traversal still has width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We continue with (si+1 A , ℓi+1 B ) and (si+1 A , si+2 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Klaus Heeger, André Nichterlein, Rolf Niedermeier 17 Finally, we are at st+1 A and the end of P t+1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We go to (st+1 A , ℓt+1 B ), followed by (st+1 A , cB) and afterwards traverse P t+1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 5, the constructed traversal still has width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, we found a traversal of (PA, PB) of width at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' It is easy to verify that this traversal has a maximum shift of 2 maxi∈[t+1] nDFD i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◁ ▷ Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If there is no i∗ ∈ [t] such that there are a ∈ Ai∗ and b ∈ Bi∗ which are orthogonal, then the Fréchet distance of PA and PB is larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assume towards a contradiction that there is a traversal T of width 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First, we show that T does not reach the first point si B of P i B for each i ∈ [t + 1] before it reaches the first point si A of P i A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assume towards a contradiction that si B is reached before si A and that i is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Clearly i > 1 as T starts with (s1 A, s1 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By choice of i we have that si−1 A is reached before or at the same time as si−1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that before si B is reached (which is before si A is reached) ℓi−1 B has to be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By Property 7, the only points from PA at distance at most one to ℓi−1 B are ℓj A and sj A for j ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By choice of i it follows that (si−1 A , ℓi−1 B ) or (ℓi−1 A , ℓi−1 B ) is part of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We show contradictions in both cases: Since si−1 A is reached before or simultaneously to si−1 B , it follows from Property 8 that (si−1 A , ℓi−1 B ) is not part of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that (ℓi−1 A , ℓi−1 B ) cannot be part of T as the immediate predecessor consists of the respective last points in PAi and PBi: As si−1 A is reached before si−1 B , this would imply PAi and PBi having a Fréchet distance of at most 1, contradicting Property 1 and PAi and PBi containing no orthogonal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, for each j ∈ [t + 1] we have that T reaches sj A before sj B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Property 1 implies that there is no traversal of width 1 that reaches the last point of P t+1 B as well as the last point of P t+1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Thus, T must reach the last point of P t+1 B while only partially traversing P t+1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Afterwards, T must move to ℓt+1 B , implying that T is at st+1 A at this point (as all other points have distance larger than 1 to ℓt+1 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' However, by Property 8, there is some point in P t+1 B of distance larger than 1 to st+1 A , a contradiction to T traversing through P t+1 B while staying at st+1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◁ The OR-cross-composition can clearly be computed in �O(t maxi∈[t] ni) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The correct- ness follows from Claims 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The maximum shift of the constructed instance is O(n) by Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consequently, it is an (1, 1)-OR-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7, yields the following: ▶ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the SETH fails, there is no �O(nγ + ℓβ)-time algorithm Discrete Fréchet Distance parameterized by maximum shift ℓ for 1 < γ < 2 and β < γ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the SETH fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for Discrete Fréchet Distance parameterized by maximum shift for 1 < γ < 2 and β < γ 2·(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' It is easy to extend the O(n2)-algorithm for Discrete Fréchet Distance [23] to run in O(nℓ) time, where ℓ is the minimal (over all solutions) maximum shift of an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This together with Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 shows that the running time lower bound from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='11 is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We now give lower bounds for another problem from computational geometry, Planar Motion Planning, based not on the hardness of Orthogonal Vectors but on the hardness of 3-Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Planar Motion Planning can be solved in �O(n2) time [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Assum- ing the 3-Sum conjecture, Planar Motion Planning cannot be solved in O(n2−ϵ) for any ϵ > 0 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We say a segment is in the vicinity of another segment if they have distance at most the length of the rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 18 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Figure 2 Left: Exemplary illustration of an instance of Planar Motion Planning constructed by the reduction from 3-Sum from Gajentaan and Overmars [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Right: An example for the OR-cross-composition of three instances of 3-Sum into Planar Motion Planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The goal position of the rod is surrounded by red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 3-Sum (1, 1)-OR-cross-composes into Planar Motion Planning para- meterized by the maximum number of segments any segment has in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let I3-Sum 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , I3-Sum t be instances of 3-Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We denote by ni the number of numbers of I3-Sum 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Gajentaan and Overmars [31] gave a reduction from 3-Sum to Planar Motion Planning which, given an instance of 3-Sum with n numbers, constructs in �O(n) time an instance of Planar Motion Planning where the rod initially is in a large “upper” rectangle and has to reach a “lower” rectangle through a narrow passage (see Figure 2 for a proof by picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We apply this reduction to each instance I3-Sum i of 3-Sum, giving us an instance IPMP i with �O(ni) many segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' From these instances, we construct an instance of Planar Motion Planning as follows: We identify the large rectangles in which the rod starts from each IPMP i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The narrow passages are copied next to each other (if the large starting rectangle is not wide enough, then we make it wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The correctness of the reduction is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Any segment from an instance I3-Sum i has distance at most the length of the rod except for the bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' By splitting the segments of the bounding box into many small segments not longer than the rod, we get that for each segment s there are at most O(n) segments whose distance to s is at most the length of the rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='12 yields the following: ▶ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the 3SUM-hypothesis fails, there is no �O(nγ + ℓβ)-time algorithm Planar Motion Planning parameterized by the maximum number ℓ of segments any segment has in its vicinity for β < γ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the 3SUM-hypothesis fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for Planar Motion Planning parameterized by the maximum number ℓ of segments any segment has in its vicinity for β < γ 2·(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The lower bound for the running time is tight by the O(K2 log n) time algorithm [49] (where K is the number of segment pairs whose distance is less than the length of the rod), the observation that K2 ≤ n · ℓ, and Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='3 Graph Problems We now turn to graph problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' First, we consider Minimum Weight k-Clique paramet- erized by the maximum size of a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Klaus Heeger, André Nichterlein, Rolf Niedermeier 19 ▶ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Minimum Weight k-Clique (1, 1)-OR-cross-composes into Minimum Weight k-Clique parameterized by the maximum size of a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Gt be instances of Minimum Weight k-Clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The cross-composition just computes the disjoint union G of G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Clearly, the cross-composition is correct, runs in linear time, and the maximum size of a connected component is bounded by the maximum size of G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ As Negative Triangle is the special case of Minimum Weight k-Clique, combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='14 yields the following lower bound: ▶ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the APSP-hypothesis fails, there is no �O(nγ + ℓβ)-time algorithm for Negative Triangle parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 and β < 2·γ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the APSP-hypothesis fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for Negative Triangle parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 and β < 2·γ 3·(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' An algorithm running in O(ℓ2n) where ℓ is the maximum size of a connected component is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This together with Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 implies that the running time lower bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Next, we turn to 2nd Shortest Path which can be solved in �O(mn) [44] or in O(Mnω) time (where M is the largest edge weight) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' If the graph is undirected [39] or one aims to find a 2nd shortest walk [24], then there is a quasi-linear-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For unweighted directed graphs, the problem can be solved in �O(m√n) time [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' An ϵ-approximation can be computed in �O( m ϵ ) time [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Negative Triangle (1, 1)-OR-cross-composes into 2nd Shortest Path parameterized by directed feedback vertex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let (INT 1 , n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , (INT t , nt) be instances of Negative Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' We assume without loss of generality that each instance of Negative Triangle has the same num- ber n of vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', ni = n for all i ∈ [t], and that the largest absolute value of an edge weight is the same for all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Vassilevska Williams and Williams [51] gave a linear-time reduction from Negative Triangle to 2nd Shortest Path which, given an instance (INT i = (Gi, wi), n) of Negative Triangle, creates an instance I2SP i of 2nd Shortest Path as follows: For each vertex v ∈ V (Gi), add three vertices av, bv, and cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, add two vertices si and ti as well as a path si, pi 1, pi 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , pi n, ti, all of whose edges have weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, there are edges (pi, av), (av, bu), (bu, cv), and (cv, pi) for every i ∈ [n], u, v ∈ V (Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' All these edges have positive weight (depending on the edge- weights in E(Gi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Identifying si with sj, ti with tj, and pi ℓ with pj ℓ for all i, j ∈ [t] and j ∈ [ℓ] then results in one instance of 2nd Shortest Path with a directed feedback vertex set of size n, namely {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , pn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' As the constructed instance is equivalent to the disjunction of I2SP 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , I2SP t (it is never beneficial to leave s, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , pn, t more than once as all edges leaving this path have positive weight), we have a (1, 1)-OR-cross-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ Combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='7 with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='16 yields the following: ▶ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless the APSP-hypothesis fails, there is no �O(nγ + ℓβ)-time algorithm for 2nd Shortest Path parameterized by directed feedback vertex set ℓ parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 and β < 2γ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 20 Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions Unless the APSP-hypothesis fails, there is no �O(nγ)-time, �O(ℓβ)-size kernel for 2nd Shortest Path parameterized by directed feedback vertex set parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 and β < 2γ 3(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In contrast to the other problems studied in this paper, we do not know whether the running time lower bounds are tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='4 Triangle Collection Last, we consider the Triangle Collection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Triangle Collection can trivially be solved in O(n3) time, but does not admit an O(n3−ϵ) time algorithm assuming SETH, the 3-Sum conjecture, or the APSP conjecture [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For this problem, we were unable to apply our framework directly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=', find an OR-cross-composition from an OR-decomposable problem to Triangle Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' However, we can still get a lower bound in a very similar fashion by combining decomposition and composition into one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The difference is that the decomposition of Triangle Collection that we use in the proof of the following result does not admit the “OR”-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ▶ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless Orthogonal Vectors, 3-Sum, or APSP-hypothesis fails, there is no �O(nγ+ℓβ)-time algorithm for Negative Triangle parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 and β < 2·γ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Unless Orthogonal Vectors, 3-Sum, or APSP-hypothesis fails, there is no �O(nγ)- time, �O(ℓβ)-size kernel for Triangle Collection parameterized by the maximum size ℓ of a connected component for 1 < γ < 3 and β < 2·γ 3·(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Fix 1 < γ < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Let G be in an instance of Triangle Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Partition V (G) into z := n λ/3+λ sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' , Vz of size q := n 3/(3+λ), where λ := β/γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For each (i, j, k) ∈ [z]3, let G(i,j,k) be the graph induced by Vi ∪ Vj ∪ Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Finally, let H be the union of all G(i,j,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that H corresponds to the output of the OR-decomposition in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Clearly G has a triangle collection if and only if H has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, H can be computed in �O(q · z3) = �O(n 3(λ+1)/(3+λ)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Now assume that there is a �O(nγ + ℓβ)-algorithm for Triangle Collection with β < 2·γ γ−1 and apply this algorithm to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' This clearly solves H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The running time for this step is �O((z3 · q)γ + qβ) = �O(nγ·(3·(λ+1)/(3+λ)) + nβ·3/(3+λ)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Note that γ·3·(λ+1)/(3+λ) = 3 · β/γ 2+β/γ < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Further, it holds that β · 3/(3+λ) = 3· β 2 + β/γ = 3· βγ 2γ + β = 3· γ 2γ/β + 1 < 3· γ 2γ/ � 2γ (γ−1) � + 1 = 3· γ (γ − 1) + 1 = 3 where we used the assumption β < 2·γ γ−1 for the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Consequently, we can solve Triangle Collection in O(n3−ϵ) time for some ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' The statement for kernels follows from the observation that an O(ℓβ)-size, O(nγ)-time kernel for Triangle Collection directly implies an O(nγ + ℓ3·β)-time algorithm for Triangle Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ◀ As an O(nℓ2) time algorithm for Triangle Collection is trivial, it follows from Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1 that the running time lower bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 5 Conclusion We introduced a framework for extending conditional running time lower bounds to para- meterized running time lower bounds and applied it to various problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Beyond the clear REFERENCES 21 task to apply the framework to further problems there are further challenges for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' For example, can we get “AND-hard” problems so that we can use AND-cross compositions similar to the ones used to exclude compression [19]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Moreover, can the framework be adapted to cope with dynamic, counting, or enumerating problems?' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 7 Anadi Agrawal and Pawel Gawrychowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' A faster subquadratic algorithm for the longest common increasing subsequence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC 2020), volume 181 of LIPIcs, pages 4:1–4:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' URL https: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Parameterized aspects of triangle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Journal of Computer and System Sciences, 103:61–77, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='jcss.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 10 Hans L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Bodlaender, Rodney G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Downey, Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Fellows, and Danny Hermelin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' On problems without polynomial kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Journal of Computer and System Sciences, 75(8): 423–434, 2009.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 13 Karl Bringmann and Marvin Künnemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Quadratic conditional lower bounds for string problems and dynamic time warping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS 2015), pages 79–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' IEEE Computer Society, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' URL https://doi.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' ACM Transactions on Algorithms, 17 (1):2:1–2:14, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1145/3402926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 15 Yijia Chen, Jörg Flum, and Moritz Müller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Lower bounds for kernelizations and other preprocessing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Theory of Computing Systems, 48(4):803–839, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 16 Thomas H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Cormen, Charles E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Leiserson, Ronald L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1137/ 130927115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 20 Guillaume Ducoffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Maximum matching in almost linear time on graphs of bounded clique-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of LIPIcs, pages 15:1–15:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1007/3-540-52846-6_81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' 53 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Ryan Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' Faster all-pairs shortest paths via circuit complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' SIAM Journal on Computing, 47(5):1965–1985, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} +page_content='1137/15M1024524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQf4_q8/content/2301.00797v1.pdf'} diff --git a/FtE2T4oBgHgl3EQfTAfF/content/tmp_files/2301.03799v1.pdf.txt b/FtE2T4oBgHgl3EQfTAfF/content/tmp_files/2301.03799v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33f9960d5a8ded47a6f9121c47b8a57239b54a18 --- /dev/null +++ b/FtE2T4oBgHgl3EQfTAfF/content/tmp_files/2301.03799v1.pdf.txt @@ -0,0 +1,388 @@ +Tensor Formulation of the General Linear Model with Einstein Notation +Gavin T. Kress, MS* +Author Affiliations: +* Corresponding Author +Author of Correspondence: Gavin Kress, MS, Email: gkress@usc.edu, Tel: (865) 804- +2847 +Keywords: General Liner Model, Tensor, Multidimensional Array, Computational +Efficiency, Einstein Notation +Author Contributions: +Gavin T. Kress (ORC ID: 0000-0001-5152-1170, gkress@usc.edu) – +Conceptualization, Formal Analysis, Investigation, Methodology, Visualization, +Writing – Original Draft Preparation, Writing – Review & Editing +Funding Source: This research received no specific grant from any funding agency in +the public, commercial, or not-for-profit sectors +Conflict of Interest Disclosure: None +Acknowledgments: None + + + +Abstract +The general linear model is a universally accepted method to conduct and test +multiple linear regression models. Using this model one has the ability to +simultaneously regress covariates among different groups of data. Moreover, there are +hundreds of applications and statistical tests associated with the general linear model. +However, the conventional matrix formulation is relatively inelegant which yields +multiple difficulties including slow computation speed due to a large number of +computations, increased memory usage due to needlessly large data structures, and +organizational inconsistency. This is due to the fundamental incongruence between +the degrees of freedom of the information the data structures in the conventional +formulation of the general linear model are intended to represent and the rank of the +data structures themselves. +Here, I briefly suggest an elegant reformulation of the general linear model which +involves the use of tensors and multidimensional arrays as opposed to exclusively flat +structures in the conventional formulation. To demonstrate the efficacy of this +approach I translate a few common applications of the general linear model from the +conventional formulation to the tensor formulation. +Keywords: General Liner Model, Tensor, Multidimensional Array, Computational +Efficiency, Einstein Notation +Introduction +The general linear model (GLM) or general multivariate regression model is a widely +accepted technique across multiple fields to perform several multiple linear regression +models. It offers advantages such as the ability to simultaneously regress covariates +among different groups of data, among others. The applications and statistical tests +derived from and expressed using the conventional matrix formulation of the GLM +are numerous and multifaceted.1,2 +However, the conventional matrix formulation is relatively inelegant in some +embodiments, yielding compromised computational efficiency and increased order of +complexity in automation of statistical tests. For example, in cases in which multiple +groups are modeled, the matrix formulation lacks the dimensionality to encode the + +relevant linear coefficients and variables. The brute force solution to this in the +conventional formulation is to simply stagger the indices corresponding to the various +groups such that the relevant parameters and variables are all encoded in a sparse, +flat data structure. +Here, I briefly suggest an elegant reformulation of the GLM, such that the data +structures describing the important parameters and variables are tensors represented +in Einstein notation.3 To demonstrate the efficacy of this approach, a general +description of the formulation will precede a few brief examples of applications for +which this formulation is more elegant than the matrix formulation. +Conventional Formulation of the GLM +The GLM most generally consists of N domain variables from which a linear atlas is +generated which maps this domain space of ℝN to a linear manifold in ℝ1 defined by +an outcome variable. Moreover, the parameters defining such atlas depend on the +group from which the domain variables are derived. This linear atlas is conventionally +expressed as described in Eq. 1. +Eq. 1 +������������ = ������������������������ + + +Where y is the outcome variable, X is the covariant vector consisting of the domain +variables with first entry equal to one corresponding to the intercept, and ������������ is the +contravariant vector consisting of the coefficients of said domain variables. This map +is generated with a function of a series of residuals defined in Eq. 2. +Eq. 2 +������������ = ������������������������ + ������������ + +In this ������������ is the contravariant vector representing samples of the linear outcome +manifold in ℝ1 for a particular group, while X is a matrix describing a series of the +same covariant vectors in Eq. 1, which were experimentally determined to map to +said samples of the outcome manifold. By choosing the parameters of ������������, the linear +map will, at best, approximate the experimentally defined atlas, implying the +existence of residuals N. +The matrices are written out explicitly for only one regressor or domain variable in +Eq. 3. + +Eq. 3 +� +������������0 +������������1 +������������2 +⋮ +� = � +1 +������������0 +1 +������������1 +1 +⋮ +������������2 +⋮ +� � ������������ +������������� + � +������������0 +������������1 +������������2 +⋮ +� + +At this point, the model is inconspicuously inelegant. However, upon the introduction +of numerous groups from which the experimental data is sampled, it becomes evident. +Expanding this example to one regressor in two groups is shown in Eq. 4. +Eq. 4 +⎣ +⎢ +⎢ +⎢ +⎢ +⎡������������1,0 +������������1,2 +⋮ +������������2,0 +������������2,1 +⋮ ⎦ +⎥ +⎥ +⎥ +⎥ +⎤ += +⎣ +⎢ +⎢ +⎢ +⎢ +⎡1 +0 +������������1,0 +0 +1 +0 +������������1,1 +0 +⋮ +0 +0 +0 +0 +1 +1 +⋮ +⋮ +0 +0 +0 +0 +������������2,0 +������������2,1 +⋮ ⎦ +⎥ +⎥ +⎥ +⎥ +⎤ +� +������������1 +������������2 +������������1 +������������2 +� + ������������ + +Clearly in this staggered configuration, as more groups are introduced into the model +the matrix continues to grow, with the majority of entries being equal to zero, which +yields an unnecessarily large number of computations and quantity of memory usage. +Moreover, without a priori knowledge of both the number of groups and number of +regressors, it is impossible to predict the organizational structure of the matrix. +Tensor Formulation of the GLM +These problems are avoided with an alternative tensor formulation of the model +expressed in Einstein notation. Notably, Eq. 2 Can be written as Eq. 5. +Eq. 5 +������������������������,������������ = ������������������������ +������������,������������������������������������,������������ + ������������������������,������������ + +Here, k indexes the samples of the experimental mapping, ������������ indexes over the group, +and ������������ indexes over the intercept and each regression variable or parameter. From +this, the extension to the translation of Eq. 1 is trivial. An example of such +formulation in hybrid matrix Einstein notation with one regression parameter is +shown in Eq. 6. +Eq. 6 +� +������������0,������������ +������������1,������������ +������������2,������������ +⋮ +� = +⎣ +⎢ +⎢ +⎢ +⎡������������0 +0,������������ = 1 +������������1 +0,������������ +������������0 +1,������������ = 1 +������������1 +1,������������ +1 +⋮ +������������1 +2,������������ +⋮ ⎦ +⎥ +⎥ +⎥ +⎤ +������������������������� +������������������������� + � +������������0,������������ +������������1,������������ +������������2,������������ +⋮ +� + +GLM Contrast Matrix in Tensor Notation + +The null hypothesis (������������0) statements to test in the GLM take the form of a linear +combination of the atlas parameters in ������������ is equal to 0. This linear combination is +conventionally expressed in a manner outlined in Eq. 7. +Eq. 7 +������������ = ������������������������ + +Where, g corresponds to the value ������������0 asserts is equal to zero and ������������ is the GLM +contrast matrix, which is a covariant vector with indices corresponding to the atlas +parameters in ������������ which serves as their linear coefficients in the ������������0 statement. This +expression is identically inelegant and separate statements are constructed for +multiple hypothesis testing such as F-testing. The compatible expression in the tensor +formulation is shown in Eq. 8. +Eq. 8 +������������������������ = ������������������������,������������ +������������ ������������������������,������������ + +This model is compatible with multiple ������������0, which ������������ indexes over, for F-testing or +multiple t-tests. +An example of such expression with two ������������0 statements in a model with two separate +groups and a single regressor is outlined component-wise in Eq. 9-22. +Eq. 9-10 +������������0,0 = ������������1, ������������1,0 = ������������1 + +Eq. 11-12 +������������0,1 = ������������2, ������������1,1 = ������������2 + +Eq. 12-13 +������������0,0 +0 = ������������������������1 +0 , ������������1,0 +0 = ������������������������1 +0 + +Eq. 13-14 +������������0,1 +0 = ������������������������2 +0 , ������������1,1 +0 = ������������������������2 +0 + +Eq. 14-15 +������������0,0 +1 = ������������������������1 +1 , ������������1,0 +1 = ������������������������1 +1 +Eq. 15-16 +������������0,1 +1 = ������������������������2 +1 , ������������1,1 +1 = ������������������������2 +1 + +Eq. 17 +������������0 = ������������������������1 +0 ������������1 + ������������������������1 +0 ������������1 + ������������������������2 +0 ������������2 + ������������������������2 +0 ������������2 + +Eq. 18 +������������1 = ������������������������1 +1 ������������1 + ������������������������1 +1 ������������1 + ������������������������2 +1 ������������2 + ������������������������2 +1 ������������2 + + +GLM Multiple T-Test in Tensor Notation + +The justification for representing the various applications of the GLM in a tensor +formulation is self-evident at this point, and in most cases it is straightforward to +infer such representations from the conventional notation. However, this is not always +true, especially in embodiments which require inverting matrices. +The t statistic is of the most important of these embodiments, which is represented in +the conventional matrix notation in Eq. 19. +Eq. 19 +������������ = +������������������������ +�������������2������������(������������������������������������)−1������������������������ + +Where the t statistic is generated separately for each ������������0, and ������������2 is the variance of +the experimental outcome measure in most cases. +To express this in the tensor formulation, it is evident that the numerator is g, which +is indexed for each hypothesis and, consequently, so is t. Moreover, it is clear that +contracting a matrix with its transposed self can be expressed as shown in Eq. 20. +Eq. 20 +������������������������ +������������′,������������ = ������������������������������������ = ������������������������ +������������′,������������������������������������ +������������,������������ + +Moreover, the inverse of a matrix expressed in tensor notation is computed as Dr. +Roger Penrose puts forth4 and as is shown in Eq. 21. +Eq. 21 +[������������−1]������������ +������������,������������ = 2[������������������������1,������������2������������������������′1,������������′2������������������������1 +������������′1,������������������������������������2 +������������′2,������������]−1������������������������,������������2������������������������,������������′2������������������������2 +������������′2,������������ + +Where ������������ is the totally antisymmetric Levi-Civita symbol which is defined from the +sign by the permutation of its indices such that each value is a power of (-1) which +matches the parity of the permutation, otherwise the value is zero. +∴ +Eq. 22 +(������������������������������������)−1 = 2[������������������������1,������������2������������������������′1,������������′2(������������������������ +������������′1,������������������������������������1 +������������,������������)(������������������������ +������������′2,������������������������������������2 +������������,������������)]−1������������������������,������������2������������������������,������������′2(������������������������ +������������′2,������������������������������������2 +������������,������������) + +∴ +Eq. 23 ������������������������ = +������������������������,������������ +������������ ������������������������,������������ +�(������������������������)2������������������������,������������ +������������ 2[������������������������1,������������2������������������������′1,������������′2(������������������������ +������������′1,������������������������������������1 +������������,������������)(������������������������ +������������′2,������������������������������������2 +������������,������������)]−1������������������������,������������2������������������������′,������������′2(������������������������ +������������′2,������������������������������������2 +������������,������������)������������ ������������ +������������′,������������ + + + + +Results and Discussion +The tensor formulation of the GLM drastically decreases the number of elements in +the data structures and reduces the quantity of operations required to perform +computations with said data structures, especially as more groups, regressors, and +hypotheses are incorporated in the model. This has the potential to significantly +reduce the time required to test various hypotheses. +Moreover, the automation of hypothesis testing with the GLM is significantly +simplified in the tensor formulation by the property that no a priori knowledge of the +number groups, regressors, and hypotheses is needed to infer the structural +organization of the data. +Finally, this solution is simply more elegant, as the rank of the tensors is +complementary to the degrees of freedom of the information which the data structure +in the GLM is designed to interact with. +There are hundreds of unique applications of the GLM, each of which can be +formulated in this proposed manner. Here, I have set forth the general structure of +such formulations with a few examples, but the literature would benefit from further +translation of other applications. +References +1. +Mccullagh, P. Generalized linear models. (1984). +2. +Khuri, A. I., Mukherjee, B., Sinha, B. K. & Ghosh, M. Design Issues for +Generalized Linear Models: A Review. +https://doi.org/10.1214/088342306000000105 21, 376–399 (2006). +3. +Åhlander, K. Einstein summation for multidimensional arrays. Computers & +Mathematics with Applications 44, 1007–1017 (2002). +4. +Roger Penrose. Applications of Negative Dimensional Tensors. Birkbeck +College, University of London. + + diff --git a/FtE2T4oBgHgl3EQfTAfF/content/tmp_files/load_file.txt b/FtE2T4oBgHgl3EQfTAfF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..80a0b809f7ed0d3b6ef35966ece2e33d709b4e0b --- /dev/null +++ b/FtE2T4oBgHgl3EQfTAfF/content/tmp_files/load_file.txt @@ -0,0 +1,177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf,len=176 +page_content='Tensor Formulation of the General Linear Model with Einstein Notation Gavin T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Kress, MS* Author Affiliations: Corresponding Author Author of Correspondence: Gavin Kress, MS, Email: gkress@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='edu, Tel: (865) 804- 2847 Keywords: General Liner Model, Tensor, Multidimensional Array, Computational Efficiency, Einstein Notation Author Contributions: Gavin T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Kress (ORC ID: 0000-0001-5152-1170, gkress@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='edu) – Conceptualization, Formal Analysis, Investigation, Methodology, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Funding Source: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors Conflict of Interest Disclosure: None Acknowledgments: None Abstract The general linear model is a universally accepted method to conduct and test multiple linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Using this model one has the ability to simultaneously regress covariates among different groups of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Moreover, there are hundreds of applications and statistical tests associated with the general linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' However, the conventional matrix formulation is relatively inelegant which yields multiple difficulties including slow computation speed due to a large number of computations, increased memory usage due to needlessly large data structures, and organizational inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' This is due to the fundamental incongruence between the degrees of freedom of the information the data structures in the conventional formulation of the general linear model are intended to represent and the rank of the data structures themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Here, I briefly suggest an elegant reformulation of the general linear model which involves the use of tensors and multidimensional arrays as opposed to exclusively flat structures in the conventional formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' To demonstrate the efficacy of this approach I translate a few common applications of the general linear model from the conventional formulation to the tensor formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Keywords: General Liner Model, Tensor, Multidimensional Array, Computational Efficiency, Einstein Notation Introduction The general linear model (GLM) or general multivariate regression model is a widely accepted technique across multiple fields to perform several multiple linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' It offers advantages such as the ability to simultaneously regress covariates among different groups of data, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' The applications and statistical tests derived from and expressed using the conventional matrix formulation of the GLM are numerous and multifaceted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='1,2 However, the conventional matrix formulation is relatively inelegant in some embodiments, yielding compromised computational efficiency and increased order of complexity in automation of statistical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' For example, in cases in which multiple groups are modeled, the matrix formulation lacks the dimensionality to encode the relevant linear coefficients and variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' The brute force solution to this in the conventional formulation is to simply stagger the indices corresponding to the various groups such that the relevant parameters and variables are all encoded in a sparse, flat data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Here, I briefly suggest an elegant reformulation of the GLM, such that the data structures describing the important parameters and variables are tensors represented in Einstein notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='3 To demonstrate the efficacy of this approach, a general description of the formulation will precede a few brief examples of applications for which this formulation is more elegant than the matrix formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Conventional Formulation of the GLM The GLM most generally consists of N domain variables from which a linear atlas is generated which maps this domain space of ℝN to a linear manifold in ℝ1 defined by an outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Moreover, the parameters defining such atlas depend on the group from which the domain variables are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' This linear atlas is conventionally expressed as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 1 ������������ = ������������������������ Where y is the outcome variable, X is the covariant vector consisting of the domain variables with first entry equal to one corresponding to the intercept, and ������������ is the contravariant vector consisting of the coefficients of said domain variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' This map is generated with a function of a series of residuals defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 2 ������������ = ������������������������ + ������������ In this ������������ is the contravariant vector representing samples of the linear outcome manifold in ℝ1 for a particular group, while X is a matrix describing a series of the same covariant vectors in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 1, which were experimentally determined to map to said samples of the outcome manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' By choosing the parameters of ������������, the linear map will, at best, approximate the experimentally defined atlas, implying the existence of residuals N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' The matrices are written out explicitly for only one regressor or domain variable in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 3 � ������������0 ������������1 ������������2 ⋮ � = � 1 ������������0 1 ������������1 1 ⋮ ������������2 ⋮ � � ������������ ������������� + � ������������0 ������������1 ������������2 ⋮ � At this point, the model is inconspicuously inelegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' However, upon the introduction of numerous groups from which the experimental data is sampled, it becomes evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Expanding this example to one regressor in two groups is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 4 ⎣ ⎢ ⎢ ⎢ ⎢ ⎡������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='0 ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='2 ⋮ ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='0 ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='1 ⋮ ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡1 0 ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='0 0 1 0 ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='1 0 ⋮ 0 0 0 0 1 1 ⋮ ⋮ 0 0 0 0 ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='0 ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='1 ⋮ ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ � ������������1 ������������2 ������������1 ������������2 � + ������������ Clearly in this staggered configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' as more groups are introduced into the model the matrix continues to grow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' with the majority of entries being equal to zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' which yields an unnecessarily large number of computations and quantity of memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Moreover, without a priori knowledge of both the number of groups and number of regressors, it is impossible to predict the organizational structure of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Tensor Formulation of the GLM These problems are avoided with an alternative tensor formulation of the model expressed in Einstein notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Notably, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 2 Can be written as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 5 ������������������������,������������ = ������������������������ ������������,������������������������������������,������������ + ������������������������,������������ Here, k indexes the samples of the experimental mapping, ������������ indexes over the group, and ������������ indexes over the intercept and each regression variable or parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' From this, the extension to the translation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 1 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' An example of such formulation in hybrid matrix Einstein notation with one regression parameter is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 6 � ������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ⋮ � = ⎣ ⎢ ⎢ ⎢ ⎡������������0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ = 1 ������������1 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ = 1 ������������1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ 1 ⋮ ������������1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ⋮ ⎦ ⎥ ⎥ ⎥ ⎤ ������������������������� ������������������������� + � ������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ⋮ � GLM Contrast Matrix in Tensor Notation The null hypothesis (������������0) statements to test in the GLM take the form of a linear combination of the atlas parameters in ������������ is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' This linear combination is conventionally expressed in a manner outlined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 7 ������������ = ������������������������ Where, g corresponds to the value ������������0 asserts is equal to zero and ������������ is the GLM contrast matrix, which is a covariant vector with indices corresponding to the atlas parameters in ������������ which serves as their linear coefficients in the ������������0 statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' This expression is identically inelegant and separate statements are constructed for multiple hypothesis testing such as F-testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' The compatible expression in the tensor formulation is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 8 ������������������������ = ������������������������,������������ ������������ ������������������������,������������ This model is compatible with multiple ������������0, which ������������ indexes over, for F-testing or multiple t-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' An example of such expression with two ������������0 statements in a model with two separate groups and a single regressor is outlined component-wise in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 9-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 9-10 ������������0,0 = ������������1, ������������1,0 = ������������1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 11-12 ������������0,1 = ������������2, ������������1,1 = ������������2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 12-13 ������������0,0 0 = ������������������������1 0 , ������������1,0 0 = ������������������������1 0 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 13-14 ������������0,1 0 = ������������������������2 0 , ������������1,1 0 = ������������������������2 0 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 14-15 ������������0,0 1 = ������������������������1 1 , ������������1,0 1 = ������������������������1 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 15-16 ������������0,1 1 = ������������������������2 1 , ������������1,1 1 = ������������������������2 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 17 ������������0 = ������������������������1 0 ������������1 + ������������������������1 0 ������������1 + ������������������������2 0 ������������2 + ������������������������2 0 ������������2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 18 ������������1 = ������������������������1 1 ������������1 + ������������������������1 1 ������������1 + ������������������������2 1 ������������2 + ������������������������2 1 ������������2 GLM Multiple T-Test in Tensor Notation The justification for representing the various applications of the GLM in a tensor formulation is self-evident at this point, and in most cases it is straightforward to infer such representations from the conventional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' However, this is not always true, especially in embodiments which require inverting matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' The t statistic is of the most important of these embodiments, which is represented in the conventional matrix notation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 19 ������������ = ������������������������ �������������2������������(������������������������������������)−1������������������������ Where the t statistic is generated separately for each ������������0, and ������������2 is the variance of the experimental outcome measure in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' To express this in the tensor formulation, it is evident that the numerator is g, which is indexed for each hypothesis and, consequently, so is t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Moreover, it is clear that contracting a matrix with its transposed self can be expressed as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 20 ������������������������ ������������′,������������ = ������������������������������������ = ������������������������ ������������′,������������������������������������ ������������,������������ Moreover, the inverse of a matrix expressed in tensor notation is computed as Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Roger Penrose puts forth4 and as is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 21 [������������−1]������������ ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ = 2[������������������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������2������������������������′1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������′2������������������������1 ������������′1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������2 ������������′2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������]−1������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������2������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������′2������������������������2 ������������′2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ Where ������������ is the totally antisymmetric Levi-Civita symbol which is defined from the sign by the permutation of its indices such that each value is a power of (-1) which matches the parity of the permutation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' otherwise the value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' ∴ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 22 (������������������������������������)−1 = 2[������������������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������2������������������������′1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������′2(������������������������ ������������′1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������1 ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������)(������������������������ ������������′2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������2 ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������)]−1������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������2������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������′2(������������������������ ������������′2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������2 ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������) ∴ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 23 ������������������������ = ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������ ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ �(������������������������)2������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ ������������ 2[������������������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������2������������������������′1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������′2(������������������������ ������������′1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������1 ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������)(������������������������ ������������′2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������2 ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������)]−1������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������2������������������������′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������′2(������������������������ ������������′2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������������������������������2 ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������)������������ ������������ ������������′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='������������ Results and Discussion The tensor formulation of the GLM drastically decreases the number of elements in the data structures and reduces the quantity of operations required to perform computations with said data structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' especially as more groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' regressors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' and hypotheses are incorporated in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' This has the potential to significantly reduce the time required to test various hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Moreover, the automation of hypothesis testing with the GLM is significantly simplified in the tensor formulation by the property that no a priori knowledge of the number groups, regressors, and hypotheses is needed to infer the structural organization of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Finally, this solution is simply more elegant, as the rank of the tensors is complementary to the degrees of freedom of the information which the data structure in the GLM is designed to interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' There are hundreds of unique applications of the GLM, each of which can be formulated in this proposed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Here, I have set forth the general structure of such formulations with a few examples, but the literature would benefit from further translation of other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Mccullagh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Generalized linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Khuri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=', Mukherjee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=', Sinha, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' & Ghosh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Design Issues for Generalized Linear Models: A Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content='1214/088342306000000105 21, 376–399 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Åhlander, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Einstein summation for multidimensional arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Computers & Mathematics with Applications 44, 1007–1017 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Roger Penrose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Applications of Negative Dimensional Tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} +page_content=' Birkbeck College, University of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf'} diff --git a/H9E4T4oBgHgl3EQfgw3C/content/tmp_files/2301.05120v1.pdf.txt b/H9E4T4oBgHgl3EQfgw3C/content/tmp_files/2301.05120v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..79b850c543109fb671a7aea61a2f60c209f95151 --- /dev/null +++ b/H9E4T4oBgHgl3EQfgw3C/content/tmp_files/2301.05120v1.pdf.txt @@ -0,0 +1,1113 @@ +arXiv:2301.05120v1 [math.PR] 12 Jan 2023 +STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES +RELATED TO PSEUDO DIFFERENTIAL EQUATIONS +VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER +Abstract. This is a review article which presents part of the contribution of Sergio +Albeverio to the study of existence and uniqueness of solutions of SPDEs driven +by jump processes and their stability properties. The results on stability properties +obtained in Albeverio et al. [1] are presented in a slightly simplified and different way. +stochastic partial differential equations; non -Gaussian additive noise; existence; +uniqueness; Itˆo formula, invariant measures for infinite-dimensional dissipative sytems +AMS classification 2020: +60H15, 60G51, 37L40, 60J76 +1. Introduction +The theory of SPDE’s driven by Brownian motion was studied for a long time and +solutions taking values in a Hilbert space are described in [17], [8] based on previ- +ous work. Sergio Albeverio was among the first mathematicians to initiate the study +of SPDE driven by jump processes [4] with solutions in Hilbert spaces in contrast to +Kallianpur and Xiong [10] who studied generalized solutions. In order to study these +equations in general, Sergio et al. +provided the L´evy -Itˆo decomposition in Banach +spaces [3]. There was a previous approach by E. Dettweiler [6], where the stochastic +integrals are defined differently from those of Itˆo. In [3] it was however proven that +the definitions are equivalent. Starting from [3](M- type 2 and type 2) Banach valued +stochastic integrals with respect to L´evy processes and compensated Poisson random +measures associated to additive processes were defined in [18], [11], [12], including the +case of separable Hilbert space valued stochastic integrals. (For the theory of stochastic +integration on Banach spaces see also [16], [21] and references there) Following this, +the article [2] establishes the basic generalization of classical work for mild solutions of +SPDE’s driven by L´evy processes and associated Poisson random noise. An Itˆo-formula +was proved in this case [19] which was later generalized in [13] and further in [1]. It has +interesting applications to stability of solutions of such SPDEs which originated in [13] +and have been continued in [12] and by Albeverio et al. in [1]. +Our project in this paper is to present first a review of the work mentioned. The results +related to the stability properties obtained in [1] are presented in Section 4 in a sim- +plified and slightly different way, by involving a dissipativity condition (condition i) in +Theorem 4.4). These motivated further investigations of stability properties of SPDEs +with multiple invariant measures in [7], which introduces a ”generalized dissipativity +condition”, that are not reported in this paper due to stipulated page limitations for +this article. +1 + +2 +VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER +2. Stochastic Integrals and Itˆo -formula +Consider a filtered probability space (Ω, F, {Ft}t≥0, P) satisfying the usual conditions. +Let H be a separable Hilbert space with norm ∥·∥H and scalar product < ·, · >H, which +for simplicity we will often denote with ∥ · ∥ and < ·, · >. Let {Lt}t≥0 be an H - valued +L´evy process on (Ω, F, {Ft}t≥0, P). Let B(H) denote the Borel -σ - Algebra on H. For +B∈ B(H) with 0 /∈ B, we define +N((0, t] × B) = +� +0 0 such that +h(x + y) +≤ C +� +h(x) + h(y) +� +, +x, y ∈ R+, +h(xy) +≤ Ch(x)h(y), +x, y ∈ R+. +In [13] the following was proved: +Theorem 2.3. Let us assume +a) H ∈ C1,2(R+ × H; F) is a function such that +∥∂yH(s, y)∥ ≤ h1(∥y∥), +(s, y) ∈ R+ × H +∥∂yyH(s, y)∥ ≤ h2(∥y∥), +(s, y) ∈ R+ × H +for quasi-sublinear functions h1, h2 : R+ → R+. +b) f : H × R+ × Ω → F is a progressively measurable process such that for all +t ∈ R+ we have P–almost surely +� t +0 +� +A ∥f(s, x)∥2ν(ds, dx) + +� t +0 +� +A h1(∥f(s, x)∥)2∥f(s, x)∥2ν(ds, dx) ++ +� t +0 +� +A h2(∥f(s, x)∥)∥f(s, x)∥2ν(ds, dx) < ∞. +Then the Itˆo -Formula 1. with 2. holds. +Remark 2.4. We remark that H(y) = ∥y∥2 is of class C2(H; R) and +Hy(y)v = 2 < y, v > +and +Hyy(y)(v)(w) = 2 < v, w > . +(2.3) +so that if for all t ∈ R+ we have P–almost surely +� t +0 +� +A ∥f(s, x)∥2ν(ds, dx) < ∞ and +� t +0 +� +A ∥f(s, x)∥4ν(ds, dx) < ∞, then Theorem 2.3 can be applied to H(s, y) := H(y) = +∥y∥2 . +3. SPDEs on Hilbert spaces +In this section we shall be studying Stochastic Partial Differential Equations (SPDEs) +driven by L´evy processes. Let (H, ∥ · ∥H) be a Hilbert space and A be an infinitesimal +generator of a semigroup {St, t ≥ 0} on H to H. This means +i) S0 = I +ii) Ss+t = SsSt +∀s, t ≥ 0 +We also assume that {St, t ≥ 0} is strongly continuous, i.e. +iii) limt→0 Stx = 0 (in norm ∥ · ∥H) + +STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES RELATED TO PSEUDO DIFFERENTIAL EQUATIONS5 +If {St, t ≥ 0} is a semigroup satisfying the above properties , we call it a ”strongly +continuous semigroup” (C0 - Semigroup). +For such a semigroup we note that there +exists α ≥ 0 and M ≥ 1 such that the operator norm in the space L(H) of bounded +linear operators from H to H satisfies +∥St∥L(H) ≤ Meαt +t ≥ 0. +We call the semigroup {St, t ≥ 0} ”pseudo -contraction” semigroup if M = 1, ”uni- +formly bounded -semigroup” if α = 0 and ”contraction semigroup” if M = 1 and α = 0. +If t → St is differentiable for all x ∈ H then the semigroup {St, t ≥ 0} is differentiable. +Let {St}:= {St, t ≥ 0} be a C0 -semigroup on H. The linear operator A with domain +D(A) := {x ∈ H, lim +t→0+ +Stx − x +t +exists} +defined by +Ax = lim +t→0+ +Stx − x +t +is called the infinitesimal generator (i.g.) of {St}. +The following facts for an i.e. A of a C0 -semigroup {St} are well known (see e.g. [15]): +(1) +For x ∈ H limh→0 1 +h +� t+h +t +Ssxds = Stx. +(2) +For x ∈ D(A), Stx ∈ D(A) and +d +dtStx = AStx = StAx. +(3) +For x ∈ H, +� t +0 Ssxds ∈ D(A) and A +� t +0 Ssxds = Stx − x. +(4) +D(A) is dense in H and A is a closed operator. +(5) +Let f : [0, T] → D(A) be a measurable function with +� T +0 ∥f(s)∥D(A)ds < ∞, +then +� T +0 f(s)ds ∈ D(A) and +� T +0 Af(s)ds = A +� T +0 f(s)ds. +We associate with A the resolvent set ρ(A) as the set of complex numbers λ for which +λI − A has bounded inverse +R(λ, A) := (λI − A)−1 ∈ L(H) +and we call R(λ, A), λ ∈ ρ(A) the resolvent of A. +We note that R(λ, A) : H → D(A) is one- to -one, i.e. +(λI − A)R(λ, A)x = x, +x ∈ H +and +R(λ, A)(λI − A)x = x, +x ∈ D(A), +giving +AR(λ, A)x = R(λ, A)Ax, +x ∈ D(A) +Remark that R(λ1, A)R(λ2, A)= R(λ2, A)R(λ1, A) for λ1, λ2 ∈ D(A). +Lemma 3.1. Let {St} be C0 -semigroup with infinitesimal generator A. Let +α0 := lim +t→∞ t−1ln(∥St∥L(H)), +then any real number λ > α0 belongs to the resolvent set ρ(A) and +R(λ, A)x = +� ∞ +0 +e−λtStxdt +x ∈ E +In addition for x ∈ H +lim +λ→∞ ∥λR(λ, A)x − x∥H = 0 + +6 +VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER +Theorem 3.2. Hille - Yosida Theorem Let A : D(A) ⊂ H → H be a linear +operator on a Hilbert space H. Necessary and sufficient conditions for A to generate a +C0 -semigroup is +(1) +A is closed and D(A)= H +(2) +There exists α, M ∈ R such that for λ > α, λ ∈ ρ(A) +∥R(λ, A)r∥L(H) ≤ M(λ − α)−r, +r = 1, 2, ... +In this case ∥St∥L(H) ≤ Meαt, t ≥ 0. +For λ ∈ ρ(A), consider the family of operators +Rλ := λR(λ, A). +Since the range R(R(λ, A)) of R(λ, A) is such that R(R(λ, A)) ⊂ D(A), we define the +”Yosida approximation” of A by +Aλx = ARλx, +x ∈ H +Using λ(λI − A)R(λ, A) = λI it is easy to prove +Aλx = λ2R(λ, A) − λI, +Aλ ∈ L(H) +Denote by Sλ +t the uniformly continuous semigroup +Sλ +t x = etAλx, +x ∈ H +Using the commutativity of the resolvent, we get Aλ1Aλ2 = Aλ2Aλ1, and clearly +AλSλ +t = Sλ +t Aλ +Theorem 3.3. Yosida approximation Let A be an infinitesimal generator of a C0 +-semigroup {St} on a Hilbert space H. Then +a) +limλ→∞ Rλx = x, +x ∈ H +b) +limλ→∞ Aλx = Ax, +for +x ∈ D(A) +c) +limλ→∞ Sλ +t x = Stx, +x ∈ H +The convergence in c) is uniform on compact subsets of R+ and +∥Sλ +t ∥L(H) ≤ Mexp +� t ∧ α +λ − α +� +with constants M, α as in Hille -Yosida Theorem +We conclude this section by introducing a concept of solution. +Let us look at the +deterministic problem +du(t) +dt += Au(t), u(0) = x, +x ∈ H +Here H is a real separable Hilbert space and A is an unbounded operator generating a +C0 -semigroup. +A classical solution u : [0, T] → H of the above equation will require a solution to be +continuously differentiable and u(t)∈ D(A). However, +ux(t) = Stx, +t ≥ 0 +is considered as a (mild) solution to the equation ([15], Capt.4). +One can consider the non -homogeneous equation +du(t) +dt += Au(t) + f(t, u(t)), u(0) = x, +x ∈ H + +STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES RELATED TO PSEUDO DIFFERENTIAL EQUATIONS7 +then for f ∈ L1([0, T], H), Bochner integrable, one can consider the integral equation +ux(t) = Stx + +� t +0 +St−sf(s, u(s))ds +(3.1) +A solution of (3.1) is called a ”mild solution”, if u ∈ C([0, T], H). +Motivated by the initial work of Sergio Albeverio with Wu and Zhang [4], we con- +tinued with Sergio [2] and further in [12] to analyze mild solutions of stochastic par- +tial differential equations (SPDEs) with Poisson noise on any filtered probability space +(Ω, F, {Ft}t≥0, P), satisfying the usual conditions with values on a separable Hilbert +space H. (For this topic see also the monograph by Peszat and Zabczyk [16] and refer- +ences there.) Remark that the stochastic integral +� t +0 St−sf(s, x)q(ds, dx), which appears +in such SPDEs, is in general not a martingale. However similar to Doob inequalities +the following Lemma holds. +Lemma 3.4. [Lemma 5.1.9 [12]] Assume {St}t≥0 is pseudo-contractive. Let q(ds, dx) be +a compensated Poisson random measure on R+×E, for some Hilbert space E, associated +to a Poisson random measure N with compensator dt⊗β(dx) on (Ω, F, {Ft}t≥0, P). For +each T ≥ 0 the following statements are valid: +(1) There exists a constant C > 0 such that for each f ∈ L2 +ad(H) we have +E +� +supt∈[0,T] +���� +� t +0 +� +E St−sf(s, x)q(ds, dx) +���� +2� +≤ +Ce2αT E +� � T +0 +� +E ∥f(s, x)∥2β(dx)ds +� +. +(3.2) +(2) For all f ∈ L2 +ad(H) and all ǫ > 0 we have +P +� +supt∈[0,T] +���� +� t +0 +� +E St−sf(s, x)q(ds, dx) +���� > ǫ +� +(3.3) +≤ 4e2αT +ǫ2 +E +� � T +0 +� +E ∥f(s, x)∥2β(dx)ds +� +. +where +� t +0 St−sf(s, x)q(ds, dx) is well defined, if the right side is finite. +� t +0 St−sf(s, x)q(ds, dx) +is c`ad`ag. +Let us assume that we are given +F : H → H , +(3.4) +f : H × H → H , +(3.5) +Assume +A) +f(u, z) is jointly measurable, +B) +F(z) is measurable, +C) +there exist constants Lf and LF > 0, s.th. +∥F(z) − F(z′)∥2 ≤ LF ∥z − z′∥2 +� +H ∥f(u, z) − f(u, z′)∥2β(du) ≤ Lf∥z − z′∥2 +for +all +z, z′ ∈ H + +8 +VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER +D) +� +H +∥f(u, 0)∥2β(du) < ∞ +(3.6) +E) +A is the infinitesimal generator of a pseudo - contraction semigroup {St}t∈[0,T] . +Remark that Assumptions C) and D) imply that there is a constant K > 0 such that +� +H +∥f(u, z)∥2β(du) ≤ K(1 + ∥z∥2) < ∞, +(3.7) +since +� +H ∥f(u, z)∥2β(du) ≤ 2 +� +H ∥f(u, z) − f(u, 0)∥2β(du) + 2 +� +H ∥f(u, 0)∥2β(du) +≤ 2max{Lf, +� +H ∥f(u, 0)∥2β(du)}(1 + ∥z∥2) < ∞ +In Albeverio et al. +[2] and [12], we analyzed (in more generality than in Theorem +3.8 below) the existence and uniqueness of mild solutions of the stochastic differential +equation on intervals [0, T], T > 0, like e.g. +dXt += +(AXt + F(Xt))dt + +� +H +f(u, Xt)q(dt, du) +(3.8) +X0 += +ξ. +(3.9) +where q(dt, du) := N(dt, du) − dtβ(du) is a compensated Poisson random measure with +compensator ν(dt, du) := dtβ(du). +In other words, we looked at the solution of the integral equation +Xt = StX0 + +� t +0 +St−sF(Xs)ds + +� t +0 +� +H +St−sf(u, Xs)q(ds, du) +(3.10) +where integrals on the r.h.s. are well defined [12]. +Definition 3.5. A stochastic process X· is called a mild solution of (3.8), if for all +t ≤ T +(i) Xt is Ft-adapted on a filtered probability space (Ω, F, {Ft}t≤T , P), +(ii) {Xt, t ≥ 0} is jointly measurable and +� T +0 E ∥Xt∥2 +H dt < ∞, +(iii) X· satisfies (3.10) P -a.s. on [0, T]. +Definition 3.6. A stochastic process X· is called a strong solution of (3.8), if for all +t ≤ T +(i) Xt is Ft-adapted on a filtered probability space (Ω, F, {Ft}t≤T , P), +(ii) X· is c`adl`ag with probability one, +(iii) Xt ∈ D(A), dt ⊗ dP a.e., +� T +0 ∥AXt∥H dt < ∞ P -a.s., +(iv) X· satisfies (3.8) P -a.s. on [0, T]. +Obviously, a strong solution X· of (3.8) is a a mild solution of (3.8). The contrary is +not neccesserily true, since e.g. Xt ∈ D(A) might not be true. (See e.g. Section 2.2 in +Albeverio et al. [1] where sufficient conditions for a mild solution X· of (3.8) are listed, +for X· to be also a strong solution.) +Let S2 +T be the linear space of all c`adl`ag, adapted processes X· such that +E +� +sup +t∈[0,T] +∥Xt∥2 +F +� +< ∞, +(3.11) + +STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES RELATED TO PSEUDO DIFFERENTIAL EQUATIONS9 +where we identify processes whose paths coincide almost surely. +Note that, by the +completeness of the filtration, adaptedness does not depend on the choice of the repre- +sentative. +Lemma 3.7. [Lemma 4.2.1 [12]] The linear space S2 +T, equipped with the norm +∥X·∥S2 +T = E +� +sup +t∈[0,T] +∥Xt∥2 +�1/2 +, +(3.12) +is a Banach space. +Theorem 3.8. [Theorem 5.3.1 [12]] Suppose assumptions A) -E) are satisfied. Then +for ξ∈ L2(Ω, F0, P; H) and T > 0, there exists a unique mild solution Xξ +· in S2 +T to (3.8) +with initial condition ξ, and satisfying Xξ +t is Ft -measurable. +Remark 3.9. For each ξ, η, ∈ L2(Ω, F0, P; H), the corresponding unique solutions Xξ +· +and Y η +· +to (3.8) in Theorem 3.8 satisfy +E +� +∥Xt − Yt∥2 +H +� +≤ C(T)E +� +∥ξ − η∥2 +H +� +, +t ∈ [0, T]. +(3.13) +for some constant C(T) depending on T > 0 (See Section 5.7 in [12]). +If X0 ≡ x ∈ H, then the corresponding solution Xx +· to (3.8) in Theorem 3.8 is Markov +(See Section 5.4 in [12]) . Such solution constitutes a Markov process whose transition +probabilities pt(x, dy) = P[Xx +t ∈ dy] are measurable with respect to x. By slight abuse of +notation we denote by (pt)t≥0 its transition semigroup, i.e., for each bounded measurable +function f : H −→ R, ptf is given by +ptf(x) = E [f(Xx +t )] = +� +H +f(y)pt(x, dy), +t ≥ 0, +x ∈ H. +(3.14) +Since due to (3.13) the solution dependences continuosly on the initial condition, it +can be shown that ptf ∈ Cb(H) for each f ∈ Cb(H), i.e. the transition semigroup is +Cb-Feller. +Let Rn = nR(n, A), with n ∈ N, n ∈ ρ(A), the resolvent set of A, R(n, A) = (nI −A)−1 +. The (SPDE) +dXt += +(AXt + RnF(Xt))dt + +� +H +Rnf(u, Xt)q(dt, du) +(3.15) +X0 += +Rnξ(ω). +obtained by Yosida Approximation of (3.8) has a unique strong solution Xn,ξ +· +which +approximates its mild solution Xξ +· of (3.8) with initial condition Xξ +0 = ξ. The precise +statement is given in the following Theorem: +Theorem 3.10. Suppose assumptions A) -E) are satisfied. Then for ξ∈ L2(Ω, F0, P; H) +and T > 0, there exists a unique strong solution Xn,ξ +· +:= +� +Xn,ξ +t +, t ≥ 0 +� +in S2 +T to (3.15) +with initial condition ξ, and satisfying Xn,ξ +t +is Ft -measurable ∀t ≥ 0. Moreover, +lim +n→∞ E +� +sup +0≤t≤T +���Xn,ξ +t +− Xξ +t +��� +2 +H +� += 0, +(3.16) +where Xξ +· := +� +Xξ +t , t ≥ 0 +� +is the mild solution of equation (3.8) with initial condition ξ. +For the proof see Theorem 2.9 of Albeverio et al. [1]. + +10 +VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER +Definition 3.11. Xn,ξ +· +is called “the Yosida approximation of Xξ +· ”. +Remark 3.12. Let A be the infinitesimal generator of a pseudo - contraction semi- +group {St}t∈[0,T] . Assume that X· is a strong solution of (3.8) and all the hypotheses +in Theorem 2.3 are satisfied. Then the Itˆo -Formula holds and can be written in the +following way: +P–almost surely +H(t, Xt) = H(0, X0) + +� t +0 ∂sH(s, Xs)ds + +� t +0 LH(s, Xs)ds + +� t +0 +� +A +� +H(s, Xs− + f(s, u)) − H(s, Xs−) +� +q(ds, du) +with +LH(s, x) :=< ∂xH(s, x), Ax + F(x) > + +(3.17) +� +H +� +H(s, x + f(s, u)) − H(s, x)− < ∂xH(s, x), f(s, u) > +� +β(du) +Remark 3.13. Assume that hypotheses A)-E) and all hypotheses a) and b) in Theorem +2.3 are satisfied. Then the Itˆo -Formula for the Yosida approximation Xn,ξ +· +of the mild +solution Xξ +· of (3.8) holds and can be written in the following way: +H(t, Xn,ξ +t +) = H(0, Xn,ξ +0 +) + +� t +0 ∂sH(s, Xn,ξ +s +)ds + +� t +0 LnH(s, Xn,ξ +s +)ds + +� t +0 +� +A +� +H(s, Xn,ξ +s− + Rnf(s, u)) − H(s, Xn,ξ +s− ) +� +q(ds, du) +with +LnH(s, x) :=< ∂xH(s, x), Ax + RnF(x) > + +� +H +� +H(s, x + Rnf(s, u)) − H(s, x)− < ∂xH(s, x), Rnf(s, u) > +� +β(du) +This follows directly from Theorem 3.10 and Remark 3.12. +In the next Section we will use the following result, which was obtained in [1] as a +consequence of an Itˆo -formula for mild solutions of SPDEs, introduced in Albeverio et +al. [1] and written in terms of Yosida approximation +Theorem 3.14. [Corollary 3.7. [1] ] Assume conditions A)- E) and all the hypotheses +in Theorem 2.3 are satisfied. Then +lim +n→∞ |LH(s, Xn,ξ +s +) − LnH(s, Xn,ξ +s +)| +P − a.s. +(3.18) +4. Some stability properties for solutions of SPDEs on Hilbert spaces +In this Section we discuss how the Itˆo Formula in Theorem 2.3 was applied by Albev- +erio et al. [1] to establish through a Lyapunov function approach stability properties +for the mild solution of (3.8) converging to a unique invariant measure. +Assumption We assume in the whole Section that conditions A) -E) are satisfied. +The mathematical tools introduced in [1] have been later extended in [7] to analyze the +limiting behaviour of mild solutions of SPDEs with multiple invariant measure. This +will however not be discussed here, due to a problem of space. + +STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES RELATED TO PSEUDO DIFFERENTIAL EQUATIONS +11 +We start to recall some definition related to the Lyapunov function approach presented +in [14] (see also the PhD thesis of the second author L. Wang) as well as [8], [12], [1]. +Definition 4.1. We say that the solution of (3.8) is exponentially stable in the mean +square sense if there exists c, ǫ > 0 such that for all t > 0 and ξ∈ L2(Ω, F0, P; H) +E[∥Xξ +t ∥2] ≤ ce−ǫtE[∥ξ∥2] +(4.1) +Definition 4.2. Let L be defined as in (3.17). A function H : H → R ∈ C2(H; R) is +a Lyapunov function for the SPDE (3.8) if it satisfies the following conditions: +I. There exist finite constants c1, c2> 0 such that for all x ∈ H +c1∥x∥2 ≤ H(x) ≤ c2∥x∥2 +II. There exists a constant c3 > 0 such that +LH(x) ≤ −c3H(x) +∀x ∈ D(A) +In Albeverio et al. [1] we proved the following Theorem +Theorem 4.3. [[1] ] Assume that there exists a function H ∈ C2(H; R) which is a +Lyapunov function for the SPDE (3.8) and the hypotheses a) and b) in Theorem 2.3 +are satisfied. Then the mild solution of (3.8) is exponentially stable in the mean square +sense. Moreover the constants in (4.1) can be chosen so that c = c2 +c1 and ǫ = c3. +Remark that for the case H ∈ C2 +b (H; R) a proof can be found in [[14] Theorem 4.2] +(see also [[19] Section 7.1] and for the Gaussian case [[8] Theorem 6.4]. The results are +stated there for the Yosida approximants. +Proof. Since all the hypotheses of Theorem 2.3 are satisfied, Itˆo formula can be applied +to the Yosida approximation. +ec3tE[H(Xn,ξ +t +) − H(Rnξ)] = E +�� t +0 +ec3sc3(H(Xn,ξ +s +) + LnH(Xn,ξ +s +))ds +� +(4.2) +From condition I it follows +c3H(Xn,ξ +s +) + LnH(Xn,ξ +s +) ≤ −LH(Xn,ξ +s +) + LnH(Xn,ξ +s +) +(4.3) +ec3tE[H(Xn,ξ +t +) − H(Rnξ)] ≤ E +�� t +0 +ec3s(−LH(Xn,ξ +s +) + LnH(Xn,ξ +s +))ds +� +(4.4) +From Theorem 3.10 and Theorem 3.14 it follows ec3tE[H(Xξ +t )] ≤ E[H(ξ)]. Condition II +implies then +c1E[∥Xξ +t ∥2] ≤ E[H(Xξ +t )] ≤ e−c3tE[H(ξ)] ≤ c2e−c3tE[∥ξ∥2] +(4.5) +and hence +E[∥Xξ +t ∥2] ≤ c2 +c1 +e−c3tE[∥ξ∥2] +(4.6) +The statement follows by choosing c = c2 +c1 and ǫ = c3. +□ +Using Theorem 4.3 we can provide an easy proof of the following statement, known in +the literature from e.g. [Section 16, [17]] and [Chapter 11, Section 5, [16]]. +Theorem 4.4. Assume that the conditions A) - E) are satisfied for (3.8), and the +following conditions hold +i) A satisfies the ”dissipativity condition” , i.e there exists α > 0 such that +< Ax − Ay, x − y > + < F(x) − F(y), x − y > +≤ −α∥x − y∥2 +∀x, y ∈ D(A); +(4.7) + +12 +VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER +ii) ǫ := 2α − Lf > 0. +iii) ∀z ∈ H +� +A ∥f(u, z)∥4β(du) < ∞ +Then for all ξ, η ∈ L2(Ω, F0, P; H) +E[∥Xξ +t − Xη +t ∥2] ≤ e−ǫtE[∥ξ − η∥2] +∀t > 0 +(4.8) +Proof. The stochastic process Xξ +· − Xη +· is the mild solution of +d(Xξ +t − Xη +t ) += +A(Xξ +t − Xη +t )dt + (F(Xξ +t ) − F(Xη +t ))dt ++ +� +H +(f(u, Xξ +t ) − f(u, Xη +t ))q(dt, du) +(4.9) +Xξ +0 − Xη +0 += +ξ − η. +(4.10) +Condition iii) implies that all hypothesis of Theorem 2.3 are satisfied for H(x, y) := +∥x − y∥2. Moreover, according to the definition of L in (3.17), we have +L∥x − y∥2 := 2 < x − y, A(x − y) > +2 < x − y, F(x) − F(y) > ++ +� +H ∥f(u, x) − f(u, y)∥2β(du) +where we used that +∥x − y + f(u, x) − f(u, y)∥2 − ∥x − y∥2 +−2 < x − y, f(u, x) − f(u, y) >= ∥f(u, x) − f(u, y)∥2 +Conditions i) and ii) imply that the function H(x, y) := ∥x−y∥2 is a Lyapunov function +for (4.9) with c1 = c2 = 1 and c3 = ǫ. Hence Xξ +· − Xη +· is exponentially stable in the +mean square sense. +□ +We denote by p∗ +t the adjoint operator to pt defined in (3.14), i.e. +p∗ +tρ(dx) = +� +H +pt(y, dx)ρ(dy), +t ≥ 0. +Recall that a probability measure π on (H, B(H)) is called invariant measure for the +semigroup (pt)t≥0 if and only if p∗ +tπ = π holds for each t ≥ 0. Let P2(H) be the space +of Borel probability measures ρ on (H, B(H)) with finite second moments. Recall that +P2(H) is separable and complete when equipped with the Wasserstein-2-distance +W2(ρ, �ρ) = +inf +G∈H(ρ,�ρ) +�� +H×H +∥x − y∥2 +HG(dx, dy) +� 1 +2 +, +ρ, �ρ ∈ P2(H). +(4.11) +Here H(ρ, �ρ) denotes the set of all couplings of (ρ, �ρ), i.e. Borel probability measures +on H × H whose marginals are given by ρ and �ρ, respectively, see [22, Section 6] for a +general introduction to couplings and Wasserstein distances. +As a consequence of our key stability estimate (4.8) we can provide, by following the +proof of Theorem 4.1 in [7], a proof for the existence and uniqueness of a unique limiting +distribution in the spirit of classical results such as [16, Section 16], [17, Chapter 11, +Section 5], and [20]. +Theorem 4.5. Assume that the conditions A) - E) are satisfied for (3.8), and the +conditions i)-iii) in Theorem 4.4 hold. Then +W2(p∗ +t ρ, p∗ +t �ρ) ≤ W2(ρ, �ρ)e−ǫt/2, +t ≥ 0, +(4.12) + +STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES RELATED TO PSEUDO DIFFERENTIAL EQUATIONS +13 +holds for any ρ, �ρ ∈ P2(H). In particular, the Markov process determined by (3.8) has a +unique invariant measure π. This measure has finite second moments and it holds that +W2(p∗ +tρ, π) ≤ W2(ρ, π)e−ǫt/2, +t ≥ 0, +(4.13) +for each ρ ∈ P2(H). +Proof. From Theorem 4.4 it follows +E[∥Xx +t − Xy +t ∥2 +H] ≤ e−ǫt∥x − y∥2 +H, +x, y ∈ H. +Using the definition of the Wasserstein distance, we conclude that +W2(p∗ +tδx, p∗ +t δy) ≤ +� +E[∥Xx +t − Xy +t ∥2 +H] +�1/2 ≤ ∥x − y∥He−ǫt/2. +The latter one readily yields (4.12). Finally, the existence and uniqueness of an invariant +measure as well as (4.13) can be derived from (4.12) combined with a standard Cauchy +argument. +□ +In [7] we introduced a “generalized dissipativity condition” and studied SPDEs with +multiple invariant measures. There we developed further the methods presented in this +Section, which have been mainly derived from Albeverio et al. [1] in combination with +the results obtained in [2], [12]. +Acknowledgment. I thank Peter Kuchling and Baris Ugurcan for a careful reading of +part of this article. +Comment by Barbara R¨udiger. My co-author and friend V. Mandrekar (Atma) +passed away the 23 June 2021. A couple of days before his departure he contacted me +through email to make sure the procedure for the submission of this article would be +successful. The invitation to contribute to this Volume, dedicated to Sergio Albeverio, +was accepted by him with enthusiasm. +Atma and Sergio had, to my feeling, a deep respect for each other and, despite the +geographic distance, a solid friendship. I think that this friendship and respect is also +due to common aspects they have in their character and soul: both are very generous +in sharing with other scientists their original ideas. Both trust in youngsters and enjoy +knowing that they can contribute to these with their own developments and ideas, as +well. +This way they both are friends, supporters, coaches and co -authors to many +young (and in the meanwhile older) mathematicians and physists. I feel very lucky to +be among them. +References +1. Sergio Albeverio, Leszek Gawarecki, Vidyadhar Mandrekar, Barbara R¨udiger, and Barun Sarkar, +Itˆo formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications +to stability properties, Random Oper. Stoch. Equ. 25 (2017), no. 2, 79–105. +2. Sergio Albeverio, Vidyadhar Mandrekar, and Barbara R¨udiger, Existence of mild solutions for sto- +chastic differential equations and semilinear equations with non-Gaussian L´evy noise, Stochastic +Process. Appl. 119 (2009), no. 3, 835–863. +3. Sergio Albeverio and Barbara R¨udiger, Stochastic integrals and the L´evy- itˆo decomposition theorem +on separable Banach spaces, Stoch. Anal. 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Leszek Gawarecki and Vidyadhar Mandrekar, Stochastic differential equations in infinite dimensions +with applications to stochastic partial differential equations, Probability and its Applications (New +York), Springer, Heidelberg, 2011. +9. Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, sec- +ond ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; +Kodansha, Ltd., Tokyo, 1989. +10. Gopinath Kallianpur and Jie Xiong, Stochastic differential equations in infinite-dimensional spaces, +Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 26, Institute of Math- +ematical Statistics, Hayward, CA, 1995. +11. V. Mandrekar and B. R¨udiger, Relation between stochastic integrals and the geometry of Banach +spaces, Stoch. Anal. Appl. 27 (2009), no. 6, 1201–1211. +12. 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Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, second ed., En- +cyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, +2014. +18. Barbara R¨udiger, Stochastic integration with respect to compensated Poisson random measures on +separable Banach spaces, Stoch. Stoch. Rep. 76 (2004), no. 3, 213–242. +19. Barbara R¨udiger and Giacomo Ziglio, Itˆo formula for stochastic integrals w.r.t. compensated Poisson +random measures on separable Banach spaces, Stochastics 78 (2006), no. 6, 377–410. +20. Anna Rusinek, Mean Reversion for HJMM Forward Rate Models, Advances in Applied Probability +42 (2010), no. 2, 371–391. +21. Jan van Neerven, Mark Veraar, and Lutz Weis, Stochastic integration in Banach space a survey, +Progr. Probab., vol. 68, Birkh¨auser/Springer, Basel, 2015. +22. C´edric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental +Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009, Old and new. +(Vidyadhar Mandrekar) Department of Statistics and Probability, Michigan State Uni- +versity, East Lansing, MI, USA +(Barbara R¨udiger) School of Mathematics and Natural Sciences, University of Wupper- +tal, Germany +Email address, Barbara R¨udiger: ruediger@uni-wuppertal.de + diff --git a/H9E4T4oBgHgl3EQfgw3C/content/tmp_files/load_file.txt b/H9E4T4oBgHgl3EQfgw3C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3c04406a0c60aed20a7858ca331f3e0b8ad1230 --- /dev/null +++ b/H9E4T4oBgHgl3EQfgw3C/content/tmp_files/load_file.txt @@ -0,0 +1,544 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf,len=543 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content='05120v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content='PR] 12 Jan 2023 STABILITY PROPERTIES OF MILD SOLUTIONS OF SPDES RELATED TO PSEUDO DIFFERENTIAL EQUATIONS VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' This is a review article which presents part of the contribution of Sergio Albeverio to the study of existence and uniqueness of solutions of SPDEs driven by jump processes and their stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' The results on stability properties obtained in Albeverio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' [1] are presented in a slightly simplified and different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' stochastic partial differential equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' non -Gaussian additive noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' existence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' uniqueness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Itˆo formula, invariant measures for infinite-dimensional dissipative sytems AMS classification 2020: 60H15, 60G51, 37L40, 60J76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Introduction The theory of SPDE’s driven by Brownian motion was studied for a long time and solutions taking values in a Hilbert space are described in [17], [8] based on previ- ous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Sergio Albeverio was among the first mathematicians to initiate the study of SPDE driven by jump processes [4] with solutions in Hilbert spaces in contrast to Kallianpur and Xiong [10] who studied generalized solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' In order to study these equations in general, Sergio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' provided the L´evy -Itˆo decomposition in Banach spaces [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' There was a previous approach by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Dettweiler [6], where the stochastic integrals are defined differently from those of Itˆo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' In [3] it was however proven that the definitions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Starting from [3](M- type 2 and type 2) Banach valued stochastic integrals with respect to L´evy processes and compensated Poisson random measures associated to additive processes were defined in [18], [11], [12], including the case of separable Hilbert space valued stochastic integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' (For the theory of stochastic integration on Banach spaces see also [16], [21] and references there) Following this, the article [2] establishes the basic generalization of classical work for mild solutions of SPDE’s driven by L´evy processes and associated Poisson random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' An Itˆo-formula was proved in this case [19] which was later generalized in [13] and further in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' It has interesting applications to stability of solutions of such SPDEs which originated in [13] and have been continued in [12] and by Albeverio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Our project in this paper is to present first a review of the work mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' The results related to the stability properties obtained in [1] are presented in Section 4 in a sim- plified and slightly different way, by involving a dissipativity condition (condition i) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' These motivated further investigations of stability properties of SPDEs with multiple invariant measures in [7], which introduces a ”generalized dissipativity condition”, that are not reported in this paper due to stipulated page limitations for this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' 1 2 VIDYADHAR MANDREKAR AND BARBARA R¨UDIGER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Stochastic Integrals and Itˆo -formula Consider a filtered probability space (Ω, F, {Ft}t≥0, P) satisfying the usual conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Let H be a separable Hilbert space with norm ∥·∥H and scalar product < ·, · >H, which for simplicity we will often denote with ∥ · ∥ and < ·, · >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Let {Lt}t≥0 be an H - valued L´evy process on (Ω, F, {Ft}t≥0, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' Let B(H) denote the Borel -σ - Algebra on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf'} +page_content=' For B∈ B(H) with 0 /∈ B, we define N((0, t] × B) = � 0500 Trps).16,17 Until now, the smallest +proteins detected with UV-FCS are penicillin amidase (29 Trps) and streptavidin (24 Trps).18 However, +tryptophan is among the least abundant amino acids in eukaryotic proteins,19 and so the vast majority of +proteins possess only a few Trp residues. A UniProt database survey out of more than 20,000 human proteins +shows that on average a human protein contains about 7 Trp residues, with half of the proteins bearing +between 1 and 5 Trps (Fig. 1a and Supporting Information Fig. S1).20 This shows that in order to really exploit +the full potential of UV autofluorescence detection and explore a broad library of label-free proteins, the +sensitivity of UV-FCS must be pushed by more than one order of magnitude down to the single tryptophan +level. +Here we address this sensitivity challenge and demonstrate UV-FCS on label-free proteins bearing a single +Trp residue. This is achieved thanks to the combination of (i) nanophotonic UV antenna to enhance the signal, +(ii) detailed analysis to reduce the background intensity and (iii) chemical photostabilizing agents to avoid +fluorescence saturation. Our results provide guidelines on how to extend plasmonics into the UV regime 21– +30 and further develop label-free single molecule spectroscopy.4,5,31–33 Earlier works using UV aluminum +nanophotonics were restricted to proteins containing a large number of Trp residues such as β-galactosidase +(156 Trps) 34,35 and streptavidin (24 Trps).36 Here we improve the sensitivity by more than one order of +magnitude, down to the single tryptophan level. This technical achievement opens the UV-FCS technique to +a huge library of proteins bearing only a few Trps (Fig. 1a). +We use a 266 nm deep UV laser excitation and a fluorescence collection in the 310 to 360 nm range for our +time-resolved UV confocal microscope (Fig. 1b). To maximize the autofluorescence signal, we employ an +optical horn antenna developed recently in our group (Fig. 1c).35 This nanophotonic device combines a central +metal nanoaperture together with a conical horn microreflector. The central nanoaperture of 80 nm +diameter concentrates the light in an attoliter detection volume and enhances the autofluorescence from + +3 + +proteins diffusing across this attoliter volume,34,37 while the metallized conical reflector collects the +fluorescence emitted at high angles and steers it towards the microscope objective.38–40 +Three different proteins are investigated in this work (Fig. 1a): thermonuclease staphylococcal nuclease +(TNase) from Staphylococcus aureus, transcription antiterminator protein (LicT) from Bacillus subtilis and +streptavidin from Streptomyces avidinii (Strep). TNase is a monomer bearing a single Trp residue, LicT is a +homodimer with a total of 2 Trps on the protein dimer, and streptavidin is a homotetramer with a total of 24 +Trps. All the details about the proteins used in this work are summarized in the Supporting Information Table +S1. TNase was selected because its single Trp residue was theoretically predicted 41 to have a quantum yield +of 28% in good agreement with ensemble spectroscopy measurements.2 TNase is also a widely studied as a +model system in protein chemistry.42,43 LicT was selected for its UV signal being comparable to TNase and its +availability in purified form labeled with Cy3B to serve as a control using visible fluorescence +spectroscopy.44,45 Streptavidin was selected because of its higher number of Trps, large availability, moderate +mass and good water solubility. +To estimate the feasibility of the UV-FCS detection of proteins with a single tryptophan, we numerically +compute the evolution of the FCS correlation amplitude 𝐺(0) = (1 − +𝐵 +𝐵+𝑁∗𝐶𝑅𝑀) +2 1 +𝑁 as a function of the +background intensity 𝐵, the fluorescence brightness per molecule 𝐶𝑅𝑀, the number of molecules 𝑁 and the +number of Trp residues per protein (Fig. 1d-f & Fig. S2).46–48 The ranges of values are taken to reproduce our +experiments. For a given number of molecules, the correlation amplitude quickly drops when the signal to +background 𝑁 𝐶𝑅𝑀/𝐵 decreases owing to the quadratic exponent in the (1 − +𝐵 +𝐵+𝑁∗𝐶𝑅𝑀) +2 + term. We indicate +on Fig. 1d-f different minimum thresholds for possible FCS detection corresponding to the contours where +𝐺(0) amounts to 0.001, 0.005 or 0.01. Since there is no general consensus in FCS in defining this minimum +threshold,46 we decide to show three different values. FCS amplitudes above 0.01 should be easily detectable +on a wide range of systems, whereas values below 0.001 are highly challenging as they come close to the +electronic noise level and the residual correlation from the background. The calculations results in Fig. 1d +and S2 show that maximizing the signal to background ratio 𝑁 𝐶𝑅𝑀/𝐵 is crucial to ensure the feasibility of +UV-FCS experiments. Using typical values of UV autofluorescence brightness and background intensity +representative of our experiments, we compute the predicted correlation amplitudes for the horn antenna +(Fig. 1e) and the confocal setup (Fig. 1f) as a function of the number of tryptophan residues per protein, +assuming for simplicity that all Trp residues contribute equally to the autofluorescence signal. For the horn +antenna, detectable correlation amplitudes above 0.01 are found for a single tryptophan provided the +number of proteins in the detection volume is between 2 and 60 (Fig. 1e). UV-FCS on a single tryptophan +protein appears feasible with the horn antenna. For the confocal reference, the single tryptophan always +yields correlation amplitudes below 0.001 (Fig. 1f). Increasing the number of proteins in the detection volume + +4 + +does not compensate for the lower signal to background ratio in this case. A realistic confocal UV-FCS +experiment requires that the protein carries at least 20 Trp residues. +Reducing the background intensity is crucial for improving the sensitivity of UV-FCS (Fig. 1g-i). Figure 1d +shows that even with the brightness enhancement brought by the horn antenna, UV-FCS on a single Trp +protein would be nearly impossible if the background intensity exceeds 2,000 counts/s. Experiments +performed on different samples milled by focused ion beam (FIB) show that the implantation of gallium oxide +resulting from the FIB process 49 is the major source of background in the horn antenna (Supporting +Information Fig. S3 & S4). Gallium oxide is luminescent when excited in the UV. 50,51 This background +contribution can be controlled by FIB while selecting the proper milling depth (Fig. S3). Moreover, the gallium +oxide luminescence spectrum is shifted toward 400 nm 50,51 and can be partially separated from the protein +autofluorescence spectra with the 310-360 nm bandpass filter (Fig. 1g). The gallium oxide +photoluminescence contains a long lifetime component, significantly longer than the 12.5 ns laser repetition +period (Fig. S4). Temporal gating to select the photon arrival time in a 3 ns window immediately after the +excitation pulse further reduces the background intensity without losing too much of the protein signal (Fig. +1h). The 3 ns window is chosen to correspond to approximately 3× the tryptohan fluorescence lifetime for +the different proteins used here in the horn antenna. Altogether, the combination of spectral filtering with +temporal gating reduces the background intensity by 7× and improves the signal to background ratio by 2.7× +(Fig. 1i) opening the possibility for single Trp detection. + + +5 + + +Figure 1. Optimizing signal to background ratio to detect a single tryptophan. (a) Histograms of the number +of tryptophan residues per protein extracted from a UniProt database of 20 399 reviewed human protein +entries.20 The limits of detection for the confocal 18 and the optical horn antenna (this work) are indicated. +The top images show the 3D structures of the proteins used in this work, made using Mol* viewer, with +tryptophan residues highlighted by magenta dots on selected monomers.52 (b) Scheme of the experiment. +(c) Scanning electron microscope image of a horn antenna combining a central nanoaperture and a conical +reflector. (d) Calculation of the FCS correlation amplitude 𝐺(0) as a function of the background intensity 𝐵 +and the fluorescence brightness per molecule 𝐶𝑅𝑀. A constant number of 𝑁 = 5 proteins was assumed, +each protein carrying a single tryptophan. All 2D maps in (d-f) share the same color scale. The lines at 0.001, +0.005 and 0.01 indicate the boundary threshold for possible FCS detection. Correlation amplitudes above +0.01 can be easily detected, while values below 0.001 are highly challenging if not impossible to detect. (e,f) +Calculations of the correlation amplitude for the horn antenna and the confocal case as a function of the +number of tryptophan residues per protein and the number of diffusing proteins in the detection volume. +The values for the background intensity and the fluorescence brightness per tryptophan residue are indicated +by the markers in (d) and correspond to typical values in our experiments. (g,h) Strategies to improve the +signal to background ratio (SBR) by spectral filtering and time gating. The protein data correspond to 8 µM + +a +TNase1Trp +LicT2Trp +Streptavidin24Trp +b +aluminum +C +Horn antenna= +quartz +80nmaperture+microreflector +UV +objective +0.8 NA +Trpresidue +Nanophotonichornantenna +266nm +Human +excitation +Confocal diffractionlimited +10 +20 +30 +40 +50 +500nm +0 +310-360nm +Trpresiduesperprotein +d +collection +e +f +Horn antenna +Confocal +Horn +100 +100 +0.01 +Correlation +antenna +amplitude +1 +Number of +Numberof +0.1 +FCS +10 +0.005 +Confocal +0.01 +FCS +1 +0.001 +ON +10 +100 +10 +100 +Trpresiduesperprotein +Trpresiduesperprotein +10 +100 +1000 +Background intensity(photons/second) +g +- +Dark counts +Bufferfluo.Al+SiO,background +Intensity +Ga,O +crystal +Nofilter +FIBGa,O,background +Protein signal +Protein +SBR 0.3 +Spectralfilter +300 +Wavelength (nm) +450 +Spectralfilter +SBR0.6 +Time gating +Protein +Spectral filter +SBR0.8 +Intensity ++ time gating +Background +0 +2 +3 +4 +5 +6 +0 +Time (ns) +10 +Detectoroutput(countspersecond）6 + +TNase solution in the horn antenna (6 proteins in the detection volume). The gallium oxide +photoluminescence spectra is taken from ref 50. (i) Experimental background intensity and total signal (8 µM +TNase solution) in the horn antenna upon spectral filtering and time gating. + + +Along with the reduction of background intensity, another major element determining the UV-FCS sensitivity +is the maximization of the fluorescence signal. This involves the use of chemical photostabilizing agents to +mitigate the buildup of the radical and triplet state populations leading to fluorescence saturation +(Supporting Information Fig. S5).18,53,54,55 We use mercaptoethylamine (MEA also known as cysteamine) or +gluthatione (GSH) which are effective in improving the autofluorescence of both TNase and LicT up to 8× (Fig. +S5) without introducing any significant additional background (Fig. S6). GSH is an antioxidant naturally +present in the human body to balance oxidative stress and neutralize reactive oxygen species (ROS).56 For +TNase and strepatvidin experiments, we have decided not to use any oxygen scavenging approach as the +TNase autofluorescence was not significantly affected by the presence of the oxygen dissolved in the buffer. +For LicT experiments, oxygen was removed by bubbling the solution with argon prior to recording the data +(see Methods). That way, dissolved oxygen was removed without adding any supplementary background +(Fig. S5f-h). +Figure 2 summarizes our main experimental results aimed at pushing the UV-FCS sensitivity down to the +single tryptophan level. The linear evolutions of the measured total intensities with the TNase and LicT +protein concentrations (Fig. 2a,b) provide a direct control that our experiments are sensitive to the protein +autofluorescence signal, even with proteins bearing only one or two Trp residues. The UV-FCS is computed +and fitted (Fig. 2c, S7-S8) to extract the number of detected proteins 𝑁 and their autofluorescence brightness +𝐶𝑅𝑀 (see Methods). Even in the absence of any protein sample, we still detect a residual background +correlation from the horn antenna filled with the buffer solution (gray trace in Fig. 2c). This background +correlation appears on all our traces with a long characteristic time above 50 ms and an amplitude below +0.01 (Fig. S7-S9) which may indicate an origin related to some remaining mechanical vibrations or electric +noise on our microscope. The significant difference between the characteristic correlation time of this +background (> 50 ms) and the protein diffusion time (< 0.5 ms) enables a clear separation of their +contributions in the FCS signal so that the contribution from the diffusing proteins can be recovered. Besides, +the correlation amplitude related to the protein is always at least twice larger than this residual background +correlation (Fig. S7,S8). In the absence of spectral filtering and time gating (Fig. S9), the correlation amplitude +found with the TNase protein falls down to 0.002 and cannot be distinguished from the background anymore. +Our results for both TNase and LicT are in good agreement with the numerical calculations in Fig. S1d: using +the experimentally determined parameters for the background intensity 𝐵 and the autofluorescence + +7 + +brightness 𝐶𝑅𝑀, the observed correlation amplitudes follow the theoretical predictions of our model (Fig. +S10). +Figure 2d-f compare the statistical distributions of the UV-FCS results for the horn antennas and different +proteins. The number of TNase molecules seen by UV-FCS follows a linear dependence with the protein +concentration (Fig. 2d). Statistical T-tests confirm the difference between the data distributions. On the +contrary, when we probe similar concentrations (8 µM) of different proteins (Fig. 2e), the T-tests give 𝑝- +values above 0.05 and so the number of molecules cannot be clearly distinguished anymore. The +fluorescence brightness are different between TNase, streptavidin and LicT (Fig. 2f). However, as reported +previously,18,57–59 the UV autofluorescence brightness does not scale linearly with the number of Trp residues +as the presence of nearby aminoacids can quench the Trp emission by charge or energy transfer. Therefore, +the brightness for streptavidin is not 24× higher than the brightness for TNase. The average quantum yield +for a Trp residue in streptavidin was estimated to be 3.5 ± 1 %,18,36 while the quantum yield of Trp in TNase +was reported to be 28 ± 2 %.2,41 With these values, the brightness for streptavidin is expected to be 24*3.5/28 += 3.0 ± 0.9 times higher than the TNase brightness. Our experimental results stand in good agreement with +this estimate as we find that the streptavidin brightness is 1.9 ± 0.7 times higher as compared to TNase (Fig. +2f). The discrepancy between the expected and the measured ratios is related to the saturation of +streptavidin at slightly lower power. We did not monitor any sign of photobleaching in our experiments. + + +8 + + +Figure 2. Label-free UV-FCS on proteins with a single tryptophan residue enabled by UV horn antennas. (a) +Fluorescence time traces and background intensity after spectral filtering and time gating for a single UV +horn antenna with different proteins and different concentrations. (b) Fluorescence intensity increase with +the protein concentration. The shaded area indicates the background level ± 2 times the standard deviation +of the background intensity. (c) FCS correlation traces recorded with the horn antenna for two TNase +concentrations. The background correlation is shown in gray, it corresponds to the FCS correlation obtained +in the same experimental conditions in the absence of protein target. The fit results are detailed in Supporting +Information Fig. S6. (d) Number of TNase proteins determined by FCS in the horn antenna as a function of +the TNase concentration. Statistical T-tests have been performed to compare between the distributions in +(d-f), the resulting 𝑝-values are written on the graphs for each pair of distributions. In (d-f), color markers +represent individual measurements, gray squares represent the average ± one standard deviation. (e) +Comparison between the numbers of proteins detected by FCS for three different proteins at the same 8 µM +concentration. (f) Fluorescence brightness per molecule 𝐶𝑅𝑀 determined by FCS for the different proteins. +(g) Average number of molecules measured by FCS in the horn antenna as a function of the concentration +for the three different proteins in the UV (color markers, the black line is a fit). The error bars on the individual + +a +b +c +1.5 +/ (kcounts/s) +2 +TNase +TNase8μM +0.04 +Streptavidin8uM +Correlation +TNase15uM +TNase15uM +LicT +l intensity +TNase 8 μM +0.02 +Background +Background +Total +Background +Total +M +wloprotein +w/oprotein +OL +ob +1 +0 +10 +20 +30 +40 +0 +5 +10 +15 +0.01 +0.1 +1 +10 +100 +Time (s) +Protein concentration (μM) +Lag time (ms) +d +e10 +P<0.01 +8 μM +P< 1e-4 +P>0.05 +molecules +150 +20 +P< 1e-4 + of molecules at 8 +8 +P > 0.05 +Brightness per molecule ( +P<0.01 +100 +6 +P<1e-4 +P< 1e-3 +. +50 +Number +4 +口 +P>0.1 +P<0.05 +2 +5 +10 +15 +TNase +Strep. +LicT +TNase +Strep. +LicT +TNase concentration (μM) +g 10 +h +- +(cts/s) +Horn +UV266nm +Experim. +Number of molecules + Simul. +100 +antenna +TNase +8F +Streptavidin +6H +LicT +x8.3 +A +5 +0.5 +x9.6 +x9.5 +VIS557nm +10 +LicT-Cy3B +8F +Confocal +Alexa546 +O +4E +0 +5 +10 +15 +266 nm +557 nm +TNase +Strep. +LicT +Proteinconcentration(uM)9 + +data are similar to the ones in (d) and are not represented for clarity. Empty markers show the numbers of +molecules determined by FCS using Alexa 546 and Cy3B fluorescent dyes at 557 nm excitation. The slope of +the lines are proportional to the detection volume, which is shown in (h) for both UV (266 nm) and visible +(557 nm) laser excitation. Numerical simulations of the detection volume (patterned area) confirm the FCS +results. (i) Fluorescence brightness per protein in the horn antenna as compared to the confocal reference +for the three different proteins. The data points for the horn antenna correspond to the average 𝐶𝑅𝑀 +determined by FCS. To determine the number of molecules for the confocal data, we assume a 1.8 fL confocal +volume on our UV microscope. + + +The UV-FCS average number of molecules is plotted in Fig. 2g as a function of the protein concentration for +the different proteins used in this work (filled markers in Fig. 2g). We find that the different datasets follow +the same line whose slope is proportional to the size of the detection volume. Experimentally, we determine +a detection volume of 1.15 ± 0.07 attoliter (10-18 L), in good agreement with the numerical simulations (Fig. +2h). To further validate the UV-FCS data, the same 80 nm apertures in aluminum are probed with visible +fluorescent dyes and 557 nm laser excitation.60 We use a Cy3B label on LicT protein and the free fluorescent +dye Alexa Fluor 546 to perform visible FCS experiments recording the number of fluorescent molecules as a +function of the concentration (raw FCS data are shown in Fig. S11). The data for both LicT-Cy3B and Alexa +Fluor 546 follow the same linear relationship with the concentration defining a similar detection volume +inside the aperture (Fig. 2g). Because of the longer illumination wavelength (557 instead of 266 nm) the +penetration depth inside the nanoaperture 61 and hence the size of the detection volume are different +between the UV and the visible experiments. This difference can be accounted for by the numerical +simulations (Fig. 2h & S12), providing a supplementary control of the experimental results. +We compare the brightness between the three different proteins using the horn antenna and the confocal +reference (the brightness for the confocal case is estimated from the measured concentration and the 1.8 fL +value of the confocal volume). The horn antenna improves the brightness by about 9× for all the different +proteins (Fig. 2i). As the horn antenna is a weakly resonant structure (as compared to a dimer nanogap +antenna62,63), its gain is essentially brought by the local excitation intensity increase and the improved +collection efficiency.35 The quantum yield enhancement plays a minor role here,36 so similar net fluorescence +enhancement are expected despite proteins with different Trp quantum yields are used. The 9× +enhancement stands also in good agreement with our calibration using the UV fluorescent dye p-terphenyl.35 +If we compare between the best results found for the horn antenna (single Trp brightness 70 cts/s, +background 600 cts/s) and the confocal setup (single Trp 8 cts/s, background 2000 cts/s), our combined +solution improves the signal to background ratio by 30×. The 1000× lower detection volume with the antenna + +10 + +efficiently eliminates the background intensity stemming from the solution (Fig. S6), yet at the expense of a +supplementary background from the antenna luminescence (Fig. 1i). +In presence of background, the signal to noise ratio for determining 𝐺(0) is given by 𝑆𝑁𝑅 = +𝐶𝑅𝑀 ( +𝑆𝐵𝑅 +1+𝑆𝐵𝑅) √𝑇𝑡𝑜𝑡 Δ𝜏 (see Supporting Information section S14), where 𝑆𝐵𝑅 = (𝑁 𝐶𝑅𝑀)/𝐵 is the signal +to background ratio, 𝑇𝑡𝑜𝑡 is the total integration time and Δ𝜏 is the temporal width of the counting interval. +The 𝑆𝑁𝑅 provides a figure of merit to compare between experiments and discuss the feasibility of an FCS +experiment. For the horn antennas and the proteins used here, the 𝑆𝑁𝑅 ranges from 0.7 to 1.5 (Fig. S7,S8), +while for the confocal configuration and even with one hour integration time, the 𝑆𝑁𝑅 remains below 0.2. +This consideration further highlights the key role played by the nanophotonic antenna to enable UV-FCS on +proteins featuring a low number of Trp residues. Besides, the FCS diffusion time influences the choice of the +temporal width Δτ, as Δτ must remain significantly smaller than the FCS diffusion time. The detection of +slower diffusing species enables the use of a longer counting interval Δ𝜏 which improves the signal to noise +ratio and can partly compensate for a lower autofluorescence brightness. +As most proteins bear only a few tryptophan residues, being able to detect a single tryptophan (instead of +several tens) is a major breakthrough opening the possibility to apply the UV-FCS technique to a huge library +of label-free proteins. This challenging task requires a careful optimization of the signal to background ratio +combining approaches to maximize the signal (optical horn antenna, antioxidants) and reduce the +background intensity (FIB milling depth, spectral filtering, time gating, buffer composition). Our calculations +provide useful guidelines to predict the feasibility of the experiments based on the correlation amplitude and +the signal to noise ratio. Altogether, the data presented in Fig. 2 demonstrate that a protein bearing a single +Trp residue can be detected using UV-FCS. We envision that the methods developed here to optimize the UV +autofluorescence signal to background ratio will be useful to a wide range of future studies on label-free +single protein spectroscopy,4,5,31–33 as well as the advancement of plasmonics into the UV range.21,22,25,26 UV- +FCS can provide information about local concentration, diffusion properties, and autofluorescence brightness +per molecule to shine new light on protein interaction dynamics with ligands or other molecular partners.46– +48 While in scattering microscopy the interference signal scales with the 3rd power of the nanoparticle +diameter,4,33,64,65 UV-FCS is less sensitive to the protein size, relying more its tryptophan content. The TNase +proteins detected here have a molecular weight lesser than 20 kDa, opening the possibility to detect label- +free proteins with molecular weights in the single-digit kDa range. As supplementary advantage of the +technique, the detection volume is in the attoliter range, three orders of magnitude below that of a +diffraction-limited confocal microscope, so that single molecule detection and UV-FCS can operate at +micromolar concentrations.66,67 Being able to work at high concentrations with single molecule resolution +and/or FCS is essential to study a broad range of enzymatic reactions, protein-protein and protein-DNA/RNA +interactions with Michaelis constants or dissociation constants in the micromolar range.68,69 + +11 + + + +Supporting Information +Tryptophan occurrence in human proteins, Protein information and sequences, Calculations of FCS +correlation amplitude in presence of background, Background intensity dependence with the FIB milling +depth, Background reduction using spectral filtering, Autofluorescence signal improvement using +antioxidants, Background from the buffer solution in the confocal configuration, FCS correlation traces and +fit results, FCS correlation is observed in the absence of spectral filtering and time gating, Validation of the +experimental correlations with the calculations model, Control FCS experiments on visible fluorescent dyes, +Numerical simulations of the detection volume, Protein absorption and emission spectra, Signal to noise ratio +in FCS in presence of background, Supplementary methods. + +Data availability +All data are available from the corresponding author upon request. + +Note +The authors declare no competing interest. + +Author contributions +J.W. designed and supervised research; P.R. performed research and analyzed data; A.B. built the +microscope and contributed to preliminary experiments; J.B.C. fabricated horn antennas; S.T. and P.R. +performed electromagnetic simulations; J.W. wrote the paper. + +Acknowledgments +We thank Emmanuel Margeat, Nathalie Declerck and Caroline Clerté for providing the LicT proteins. 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Nucleic Acids Res. 2021, 49, 12348–12357. + +15 + + +Supporting Information for +Ultraviolet Nanophotonics Enables Autofluorescence Correlation Spectroscopy on +Label-Free Proteins With a Single Tryptophan +Prithu Roy,1 Jean-Benoît Claude,1 Sunny Tiwari,1 Aleksandr Barulin,1 Jérôme Wenger1,* +1 Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, AMUTech, 13013 Marseille, France +* Corresponding author: jerome.wenger@fresnel.fr + + + + +Contents: +S1. +Tryptophan occurrence in human proteins +S2. +Protein information and sequences +S3. +Calculations of FCS correlation amplitude in presence of background +S4. +Background intensity dependence with the FIB milling depth +S5. +Background reduction using spectral filtering +S6. +Autofluorescence signal improvement using antioxidants +S7. +Background from the buffer solution in the confocal configuration +S8. +FCS correlation traces and fit results +S9. +FCS correlation is observed in the absence of spectral filtering and time gating +S10. Validation of the experimental correlations with the calculations model +S11. Control FCS experiments on visible fluorescent dyes +S12. Numerical simulations of the detection volume +S13. Protein absorption and emission spectra +S14. Signal to noise ratio in FCS in presence of background +S15. Supplementary methods + + + + + + + +16 + +S1. Tryptophan occurrence in human proteins + + +Figure S1. Statistics of tryptophan distribution in human proteins. A total of 20 399 proteins was extracted +from the UniProt database corresponding to human proteins whose entries were reviewed by Swiss-Prot 1. +From the sequence information, the histograms of protein sequence length (a) and number of tryptophan +residues (b) are extracted. The scatter plot in (c) shows the correlation between the number of Trp residues +and the total sequence length and enables a better view of the extreme values of the distributions. Out of +the 20 399 human proteins with available data, 1353 do not bear a Trp residue (6.6% of total), so 93.4% of +humans proteins have at least one Trp residue. The Trp frequency measured from the entire distribution is +1.2%, which corresponds also to the ratio of the average values (on average, a human protein contains 558 +amino acids among which 6.8 (1.2%) are tryptophans). About half of the human proteins (48.9%) have +between 1 and 5 tryptophans. Only 4% of the human proteins have more than 20 Trp residues. + + + + + + +2000 +into 50 aa bins +Median415aa +Average 558 aa +1000 +Occurrences +residues +150 +Number of tryptophan r +0 +100 +0 +500 +1000 +1500 +2000 +2500 +3000 +Protein sequence length (aa) +50 +Median5Trps +Number of proteins +2000 +Average6.8Trps +0 +2000 +4000 +6000 +8000 +Protein sequence length (aa) +1000 +0 +0 +10 +20 +30 +40 +Numberoftrvptophanresidues17 + +S2. Protein information and sequences + + +Table S1. Detailed information about the proteins used in this work. +Acronym + +TNase +Strep +LicT +Name + + + +Thermonuclease +Staphylococcal +nuclease +Streptavidin +Transcription +antiterminator protein +LicT +Organism + +Staphylococcus aureus +Streptomyces avidinii +Bacillus subtilis +UniPROT reference + +P00644 +P22629 +P39805 +RCSB PDB structure ref + +1A2U +2RTR +6TWR +Sigma Aldrich +product number +N 3755 +S 4762 +n/a +Form + +Monomer +Homotetramer +Homodimer +Monomer +molecular weight (Da) +16,807 +18,834 +19,908 +Monomer +sequence length (aa) +149 +183 +175 +Monomer sequence +(tryptophan W and +tyrosine Y highlighted) +ATSTKKLHKEPATLIKAID +GDTVKLMYKGPQMTFR +LLLVDTPQTKHPKKGVEK +YGPEASAFTKKMVENAK +KIEVEFNKGQRTDKYGR +GLAYIYADGKMVNEALV +RQGLAKVAYVYKPNNTH +EQLLRKEKKSEAQAKLNI +WSENDADSGQ + +MRKIVVAAIAVSLTTVSIT +ASASADPSKDSKAQVSA +AEAGITGTWYNQLGSTFI +VTAGADGALTGTYESAV +GNAESRYVLTGRYDSAP +ATDGSGTALGWTVAWK +NNYRNAHSATTWSGQY +VGGAEARINTQWLLTSG +TTEANAWKSTLVGHDTF +TKVKPSAASIDAAKKAGV +NNGNPLDAVQQ + +MKIAKVINNNVISVVNE +QGKELVVMGRGLAFQK +KSGDDVDEARIEKVFTLD +NKDVSEKFKTLLYDIPIEC +MEVSEEIISYAKLQLGKKL +NDSIYVSLTDHINFAIQR +NQKGLDIKNALLWETKR +LYKDEFAIGKEALVMVK +NKTGVSLPEDEAGFIALHI +VNAELNELQHHHHHH + +Tryptophan count per +monomer +1 +6 +1 +Tyrosine count per +monomer +7 +6 +4 +Extinction coefficient 𝜀 +at 280 nm (M-1 cm-1) +22,050 +169,360 +12,785 + + + + + +18 + +S3. Calculations results show that the background intensity plays a major role in determining the +feasibility of UV-FCS experiments on label-free proteins + + + +Figure S2. Calculations of the FCS correlation amplitude 𝐺(0) = (1 − +𝐵 +𝐹) +2 1 +𝑁 , where 𝐵 is the background +intensity, 𝑁 the number of detected fluorescent molecules, and 𝐹 = 𝐵 + 𝑁 ∗ 𝐶𝑅𝑀 is the total detected +intensity in presence of the fluorescent molecules. 𝐶𝑅𝑀 denotes the average fluorescence brightness per +molecule (count rate per molecule). (a-c) Correlation amplitude as a function of the number of detected +molecules 𝑁 for different values of the fluorescence brightness per molecule 𝐶𝑅𝑀 and background intensity +𝐵 (as indicated on the graphs, these values are typical for the experiments performed here). The shaded area +indicates correlation amplitudes below 0.005 which are highly challenging to detect with FCS due to the +residual background correlation and the electronic noise. We define the threshold of G(0) = 0.005 as a +minimum amplitude for possible detection. This definition is an arbitrary choice, any value in the range 0.001- +0.01 would be realistic. As a consequence of the low signal to background ratio, the 𝐺(0) dependence with +𝑁 is more complex than just the classical 1/𝑁 rule. For low 𝑁 values, the background term (1 − +𝐵 +𝐹) +2 += + (1 − +𝐵 +𝐵+𝑁∗𝐶𝑅𝑀) +2 +plays a major role and the correlation amplitude increases when the number of molecules +grows. For large 𝑁 values, the background term is not so influential, and we retrieve the 1/𝑁 dependence +leading to a decrease of 𝐺(0) when 𝑁 grows. It can be shown that the correlation amplitude 𝐺(0) reaches +its maximum when the number of molecules amounts to 𝑁 = 𝐵/𝐶𝑅𝑀. In that case, the maximum amplitude +is 𝐺𝑚𝑎𝑥(0) = 𝐶𝑅𝑀/4𝐵. (d) Correlation amplitude as a function of the 𝐶𝑅𝑀 for different values of +background 𝐵. For simplicity, the number of molecules is fixed to 𝑁 = 5. (e) Maximum 𝐺𝑚𝑎𝑥(0) value at the +optimum number of molecules 𝑁 = 𝐵/𝐶𝑅𝑀 corresponding to the highest correlation amplitude achievable +for a given (𝐵, 𝐶𝑅𝑀) set of values. The calculations results throughout (a-e) show that the background +intensity plays a major role in determining the feasibility of an FCS experiment at a given brightness 𝐶𝑅𝑀. +For the range of brightness between 10 and 100 counts/s/molecule typically achievable in our experiments, + +a +0.1F +CRM = 140 cts/s +B=0.6kcts/s +b +0.1 +B = 2 kcts/s +c +0.1F +B=5kcts/s +Correlation amplitudeG(0) +70 +Correlation amplitude G(0) +Correlation amplitude G(0) +CRM=140cts/s +40 +0.01 +0.01F +0.01E +70 +CRM=140cts/s +20 +40 +10 +70 +20 +40 +5 +0.001 +0.001 +0.001 +10 +20 +5 +10 +5 +0.0001 +0.0001 +0.0001 +1 +10 +100 +10 +100 +1 +10 +100 +Numberofmolecules +Numberofmolecules +Numberofmolecules +d +N= 5 +e +N=B/CRM +f +0.1 +B = 0.2 kcts/s +B = 0.2 kcts/s +Correlation amplitude G(0) +Correlation amplitude G(0) +Correlation amplitude G(0) +0.6 +0.6 +Horn antenna +2 +B = 0.6 kcts/s +2 +0.01 +0.01F +4 +Gain = 12 +10 +10 +0.01 +Confocal +0.001 +0.001 +0.001 +B = 2 kcts/s +Gain=1 +0.0001 +0.0001 +0.0001 +10 +100 +1000 +10 +100 +1000 +10 +100 +Brightness(photons/second/molecule) +Numberoftryptophanresiduesperprotein19 + +the background intensity must remain below 2-3 kcts/s to yield a detectable FCS correlation. (f) Simulated +correlation amplitudes as a function of the number of tryptophan residues for the cases corresponding to +the horn antenna (red trace) and the confocal reference (blue). The total number of protein is set to 𝑁 = 5 +and we assume a constant brightness of 5 counts per second for each tryptophan residue, which is typical +for our UV microscope. We also assume a gain of 12 brought by the presence of the horn antenna, which +enhances the tryptophan brightness to 12*5 = 60 counts per second. The graph in (f) shows that the detection +of a protein with a single tryptophan is feasible with the horn antenna, while for the confocal case, at least +15 tryptophan residues per protein are needed to yield a detectable FCS correlation amplitude. + + + + +20 + +S4. Background intensity dependence with the FIB milling depth indicate that the gallium oxide +implanted during FIB milling has a major contribution in the total background intensity + + +Figure S3. (a) Cross-cut scanning electron microscope image of nanoapertures milled with different depths +Z. The sample has been filled with platinum for a better side-view imaging, cut by FIB up to the half of the +aperture, and tilted by 52° to enable a cross-cut view of the nanoaperture profile. The scheme on the right is +a guide to understand the SEM image. Due to the FIB milling process, the region at the bottom of the +nanoaperture is enriched in gallium, forming different alloys with the SiO2 quartz substrate, notably gallium +oxide Ga2O3 which is UV-photoluminescent 2,3. The aperture on the middle corresponds to the configuration +used for milling the central aperture of the horn antenna in this work, using a 60 nm deep undercut into the +quartz substrate to maximize the signal enhancement 4. The aperture on the left is not completely milled, +leading to a degradation of the signal. (b) Background intensity from a single nanoaperture as a function of +the 266 nm laser power. The different data points correspond to different nanoapertures milled with the +same conditions. The black lines are numerical fits using a fluorescence saturation model +A*Plaser/(1+Plaser/Psat). The saturation indicates that the background stems mostly from photoluminescence, +as scattering and backreflection scale linearly with the excitation power. (c) Evolution of the background +intensity with the undercut depth for single apertures milled with different Z parameter conditions (the Z +parameter is the input used by our FEI DB235 focused ion beam system). (d) Assuming a uniform thickness +of 20 nm for the FIB gallium oxide implant, we compute the total volume of gallium oxide implant and plot +the background intensity as a function of this volume for 80 nm apertures and 200 nm apertures with +different milling depths. The linear relationship between the total background intensity and the volume of +the gallium implanted region confirms that the gallium implantation during FIB is the main source of +background in our system. We have tried to further reduce this background by annealing to 400°C or +photobleaching with prolonged illumination with UV light but both were unsuccessful. + + +a +Platinumdepositedforcross-sectionimaging +SiO,PECVDlayer +Al layer2 +Al layer1 +Al oxide +Undercut +layer +Diam80nm +Galliumoxide +100nm +DifferentFIBmillingdepthsZ +implantduringFIB +Quartzsubstrate +b +C +d +10 +(kcnts/s) +Diam80nm +10 +Diam80nm +Diam200nm +Z0.6 +S +(kcnts/ +5 +Z0.3 +Backgroundat10μw +Z 0.6 +8 +Diam80nm +5 +Z 0.4 +Z0.6 +Y +4 +Z 0.3 +Z 0.4 +Z0.3 +B +0 +10 +20 +30 +50 +100 +150 +0.5 +1.5 +266nmlaserpower(uw) +Undercut (nm)21 + +S5. Background reduction using spectral filtering + + +Figure S4. (a) Background intensity recorded on a single horn antenna with 10 µW average excitation power +with and without spectral filtering and time gating. (b) Time-resolved TCSPC photon arrival time histograms +respective to the 80 MHz synchronization signal from the pulsed laser. All histograms are sampled into 8 ps +time bins and are integrated over the same 20 s duration so that the intensities on the left axis can be directly +compared across the different curves. Dark blue traces correspond to the background without any spectral +filter while light blue traces are with the 310-360 nm bandpass filter. +The horn antenna background intensity significantly differs from the background recorded on the bare +aluminum film. This shows that the backreflection from the laser and the residual luminescence from the +quartz substrate and the aluminum layer are not the primary source of background. On the contrary, the +horn antenna background resembles very closely to the background recorded in the center of a 2 µm +diameter aperture (no aluminum is illuminated in this case, only the photoluminescent gallium oxide +implanted during FIB remains. We used pure heavy water to reduce the background from the solution in this +experiment). +Without the spectral filter, the horn antenna background TCSPC histogram has a baseline at around 15 counts +per 8 ps binning (corresponding to 1.2 kcnts/s average intensity). This baseline is representative of a +photoluminescence with a lifetime much larger than the 12.5 ns period of the laser pulses, so that the arrival +time of the photoluminescence photons appears uncorrelated with the laser pulses synchronization signal. +The spectral filter strongly reduces this baseline, indicating that most of the baseline photoluminescence falls +outside the 310-360 nm region (as expected for gallium oxide photoluminescence2). A similar baseline is also +found on the 2 µm wide FIB-processed region where the aluminum layer has been totally removed and +gallium oxide has been implanted. + + +a +Background intensity (kcnts/s) +Hornantennabackgroundwle +proteir +6 +No filter,wlo timegating +Withfilter,w/otimegating +Withfilter+timegating +0 +10 +20 +30 +40 +Time (s) +b +1000 +Hornantennabackground +2μmmilledareabackground +Aluminumfilmbackground +100 +withoutspectralfilter +per +Counts +10 +WIt +spectralfilter +0 +2 +4 +6 +8 +10 +12 +0 +2 +6 +8 +10 +12 +2 +4 +6 +4 +0 +8 +10 +12 +Time (ns) +Time (ns) +Time (ns)22 + +S6. Autofluorescence signal improvement using antioxidants + + +Figure S5. (a) Total fluorescence intensity collected from the confocal volume as a function of the 266 nm +laser power for a 8 µM TNase solution in presence of different chemical agents (MEA mercaptoethylamine +or cysteamine; GSH glutathione; AA ascorbic acid; DABCO 1,4-diazabicyclo[2.2.2]octane). Oxygen removal +using degassing did not modify the results for TNase significantly, therefore the oxygen dissolved in the buffer +solution was not modified for the TNase experiments. (b,c) Evolution of the TNase fluorescence intensity for +different GSH concentrations and the associated fluorescence lifetime decays (d). Moderate GSH +concentrations in the 5-25 mM range improve the signal intensity by increasing the quantum yield, the +fluorescence lifetime and the excitation power leading to saturation. This indicates a reduction of the +nonradiative decay rate as well as a reduction of the radical state buildup 5,6. However, higher GSH +concentrations lead to a decrease of the signal, which may indicate the formation of GSH-TNase complex +and/or quenching of the tryptophan singlet excited state by concentrated GSH 7. (e) Simplified Jablonski +diagram of the ground state S0, the singlet excited state S1 and the radical state R1 (the triplet state is omitted +for simplicity). The presence of reactive oxygen species ROS produced by UV illumination 6,8 promote + +a +b +TNase8μM +MEA10mM+GSH5mM +TNase8μM +GSH15mM +80 +MEA10mM+GSH25mM +S +(kcnts/s +AA10mM +60 +GSH15mM +GSH10mM +60 +MEA10mM +GSH25mM +intensity +MEA5mM+GSH15mM +GSH35mM +40 +40 +GSH50mM +Fluorescence +20 +GSH5mM +20 +DABCO10mM +Hepesbuffer +Hepesbuffer +0 +0 +0 +10 +20 +30 +40 +0 +10 +20 +30 +40 +266nmlaserpower (μW) +266 nm laser power (μW) +d +e ++GSH +TNase8μM +TNase +S, +ROS +60 +lintensity +GSH10mM ++GSH +20μWexcitation +R, +40 +ROS +GSH25mM +GSH50mM +Normalize +GSH35mM +Ket +bleach +10uwexcitation +20 +red +Hepesbuffer ++GSH +ob +S. +0 +20 +40 +5 +10 +Time (ns) +GSHconcentration(mM) +f +g +h +LicTdimer3.4uM +LicT +S +■GSH25mM +Hepes +Fluorescence intensity (kcnts/s +60 +O,removed +Hepesbuffer +LicT +l intensity +MEA10mM-0, +O2 removed +O,removed +O,removed +?MEA10mM +TNase +40 ++GSH25mM +AA10 +Normalized +0.1 +O,removed +GSH25mM +MEA10 +MEA10mM-0 +Oremoved +20 +MEA10mM +MEA10mM +GSH25 ++GSH25mM ++GSH25mM +Hepesbuffer +O,removed +0.01 +MEA 10 GSH 25 +0 +0 +20 +40 +0 +2 +4 +6 +8 +1 +3 +5 +7 +266nmlaserpower (μW) +Time (ns) +Signal gain23 + +nonradiative decay and oxidation rates. The addition of antioxidant GSH or MEA neutralize the negative +effects of ROS and can also promote the reduction from R1 to S0 6. However, high concentrations of GSH +above 25 mM also tend to increase the nonradiative decay knr and quench fluorescence. (f,g) Same as (a,d) +for LicT proteins. Here, the removal of oxygen using argon degassing plays a beneficial role, improving the +signal linearity and delaying the occurrence of saturation. Oxygen removal also tends to increase the +fluorescence lifetime and promote the quantum yield, yet the addition of reductants (MEA, GSH) is further +reducing the fluorescence lifetime without significantly changing the total detected intensity. (h) Signal gain +at 10 µW power for TNase and LicT in presence of different antifading compositions as compared to the +hepes buffer reference. For LicT, oxygen is removed by argon degassing while for TNase we keep the oxygen +dissolved in the buffer as this did not modify significantly our results. The FCS experiments with the horn +antennas use 10 mM MEA and 25 mM GSH with and without oxygen for TNase and LicT respectively. +Our data in Fig. S5 show that moderate MEA and GSH concentrations in the 5-25 mM range improve the +linearity of the autofluorescence intensity with the excitation power and increase the autofluorescence +intensity and the lifetime. These features correspond to a reduction of the nonradiative decay rate as well as +a reduction of the radical state buildup, which we relate to a neutralization of ROS by the antioxidants. We +have also found that the oxygen scavenger system GODCAT (glucose oxidase and catalase enzymes) was +leading to a too high background for our goal and was prone to photopolymerization issues in the +nanoaperture. + + + + +S7. Background from the buffer solution in the confocal configuration + + +Figure S6. Background intensity for the confocal microscope (without horn antenna). Ascorbic acid yields a +high background intensity and is discarded as reducing agent. MEA and GSH do not show a significantly higher +background intensity than the normal buffer. Impurities present in sucrose and glycerol contribute to +increase significantly the background, therefore we have discarded their use to increase the buffer viscosity. + + + +Confocal +background-noprotein +Ascorbic +acid 10 mM +80 +(kcnts) +DABCO.10mM +Sucrose 50% +10 ++ MEA 10 mM +60 +MEA10mM +intensi +Hepesbuffer +Glycerol25% +40 ++ MEA 10 mM +GSH25mM +5 +MEA10mM+GSH25mM +20 +0 +20 +40 +0 +10 +20 +30 +266nmlaserpower(uW) +266 nm laserpower (uW)24 + +S8. FCS correlation traces and fit results + + +Figure S7. FCS correlation functions for different TNase concentrations in the horn antenna (orange dots) +and their numerical fits (black line) with spectral filtering and time gating. The fit parameters are indicated +for each case. We also indicate the values for the signal to background (𝑆𝐵𝑅) and the signal to noise (𝑆𝑁𝑅). +For definitions of these quantities, see section 14 page S16 of this document. The gray data trace is the +residual background correlation recorded in the absence of proteins. + + + +0.06 +0.06 +TNase4μM +TNase8μM +F=909cts/s +F=1155cts/s +B = 670 cts/s +B = 680 cts/s +P1 = 24.6e-3 +P, = 29.2e-3 +0.04 +P2 = 9.5e-3 +0.04 +P2= 6e-3 +Correlation +Correlation +T1 = 0.45 ms +T1 = 0.45 ms +T2 = 55 ms +T2=55ms +N = 2.8 +N = 5.8 +0.02 +CRM=85cts/s +0.02 +CRM = 82 cts/s +SBR = 0.36 +SBR = 0.70 +SNR=0.67 +SNR = 1.01 +0 +0 +1. +L +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +Lag time (ms) +Lag time (ms) +0.06 +0.06 +TNase12μM +TNase15μM +F=1136cts/s +F = 1204 cts/s +B=540cts/s +B=530cts/s +P1 = 34e-3 +P, = 22.5e-3 +0.04 +P2 = 8e-3 +0.04 +Correlation +Correlation +P2 = 7e-3 +T1 = 0.45 ms +T1 = 0.45 ms +T2 = 55 ms +T2= 50 ms +N = 8.1 +N= 13.9 +0.02 +CRM = 74 cts/s +0.02 +CRM = 48 cts/s +SBR= 1.11 +SBR = 1.26 +SNR = 1.17 +SNR=0.80 +0 +0 +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +Lag time (ms) +Lag time (ms)25 + + + + +Figure S8. FCS correlation functions for streptavidin and LicT in the horn antenna (blue and green markers) +and their numerical fits (black line) with spectral filtering and time gating. The fit parameters are indicated +for each case as well as the corresponding signal to background (𝑆𝐵𝑅) and signal to noise (𝑆𝑁𝑅). The gray +data trace is the residual background correlation recorded in the absence of proteins. + + + +0.06 +0.06 +Streptavidin7.8μM +LicT 3.4 μM +F=1944cts/s +F=560.5cts/s +B=1230cts/s +B = 400 cts/s +P, = 26e-3 +P1 = 39.5e-3 +0.04 +P2 = 2.5e-3 +0.04 +P2 = 1.7e-3 +Correlation +Correlation +T, = 0.25 ms +T, = 0.15 ms +T2 = 150 ms +T2 = 50 ms +N= 5.2 +N= 2.1 +0.02 +CRM=138cts/s +0.02 +CRM = 77 cts/s +SBR=0.58 +SBR=0.40 +SNR = 1.52 +SNR=0.66 +0 +0 +L +LL +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +Lag time (ms) +Lag time (ms) +0.06 +0.06 +LicT 7 μM +LicT14uM +F= 803 cts/s +F=989 cts/s +B=480 cts/s +B =440 cts/s +P1 = 31.2e-3 +P1 = 33.8e-3 +0.04 +P2 = 4e-3 +0.04 +Correlation +P2 = 3.3e-3 +Correlation +T, = 0.16 ms +T = 0.21 ms +T2 = 50 ms +T2 = 65 ms +N = 5.1 +N= 9.2 +0.02 +CRM=63cts/s +0.02 +CRM=60 cts/s +SBR=0.67 +SBR= 1.25 +SNR=0.76 +SNR = 1.00 +0 +0 +l +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +Lagtime(ms) +Lagtime (ms)26 + +S9. No FCS correlation is observed in the absence of spectral filtering and time gating + + +Figure S9. Comparison of the FCS correlation with (a) and without (b) spectral filtering and time gating. The +horn antenna and TNase concentration are identical for both cases. Due to the higher background level in +the absence of spectral filtering and time gating, the correlation amplitude is strongly decreased and +becomes undistinguishable from the background correlation. From the calculations in Fig. S1d, for a +background intensity of 4.4 kcnts/s, 5 proteins and a brightness of 70 cnts/s, the estimated G(0) amplitude is +about 0.001, in good agreement with the experimental results in (b). + + +S10. Validation of the experimental correlations with the calculations model + + +Figure S10. Comparison of the experimental correlation amplitudes for the first species 𝜌1 (markers, +corresponding to proteins contribution separated from background) with the theoretical predictions (lines) +from Fig. S1 for TNase (a) and LicT (b). For LicT, we used a specially selected sample with low background +(presumably because of a slightly reduced milling depth during the FIB process, the background values were +recorded for each structure). Two lines are shown for LicT corresponding to slightly different parameters for +the background intensity 𝐵 and the fluorescence brightness 𝐶𝑅𝑀. The dispersion remains within the +uncertainty margins on our experimental data. The shaded region corresponds to the detection limit as in +Fig. S2. + +a +Withspectralfilterandtimegating +b +Withoutspectralfilterandtimegating +0.04 +0.04 +0.006 +TNase 8 μM +0.004 +Background +lation +TNase8μM +lation +0.002 +Correla +0.02 +0.02 +Corre +0.01 +0.1 +1 +10 +100 +Background +0 +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +Lagtime(ms) +Lagtime(ms)0.1F +TNase +b +0.1F +a +LicT +B=0.40kHz +B = 0.6 kHz +CRM=77cts/s +CRM=70cts/s +Correlation amplitude +Correlation amplitude +B=0.46kHz +CRM=61.5cts/s +0.01 +0.01 +0.001 +0.001 +10 +100 +1 +10 +100 +Numberofmolecules +Numberofmolecules27 + +S11. Control FCS experiments on visible fluorescent dyes +To confirm the UV FCS experiments, we have performed FCS measurements on a visible fluorescence +microscope using Alexa 546 and Cy3B fluorescent dyes. The experimental setup is described in details in ref. +9 and features a 557 nm laser focused by a 1.2NA water immersion objective and 570-620 nm confocal +detection by an avalanche photodiode. We use 80 nm apertures milled in similar conditions to the devices +used for the UV experiments. Therefore the number of detected molecules and the size of the detection +volume can be compared considering the difference in the illumination wavelength. Figure S11 shows FCS +correlation functions recorded with various Alexa Fluor 546 concentrations. The FCS correlation data is fitted +using a three dimensional Brownian diffusion model with an additional blinking term:10 +𝐺(𝜏) = +1 +𝑁 [1 + +𝑇 +1−𝑇 exp (− +𝜏 +𝜏𝑇)] (1 + +𝜏 +𝜏𝑑) +−1 +(1 + +1 +𝜅² +𝜏 +𝜏𝑑) +−0.5 + + (S1) +where N is the total number of molecules, T the fraction of dyes in the dark state, 𝜏𝑇 the dark state blinking +time, 𝜏𝑑 the mean diffusion time and 𝜅 the aspect ratio of the axial to transversal dimensions of the nanohole +volume. While the ZMW geometry obviously does not fulfill the assumption of free 3D diffusion, the above +model equation was found to empirically describe well the FCS data inside ZMWs using an aspect ratio set to +𝜅 = 1 as found previously 9,11. The dark state contribution remains quite small and is only needed to account +for the fast dynamics below 10 µs. Owing to the larger statistical noise, this type of fluctuation is not currently +detectable in the UV. + +Figure S11. FCS correlation (green) and numerical fits (black dashed lines). The fit parameters are indicated +each panel. The 80 nm ZMW is milled with identical parameters as the one used for the UV experiments. + + + +Alexa5464μM +Alexa5466μM +0.6 +N = 1.84 +N=2.43 +T = 73 μs +Ta = 71 μs +T =0.23 +0.4 +T=0.25 +Correlation +T = 4 μs +Correlation +T = 4 μs +0.4 +0.2 +0.2 +OL +0.001 +0.01 +0.1 +1 +10 +0.001 +0.01 +0.1 +10 +Lagtime(ms) +Lagtime (ms) +0.4 +Alexa5468μM +4 μM +N = 3.26 +0.6 +Ta = 64 μs +0.3 +T = 0.17 +6μM +Correlation +T-=3μs +0.4 +0.2 +8 μM +0.2 +0.1 +11 μM +OF +0.001 +0.01 +0.1 +1 +10 +0.001 +0.01 +0.1 +1 +10 +Lagtime (ms) +Lag time (ms)28 + +S12. Numerical simulations of the detection volume +We use the wave optics module of COMSOL Multiphysics 5.5 to simulate the propagation of light inside a 80 +nm diameter nanoaperture. The excitation is a plane wave with 266 or 557 nm. The vertical profile of the +nanoaperture takes into account the tapering due to FIB milling and the 12 nm thick silica layer deposited by +PECVD. To mimick our experiments, light is incoming from the bottom of the aperture where the diameter is +the smallest. The inside volume of the aperture and the upper medium are set to a refractive index of water. +A tetrahedral mesh is used with mesh size ranging from 0.3 nm to 10 nm. Scattering boundary conditions +were used to suppress reflections from the domain boundaries. +Figure S12 show the simulation results for the two excitation wavelengths. The decay profile of the excitation +intensity along the vertical center axis of the aperture is also represented. The light with the longer 557 nm +wavelength has a shorter penetration inside the sub-wavelength aperture, explaining why the detection +volume is smaller with 557 nm as compared to 266 nm excitation (Fig. 2g,h). To compute the FCS detection +volume at each wavelength, we take into account the undercut into the quartz substrates, which add a +constant volume of 0.45 aL for both wavelengths. Then we sum the volume from the undercut to the volume +from the aperture, assuming for simplicity a monoexponential decay inside the aperture with characteristic +decay length 27 nm at λ = 557 nm and 35 nm at λ = 266 nm. For the UV illumination, a cavity-like mode is +excited, which shifts the attenuation decay by an extra 50 nm inside the aperture. With these values, we +obtain a detection volume of 1.1 aL at 266 nm and 0.7 aL at 557 nm, in very good agreement with the +experimental volumes (Fig. 2h) determined from the slope of the linear fits in Fig. 2g. + + +Figure S12. (a,b) Normalized intensity profiles at two different wavelengths 266 nm (a) and 557 nm (b) +computed for a 80 nm diameter aperture milled in aluminum and covered by a 12 nm thick silica layer aiming +at reproducing the experimental FIB-milled configuration (Fig. S3a). The peak enhancement value along the +vertical center axis is 4.6 for 266 nm and 3.7 for 557 nm. (c) Comparison of the normalized decay profiles of +the excitation intensities along the vertical center axis of the aperture. The origin (Z=0) is taken at the bottom +quartz-aluminum interface. Monoexponential fits of the evanescently decaying sections are shown in black +dashed lines. + + +a +b +C +266 nm +557 nm +Normalized intensity +[E[2/|Ecentel +266nm +1 +0.8 +5 +0.6 +0.4 +0.2 +557 nm +0 +0 +50.nm +-100 +50nm +0 +100 +200 +Z coordinate (nm)29 + +S13. Protein absorption and emission spectra + + +Figure S13. (a-c) Absorbance spectra corrected for background recorded for TNase, Streptavidin and LicT on +a Tecan Spark 10M spectrofluorometer with 0.05 cm path length. (d) Normalized autofluorescence emission +spectra for the proteins and pure tryptophan dissolved in water. The emission spectrum of tryptophan +depends on its local environment, with a tendency to red-shift when exposed to water 12. + + + + + + + +a +0.10 +b +5mmAbsorbance +TNase60μM +0.5 mm Absorbance +Streptavin80μM +0.08 +0.6 +0.06 +0.4 +0.04 +0.2 +0.5 +0.02 +0.00 +0.0L +250 +300 +350 +400 +250 +300 +350 +400 +Wavelength (nm) +Wavelength(nm) +C +0.03 +Normalized fluorescence +0.5 mm Absorbance +Tryptophan inwater +LicT18μM +LicT +Streptavidin +0.02 +TNase +0.5 +0.01 +0.00 +0 +250 +300 +350 +400 +300 +350 +400 +Wavelength(nm) +Wavelength(nm)30 + +14. Signal to noise ratio in FCS in presence of background +The signal to noise ratio 𝑆𝑁𝑅 in a FCS experiment is commonly defined by 𝑆𝑁𝑅 = +𝐺(0) +𝜎(𝐺(0)) where 𝜎(𝐺(0)) is +the standard deviation of the correlation amplitude 𝐺(0). Following Koppel’s seminal work about statistical +accuracy in FCS,13 the 𝑆𝑁𝑅 is generally derived from Eq. (40) in Ref.13 as 𝑆𝑁𝑅 = 𝐶𝑅𝑀√𝑇𝑡𝑜𝑡 Δ𝜏, where 𝐶𝑅𝑀 +denotes the fluorescence brightness (detected photons per second and per molecule), 𝑇𝑡𝑜𝑡 the total +integration time and Δ𝜏 the minimum lag time defining the time interval for computing the correlation. Note +that Koppel’s Eq. (40) contains a denominator taking the form (1 + 4 𝐶𝑅𝑀 Δ𝜏 + 2 𝐶𝑅𝑀2Δ𝜏 𝜏𝑑)1/2 where 𝜏𝑑 +is the diffusion time. However in practice the correction introduced by this denominator is small and can +generally be neglected. Note also that an extra term +1 +√1+1/𝑁 can be introduced to account for possibly low +average number of molecules in the detection volume 14 (Koppel’s formula assumes 𝑁 ≫ 1). +It is important to stress out that the generally used formula 𝑆𝑁𝑅 = 𝐶𝑅𝑀√𝑇𝑡𝑜𝑡 Δ𝜏 does not consider the +presence of background.13 In presence of an extra background intensity 𝐵 (counts per second) on the +detector, the 𝑆𝑁𝑅 expression must be modified. Combining Koppel’s Eq.(40), (63) and (64) in ref.13 and using +our notations, the signal to noise ratio in FCS in presence of background can be expressed as: +𝑆𝑁𝑅 = +𝐶𝑅𝑀 ( +𝑆𝐵𝑅 +1 + 𝑆𝐵𝑅) √𝑇𝑡𝑜𝑡 Δ𝜏 +(1 + 4 𝐶𝑅𝑀 Δ𝜏 ( +𝑆𝐵𝑅 +1 + 𝑆𝐵𝑅) + 2 𝐶𝑅𝑀2 ( +𝑆𝐵𝑅 +1 + 𝑆𝐵𝑅) +2 +Δ𝜏 𝜏𝑑)1/2 + +(S2) + +In presence of background, this equation includes an extra term ( +𝑆𝐵𝑅 +1+𝑆𝐵𝑅) where 𝑆𝐵𝑅 = (𝑁 ∗ 𝐶𝑅𝑀)/𝐵 is the +signal to background ratio. In practice, the denominator in Eq. (S2) can often be neglected so that the +simplified expression for the 𝑆𝑁𝑅 with background becomes +𝑆𝑁𝑅 = 𝐶𝑅𝑀 ( +𝑆𝐵𝑅 +1 + 𝑆𝐵𝑅) √𝑇𝑡𝑜𝑡 Δ𝜏 = 𝐶𝑅𝑀 (1 − +𝐵 +𝐵 + 𝑁 ∗ 𝐶𝑅𝑀) √𝑇𝑡𝑜𝑡 Δ𝜏 +(S3) + +For large signal to background ratios, one retrieves Koppel’s established formula, while for very low 𝑆𝐵𝑅 +values, the signal to background becomes a linear prefactor lowering the 𝑆𝑁𝑅. +In our UV-FCS experiments with the horn antennas, our typical values are 𝐶𝑅𝑀 = 70 cts/s, 𝐵 = 600 cts/s, 𝑁 = +3 to 14 molecules, 𝑇𝑡𝑜𝑡 = 90 s, Δ𝜏 = 10 µs. This gives signal to background ratios between 0.35 and 1.6, and +signal to noise ratios between 0.5 and 1.3. +In the confocal case, using 𝐶𝑅𝑀 = 8 cts/s, 𝐵 = 2000 cts/s, 𝑁 = 10 molecules, 𝑇𝑡𝑜𝑡 = 90 s, Δ𝜏 = 10 µs, the signal +to background becomes 0.04 and the signal to noise is less than 0.01, preventing any possible experiment. +Increasing the integration time to 1 hour and the minimum lag time to 100 µs, improves the 𝑆𝑁𝑅 to 0.18, yet +this low value shows that any confocal UV-FCS experiment on single Trp protein would remain highly +challenging. +Lastly, we would like to point out that maximizing the 𝑆𝑁𝑅 should not be the only consideration. As already +pointed out by Koppel, another constraint is that 𝐺(0) must remain much greater that the residual +background correlation amplitude 𝜌𝐵: “If this isn’t much greater than [𝜌𝐵], the experiment is in trouble” (ref.13 +page 1944). In our antenna experiments, we have 𝜌𝐵 = 0.01, while our different signal correlation +amplitudes range from 0.022 to 0.04 (Fig. S7-S8), satisfying the condition 𝐺(0) > 𝜌𝐵 . + + +31 + +The 𝑆𝑁𝑅 formula can be used to predict the feasibility of a UV-FCS experiment and provide guidelines so as +to set the experimental conditions. Considering the case of free tryptophan molecules in solution (quantum +yield 12%, expected brightness in the antenna 30 counts/s), the small molecular mass of 0.2 kDa implies a +diffusion time below 10 µs imposing to set Δτ in the microsecond range. In the current conditions with 20 +molecules in the antenna, the predicted 𝑆𝑁𝑅 for free tryptophan is 0.2, which remains too low to yield +relevant FCS data. However, if with further work the background can be further reduced to 200 counts/s, +and if the diffusion can be slowed down without introducing some photopolymerization artefacts, then UV- +FCS on free tryptophans would become within experimental range. + + +S15. Supplementary Methods + +Optical horn antennas +The fabrication of horn antennas builds on our recent protocol 15. We first mill the microreflector into a +NEGS1 quartz coverslip covered with 100 nm aluminum using FIB (FEI dual beam DB235 Strata, 30 kV +acceleration voltage, 300 pA ion current). The cone angle is chosen around 30° following our previous work +15. A second 100 nm thick aluminum layer is deposited on top of the sample by electron-beam evaporation +(Bühler Syrus Pro 710). Then a 80 nm diameter aperture is milled by FIB (10 pA current) in the center of the +horn antenna unit with a 60 nm undercut chosen to optimize the signal to background ratio (Fig. S3) 16. To +protect the aluminum surface against corrosion 8,17,18, the horn antennas are covered by a 12 nm-thick +conformal silica layer using plasma-enhanced chemical vapor protection (PECVD, PlasmaPro NGP80 from +Oxford Instruments). + +Protein samples +Thermonuclease staphylococcal nuclease from Staphylococcus aureus and streptavidin from Streptomyces +avidinii are purchased from Sigma-Aldrich. Transcription antiterminator protein LicT from Bacillus subtilis is +provided by Emmanuel Margeat, Nathalie Declerck and Caroline Clerté (CBS Montpellier, France) 19,20. All +details about the proteins used in this work are summarized in the Supporting Information Tab. S1. The +proteins are dissolved in a Hepes buffer (25 mM Hepes, 300 mM NaCl, 0.1 v/v% Tween20, 1 mM DTT, and +1 mM EDTA 1 mM at pH 6.9). The protein solutions are centrifuged for 12 min at 142,000 g using an air +centrifuge (Airfuge 20 psi) and the supernatants are stored in small aliquots at -20°C. The concentrations are +assessed using a spectrofluorometer (Tecan Spark 10 M, Fig. S13) using the extinction coefficients derived +from the protein sequence and summarized in Tab. S1. 10 mM of mercaptoethylamine MEA and 25 mM of +glutathione GSH (both from Sigma Aldrich) are added to the buffer just before the experiments to improve +the photostability and neutralize the reactive oxygen species (Fig. S5). Oxygen removal using degassing did +not modify significantly the TNase autofluorescence signal. Therefore, we decided to not undertake any +special action to remove the oxygen dissolved in the buffer solution for the TNase and the streptavidin +experiments. For LicT experiments, the oxygen dissolved in the solution is removed by bubbling the buffer +with argon for at least 5 minutes, as it has a significant impact on the LicT autofluorescence signal (Fig. S5f). +The solution is then quickly placed on the UV microscope and the chamber is filled with argon and covered +with a coverslip to prevent the oxygen from the air to enter into the solution. The LicT autofluorescence +remained stable for about one hour, showing no sign of oxygen rediffusion during this period. + +32 + + +Experimental setup +The laser source is a 266 nm picosecond laser (Picoquant LDH-P-FA-266, 70 ps pulse duration, 80 MHz +repetition rate) with an average power of 10 µW. The laser beam is spatially filtered by a 50 µm pinhole to +ensure a quasi-Gaussian profile. A dichroic mirror (Semrock FF310-Di01-25-D) reflects the laser beam +towards the microscope. The UV objective is a LOMO 58x 0.8 NA with water immersion. The optical horn +antenna is positioned at the laser focus with a 3-axis piezoelectric stage (Physik Instrumente P-517.3CD). The +fluorescence light is collected by the same microscope objective and focused onto a 80 µm pinhole by a +quartz lens with 200 mm focal length (Thorlabs ACA254-200-UV). A stack of three emission filters (Semrock +FF01-300/LP-25, FF01-375/110-25 and FF01-334/40-25) selects the fluorescence photons within the 310-360 +nm range. The detection is performed by a photomultiplier tube (Picoquant PMA 175) connected to a time- +correlated single photon counting TCSPC module (Picoquant Picoharp 300 with time-tagged time-resolved +mode). The integration time is 90 sec per antenna. Since we work at quasi-neutral pH, some photocorrosion +of the aluminum can still occur after a longer UV exposure 17, this is why we decide to limit the illumination +time to ensure the maximum data reproducibility. For each horn antenna used in this work, we record the +background intensity 𝐵 by replacing the protein solution by the buffer only, all the other experimental +conditions are kept identical. + +Fluorescence correlation spectroscopy +The fluorescence time traces data are analyzed with Symphotime 64 (Picoquant) and Igor Pro 7 +(Wavemetrics). For the time gating, only the photons within a 3 ns window after the fluorescence peak are +selected for the analysis. The rationale behind this choice is that for an exponential decay of characteristic +time τ, 95% of the signal is collected within a time window of width 3τ. We thus consider a time window +corresponding to 3× the fluorescence lifetime (which is about 1 ns for tryptophan in the horn antenna for the +different proteins tested here) in order to provide a good trade-off between signal collection and noise +rejection. +The FCS correlations are computed using Symphotime 64 and fitted with a two species model:21 +G(τ) = ρ1 (1 + +τ +τ1) +−1 +(1 + +1 +κ² +τ +τ1) +−0.5 ++ ρ2 (1 + +τ +τ2) +−1 +(1 + +1 +κ² +τ +τ2) +−0.5 + (S4) +where 𝜌𝑖 and τi are the amplitude and diffusion time of each species and κ is the aspect ratio of the axial to +transversal dimensions of the detection volume (set to κ = 1 for the horn antenna following our previous +works 15,16,22). The rationale behind this two species model is that the first fast-diffusing species accounts for +the protein while the second slow-diffusing species corresponds to the residual background. In the absence +of any protein sample (using only the same buffer solution), a correlation is still observed from the +background with an amplitude below 0.01 and a characteristic time of 50 ms (Fig. S7-S9). The origin of this +background correlation remains unclear, the 50 ms time (20 Hz) suggests some remaining mechanical +vibration or electric noise on our microscope. As the diffusion times of the proteins are below 500 µs, the +background contribution (𝜌2, τ2) can be readily separated on the FCS data. All the fit results are detailed in +the Supporting Information Fig. S7 & S8. From the correlation amplitude 𝜌1, the total fluorescence intensity +𝐹 and the background intensity 𝐵, we compute the background-corrected number of proteins 21 as 𝑁 = (1 − +𝐵/𝐹)² /𝜌1 and the fluorescence brightness per protein as 𝐶𝑅𝑀 = (𝐹 − 𝐵)/𝑁 16. As we consider lag times +longer than 10 µs for the UV-FCS analysis, the afterpulsing from the photomultiplier and the repetition rate +of the laser are not a problem. + + +33 + +Supplementary references +(1) +The UniProt Consortium. UniProt: The Universal Protein Knowledgebase in 2021. Nucleic Acids Res. +2021, 49, D480–D489. +(2) +Binet, L.; Gourier, D. Origin Of The Blue Luminescence Of β-Ga2O3. J. Phys. Chem. Solids 1998, 59, +1241–1249. +(3) +S. I. Stepanov; V.I. Nikolaev; V.E. Bougrov; A.E. Romanov. Gallium Oxide: Properties and Applications - +A Review. Rev.Adv.Mater.Sci. 2016, 44, 63–86. +(4) +Wu, M.; Liu, W.; Hu, J.; Zhong, Z.; Rujiralai, T.; Zhou, L.; Cai, X.; Ma, J. Fluorescence Enhancement in an +Over-Etched Gold Zero-Mode Waveguide. Opt. Express 2019, 27, 19002–19018. +(5) +Hevekerl, H.; Tornmalm, J.; Widengren, J. Fluorescence-Based Characterization of Non-Fluorescent +Transient States of Tryptophan – Prospects for Protein Conformation and Interaction Studies. Sci. Rep. +2016, 6, 35052. +(6) +Barulin, A.; Wenger, J. Ultraviolet Photostability Improvement for Autofluorescence Correlation +Spectroscopy on Label-Free Proteins. J. Phys. Chem. Lett. 2020, 11, 2027–2035. +(7) +Widengren, J.; Chmyrov, A.; Eggeling, C.; Löfdahl, P.-Å.; Seidel, C. A. M. Strategies to Improve +Photostabilities in Ultrasensitive Fluorescence Spectroscopy. J. Phys. Chem. A 2007, 111, 429–440. +(8) +Barulin, A.; Claude, J.-B.; Patra, S.; Moreau, A.; Lumeau, J.; Wenger, J. Preventing Aluminum +Photocorrosion for Ultraviolet Plasmonics. J. Phys. Chem. Lett. 2019, 10, 5700–5707. +(9) +Baibakov, M.; Patra, S.; Claude, J.-B.; Moreau, A.; Lumeau, J.; Wenger, J. Extending Single-Molecule +Förster Resonance Energy Transfer (FRET) Range beyond 10 Nanometers in Zero-Mode Waveguides. +ACS Nano 2019, 13, 8469–8480. +(10) Pramanik, A.; Widengren, J. Fluorescence Correlation Spectroscopy (FCS). In Reviews in Cell Biology +and Molecular Medicine; American Cancer Society, 2006. +(11) Wenger, J.; Gérard, D.; Dintinger, J.; Mahboub, O.; Bonod, N.; Popov, E.; Ebbesen, T. W.; Rigneault, H. +Emission and Excitation Contributions to Enhanced Single Molecule Fluorescence by Gold Nanometric +Apertures. Opt. Express 2008, 16, 3008–3020. +(12) Lakowicz, J. R. Principles of Fluorescence Spectroscopy, 3rd ed.; Springer US, 2006. pp 529-575. +(13) Koppel, D. E. Statistical Accuracy in Fluorescence Correlation Spectroscopy. Phys. Rev. A 1974, 10, +1938–1945. +(14) Wenger, J.; Gérard, D.; Aouani, H.; Rigneault, H.; Lowder, B.; Blair, S.; Devaux, E.; Ebbesen, T. W. +Nanoaperture-Enhanced Signal-to-Noise Ratio in Fluorescence Correlation Spectroscopy. Anal. Chem. +2009, 81, 834–839. +(15) Barulin, A.; Roy, P.; Claude, J.-B.; Wenger, J. Ultraviolet Optical Horn Antennas for Label-Free +Detection of Single Proteins. Nat. Commun. 2022, 13, 1842. +(16) Barulin, A.; Claude, J.-B.; Patra, S.; Bonod, N.; Wenger, J. Deep Ultraviolet Plasmonic Enhancement of +Single Protein Autofluorescence in Zero-Mode Waveguides. Nano Lett. 2019, 19, 7434–7442. +(17) Roy, P.; Badie, C.; Claude, J.-B.; Barulin, A.; Moreau, A.; Lumeau, J.; Abbarchi, M.; Santinacci, L.; +Wenger, J. Preventing Corrosion of Aluminum Metal with Nanometer-Thick Films of Al2O3 Capped +with TiO2 for Ultraviolet Plasmonics. ACS Appl. Nano Mater. 2021, 4, 7199–7205. +(18) Renard, D.; Tian, S.; Lou, M.; Neumann, O.; Yang, J.; Bayles, A.; Solti, D.; Nordlander, P.; Halas, N. J. +UV-Resonant Al Nanocrystals: Synthesis, Silica Coating, and Broadband Photothermal Response. Nano +Lett. 2021, 21, 536–542. +(19) Clerte, C.; Declerck, N.; Margeat, E. Competitive Folding of Anti-Terminator/Terminator Hairpins +Monitored by Single Molecule FRET. Nucleic Acids Res. 2013, 41, 2632–2643. +(20) Ait-Bara, S.; Clerté, C.; Declerck, N.; Margeat, E. Competitive Folding of RNA Structures at a +Termination–Antitermination Site. RNA 2017, 23, 721–734. +(21) Wohland, T.; Maiti, S.; Macháň, R. An Introduction to Fluorescence Correlation Spectroscopy; IOP +Publishing, 2020. pp. 1-357. +(22) Barulin, A.; Roy, P.; Claude, J.-B.; Wenger, J. Purcell Radiative Rate Enhancement of Label-Free +Proteins with Ultraviolet Aluminum Plasmonics. J. Phys. Appl. Phys. 2021, 54, 425101. + + + diff --git a/HtAzT4oBgHgl3EQfjf0-/content/tmp_files/load_file.txt b/HtAzT4oBgHgl3EQfjf0-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3dca90c6b0f4d17723fc06f7cf29d5177201dde --- /dev/null +++ b/HtAzT4oBgHgl3EQfjf0-/content/tmp_files/load_file.txt @@ -0,0 +1,2072 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf,len=2071 +page_content='1 Ultraviolet Nanophotonics Enables Autofluorescence Correlation Spectroscopy on Label-Free Proteins With a Single Tryptophan Prithu Roy,1 Jean-Benoît Claude,1 Sunny Tiwari,1 Aleksandr Barulin,1 Jérôme Wenger1,* 1 Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, AMUTech, 13013 Marseille, France Corresponding author: jerome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='wenger@fresnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='fr Abstract: Using the ultraviolet autofluorescence of tryptophan aminoacids offers fascinating perspectives to study single proteins without the drawbacks of fluorescence labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' However, the low autofluorescence signals have so far limited the UV detection to large proteins containing several tens of tryptophan residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This limit is not compatible with the vast majority of proteins which contain only a few tryptophans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Here we push the sensitivity of label-free ultraviolet fluorescence correlation spectroscopy (UV-FCS) down to the single tryptophan level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Our results show how the combination of nanophotonic plasmonic antennas, antioxidants and background reduction techniques can improve the signal-to-background ratio by over an order of magnitude and enable UV-FCS on thermonuclease proteins with a single tryptophan residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This sensitivity breakthrough unlocks the applicability of UV-FCS technique to a broad library of label-free proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Keywords: plasmonics, nanophotonics, ultraviolet UV, single molecule fluorescence, tryptophan autofluorescence Figure for Table of Contents TNase SingleTrp roteins 2000 Nano Confocal photonics diffractionlimited 1000 Human 0 01 10 20 30 40 N=1 Trvptophansperprotein2 Proteins containing aromatic amino-acids (tryptophan, tyrosine and phenylalanine) are naturally fluorescent when excited in the deep ultraviolet (UV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1–3 This property opens fascinating opportunities to investigate proteins in their native state without introducing any external fluorescent label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4,5 Avoiding the fluorescence labelling not only simplifies the protein preparation and purification steps, it importantly rules out all the necessary controls to ensure the fluorescent label does not affect the protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6–11 Tryptophan (Trp), the brightest of the three aromatic amino acids, has a 12% quantum yield in water and an absorption cross section 20× lower than typical fluorescent dyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1,2 As a consequence, the tryptophan UV autofluorescence signal is very dim as compared to an organic fluorescent dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Therefore, earlier works on UV (auto)fluorescence correlation spectroscopy (UV-FCS) have been restricted to large proteins featuring several tens of tryptophan residues such as β-galactosidase (156 Trps),12,13 hemocyanin (148 Trps),14 phosphofructokinase oligomers (340 Trps) 15 or protein amyloid fibrils (>500 Trps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='16,17 Until now, the smallest proteins detected with UV-FCS are penicillin amidase (29 Trps) and streptavidin (24 Trps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='18 However, tryptophan is among the least abundant amino acids in eukaryotic proteins,19 and so the vast majority of proteins possess only a few Trp residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A UniProt database survey out of more than 20,000 human proteins shows that on average a human protein contains about 7 Trp residues, with half of the proteins bearing between 1 and 5 Trps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1a and Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='20 This shows that in order to really exploit the full potential of UV autofluorescence detection and explore a broad library of label-free proteins, the sensitivity of UV-FCS must be pushed by more than one order of magnitude down to the single tryptophan level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Here we address this sensitivity challenge and demonstrate UV-FCS on label-free proteins bearing a single Trp residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This is achieved thanks to the combination of (i) nanophotonic UV antenna to enhance the signal, (ii) detailed analysis to reduce the background intensity and (iii) chemical photostabilizing agents to avoid fluorescence saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Our results provide guidelines on how to extend plasmonics into the UV regime 21– 30 and further develop label-free single molecule spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4,5,31–33 Earlier works using UV aluminum nanophotonics were restricted to proteins containing a large number of Trp residues such as β-galactosidase (156 Trps) 34,35 and streptavidin (24 Trps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='36 Here we improve the sensitivity by more than one order of magnitude, down to the single tryptophan level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This technical achievement opens the UV-FCS technique to a huge library of proteins bearing only a few Trps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We use a 266 nm deep UV laser excitation and a fluorescence collection in the 310 to 360 nm range for our time-resolved UV confocal microscope (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To maximize the autofluorescence signal, we employ an optical horn antenna developed recently in our group (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='35 This nanophotonic device combines a central metal nanoaperture together with a conical horn microreflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The central nanoaperture of 80 nm diameter concentrates the light in an attoliter detection volume and enhances the autofluorescence from 3 proteins diffusing across this attoliter volume,34,37 while the metallized conical reflector collects the fluorescence emitted at high angles and steers it towards the microscope objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='38–40 Three different proteins are investigated in this work (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1a): thermonuclease staphylococcal nuclease (TNase) from Staphylococcus aureus, transcription antiterminator protein (LicT) from Bacillus subtilis and streptavidin from Streptomyces avidinii (Strep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' TNase is a monomer bearing a single Trp residue, LicT is a homodimer with a total of 2 Trps on the protein dimer, and streptavidin is a homotetramer with a total of 24 Trps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' All the details about the proteins used in this work are summarized in the Supporting Information Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' TNase was selected because its single Trp residue was theoretically predicted 41 to have a quantum yield of 28% in good agreement with ensemble spectroscopy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 TNase is also a widely studied as a model system in protein chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='42,43 LicT was selected for its UV signal being comparable to TNase and its availability in purified form labeled with Cy3B to serve as a control using visible fluorescence spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='44,45 Streptavidin was selected because of its higher number of Trps, large availability, moderate mass and good water solubility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To estimate the feasibility of the UV-FCS detection of proteins with a single tryptophan, we numerically compute the evolution of the FCS correlation amplitude 𝐺(0) = (1 − 𝐵 𝐵+𝑁∗𝐶𝑅𝑀) 2 1 𝑁 as a function of the background intensity 𝐵, the fluorescence brightness per molecule 𝐶𝑅𝑀, the number of molecules 𝑁 and the number of Trp residues per protein (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1d-f & Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='46–48 The ranges of values are taken to reproduce our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For a given number of molecules, the correlation amplitude quickly drops when the signal to background 𝑁 𝐶𝑅𝑀/𝐵 decreases owing to the quadratic exponent in the (1 − 𝐵 𝐵+𝑁∗𝐶𝑅𝑀) 2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We indicate on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1d-f different minimum thresholds for possible FCS detection corresponding to the contours where 𝐺(0) amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='005 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Since there is no general consensus in FCS in defining this minimum threshold,46 we decide to show three different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS amplitudes above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 should be easily detectable on a wide range of systems, whereas values below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 are highly challenging as they come close to the electronic noise level and the residual correlation from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The calculations results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1d and S2 show that maximizing the signal to background ratio 𝑁 𝐶𝑅𝑀/𝐵 is crucial to ensure the feasibility of UV-FCS experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Using typical values of UV autofluorescence brightness and background intensity representative of our experiments, we compute the predicted correlation amplitudes for the horn antenna (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1e) and the confocal setup (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1f) as a function of the number of tryptophan residues per protein, assuming for simplicity that all Trp residues contribute equally to the autofluorescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For the horn antenna, detectable correlation amplitudes above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 are found for a single tryptophan provided the number of proteins in the detection volume is between 2 and 60 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' UV-FCS on a single tryptophan protein appears feasible with the horn antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For the confocal reference, the single tryptophan always yields correlation amplitudes below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Increasing the number of proteins in the detection volume 4 does not compensate for the lower signal to background ratio in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A realistic confocal UV-FCS experiment requires that the protein carries at least 20 Trp residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Reducing the background intensity is crucial for improving the sensitivity of UV-FCS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1g-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure 1d shows that even with the brightness enhancement brought by the horn antenna, UV-FCS on a single Trp protein would be nearly impossible if the background intensity exceeds 2,000 counts/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Experiments performed on different samples milled by focused ion beam (FIB) show that the implantation of gallium oxide resulting from the FIB process 49 is the major source of background in the horn antenna (Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S3 & S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Gallium oxide is luminescent when excited in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 50,51 This background contribution can be controlled by FIB while selecting the proper milling depth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Moreover, the gallium oxide luminescence spectrum is shifted toward 400 nm 50,51 and can be partially separated from the protein autofluorescence spectra with the 310-360 nm bandpass filter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The gallium oxide photoluminescence contains a long lifetime component, significantly longer than the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 ns laser repetition period (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Temporal gating to select the photon arrival time in a 3 ns window immediately after the excitation pulse further reduces the background intensity without losing too much of the protein signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The 3 ns window is chosen to correspond to approximately 3× the tryptohan fluorescence lifetime for the different proteins used here in the horn antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Altogether, the combination of spectral filtering with temporal gating reduces the background intensity by 7× and improves the signal to background ratio by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7× (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1i) opening the possibility for single Trp detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Optimizing signal to background ratio to detect a single tryptophan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a) Histograms of the number of tryptophan residues per protein extracted from a UniProt database of 20 399 reviewed human protein entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='20 The limits of detection for the confocal 18 and the optical horn antenna (this work) are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The top images show the 3D structures of the proteins used in this work, made using Mol* viewer, with tryptophan residues highlighted by magenta dots on selected monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='52 (b) Scheme of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (c) Scanning electron microscope image of a horn antenna combining a central nanoaperture and a conical reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (d) Calculation of the FCS correlation amplitude 𝐺(0) as a function of the background intensity 𝐵 and the fluorescence brightness per molecule 𝐶𝑅𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A constant number of 𝑁 = 5 proteins was assumed, each protein carrying a single tryptophan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' All 2D maps in (d-f) share the same color scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The lines at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='005 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 indicate the boundary threshold for possible FCS detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Correlation amplitudes above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 can be easily detected, while values below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 are highly challenging if not impossible to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (e,f) Calculations of the correlation amplitude for the horn antenna and the confocal case as a function of the number of tryptophan residues per protein and the number of diffusing proteins in the detection volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The values for the background intensity and the fluorescence brightness per tryptophan residue are indicated by the markers in (d) and correspond to typical values in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (g,h) Strategies to improve the signal to background ratio (SBR) by spectral filtering and time gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The protein data correspond to 8 µM a TNase1Trp LicT2Trp Streptavidin24Trp b aluminum C Horn antenna= quartz 80nmaperture+microreflector UV objective 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 NA Trpresidue Nanophotonichornantenna 266nm Human excitation Confocal diffractionlimited 10 20 30 40 50 500nm 0 310-360nm Trpresiduesperprotein d collection e f Horn antenna Confocal Horn 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 Correlation antenna amplitude 1 Number of Numberof 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 FCS 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='005 Confocal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 FCS 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 ON 10 100 10 100 Trpresiduesperprotein Trpresiduesperprotein 10 100 1000 Background intensity(photons/second) g Dark counts Bufferfluo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='Al+SiO,background Intensity Ga,O crystal Nofilter FIBGa,O,background Protein signal Protein SBR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 Spectralfilter 300 Wavelength (nm) 450 Spectralfilter SBR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 Time gating Protein Spectral filter SBR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 Intensity + time gating Background 0 2 3 4 5 6 0 Time (ns) 10 Detectoroutput(countspersecond）6 TNase solution in the horn antenna (6 proteins in the detection volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The gallium oxide photoluminescence spectra is taken from ref 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (i) Experimental background intensity and total signal (8 µM TNase solution) in the horn antenna upon spectral filtering and time gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Along with the reduction of background intensity, another major element determining the UV-FCS sensitivity is the maximization of the fluorescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This involves the use of chemical photostabilizing agents to mitigate the buildup of the radical and triplet state populations leading to fluorescence saturation (Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='18,53,54,55 We use mercaptoethylamine (MEA also known as cysteamine) or gluthatione (GSH) which are effective in improving the autofluorescence of both TNase and LicT up to 8× (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S5) without introducing any significant additional background (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' GSH is an antioxidant naturally present in the human body to balance oxidative stress and neutralize reactive oxygen species (ROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='56 For TNase and strepatvidin experiments, we have decided not to use any oxygen scavenging approach as the TNase autofluorescence was not significantly affected by the presence of the oxygen dissolved in the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For LicT experiments, oxygen was removed by bubbling the solution with argon prior to recording the data (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' That way, dissolved oxygen was removed without adding any supplementary background (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S5f-h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure 2 summarizes our main experimental results aimed at pushing the UV-FCS sensitivity down to the single tryptophan level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The linear evolutions of the measured total intensities with the TNase and LicT protein concentrations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2a,b) provide a direct control that our experiments are sensitive to the protein autofluorescence signal, even with proteins bearing only one or two Trp residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The UV-FCS is computed and fitted (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2c, S7-S8) to extract the number of detected proteins 𝑁 and their autofluorescence brightness 𝐶𝑅𝑀 (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Even in the absence of any protein sample, we still detect a residual background correlation from the horn antenna filled with the buffer solution (gray trace in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This background correlation appears on all our traces with a long characteristic time above 50 ms and an amplitude below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7-S9) which may indicate an origin related to some remaining mechanical vibrations or electric noise on our microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The significant difference between the characteristic correlation time of this background (> 50 ms) and the protein diffusion time (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 ms) enables a clear separation of their contributions in the FCS signal so that the contribution from the diffusing proteins can be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Besides, the correlation amplitude related to the protein is always at least twice larger than this residual background correlation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7,S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In the absence of spectral filtering and time gating (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S9), the correlation amplitude found with the TNase protein falls down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='002 and cannot be distinguished from the background anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Our results for both TNase and LicT are in good agreement with the numerical calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S1d: using the experimentally determined parameters for the background intensity 𝐵 and the autofluorescence 7 brightness 𝐶𝑅𝑀, the observed correlation amplitudes follow the theoretical predictions of our model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure 2d-f compare the statistical distributions of the UV-FCS results for the horn antennas and different proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The number of TNase molecules seen by UV-FCS follows a linear dependence with the protein concentration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Statistical T-tests confirm the difference between the data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' On the contrary, when we probe similar concentrations (8 µM) of different proteins (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2e), the T-tests give 𝑝- values above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='05 and so the number of molecules cannot be clearly distinguished anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The fluorescence brightness are different between TNase, streptavidin and LicT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' However, as reported previously,18,57–59 the UV autofluorescence brightness does not scale linearly with the number of Trp residues as the presence of nearby aminoacids can quench the Trp emission by charge or energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Therefore, the brightness for streptavidin is not 24× higher than the brightness for TNase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The average quantum yield for a Trp residue in streptavidin was estimated to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 ± 1 %,18,36 while the quantum yield of Trp in TNase was reported to be 28 ± 2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2,41 With these values, the brightness for streptavidin is expected to be 24*3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5/28 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='9 times higher than the TNase brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Our experimental results stand in good agreement with this estimate as we find that the streptavidin brightness is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7 times higher as compared to TNase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The discrepancy between the expected and the measured ratios is related to the saturation of streptavidin at slightly lower power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We did not monitor any sign of photobleaching in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Label-free UV-FCS on proteins with a single tryptophan residue enabled by UV horn antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a) Fluorescence time traces and background intensity after spectral filtering and time gating for a single UV horn antenna with different proteins and different concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (b) Fluorescence intensity increase with the protein concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The shaded area indicates the background level ± 2 times the standard deviation of the background intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (c) FCS correlation traces recorded with the horn antenna for two TNase concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The background correlation is shown in gray, it corresponds to the FCS correlation obtained in the same experimental conditions in the absence of protein target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The fit results are detailed in Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (d) Number of TNase proteins determined by FCS in the horn antenna as a function of the TNase concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Statistical T-tests have been performed to compare between the distributions in (d-f), the resulting 𝑝-values are written on the graphs for each pair of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In (d-f), color markers represent individual measurements, gray squares represent the average ± one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (e) Comparison between the numbers of proteins detected by FCS for three different proteins at the same 8 µM concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (f) Fluorescence brightness per molecule 𝐶𝑅𝑀 determined by FCS for the different proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (g) Average number of molecules measured by FCS in the horn antenna as a function of the concentration for the three different proteins in the UV (color markers, the black line is a fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The error bars on the individual a b c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 / (kcounts/s) 2 TNase TNase8μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 Streptavidin8uM Correlation TNase15uM TNase15uM LicT l intensity TNase 8 μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 Background Background Total Background Total M wloprotein w/oprotein OL ob 1 0 10 20 30 40 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Time (s) Protein concentration (μM) Lag time (ms) d e10 P<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 8 μM P< 1e-4 P>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='05 molecules 150 20 P< 1e-4 of molecules at 8 8 P > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='05 Brightness per molecule ( P<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 100 6 P<1e-4 P< 1e-3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 50 Number 4 口 P>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 P<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='05 2 5 10 15 TNase Strep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' LicT TNase Strep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' LicT TNase concentration (μM) g 10 h (cts/s) Horn UV266nm Experim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Number of molecules Simul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 100 antenna TNase 8F Streptavidin 6H LicT x8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 A 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 x9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 x9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 VIS557nm 10 LicT-Cy3B 8F Confocal Alexa546 O 4E 0 5 10 15 266 nm 557 nm TNase Strep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' LicT Proteinconcentration(uM)9 data are similar to the ones in (d) and are not represented for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Empty markers show the numbers of molecules determined by FCS using Alexa 546 and Cy3B fluorescent dyes at 557 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The slope of the lines are proportional to the detection volume, which is shown in (h) for both UV (266 nm) and visible (557 nm) laser excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Numerical simulations of the detection volume (patterned area) confirm the FCS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (i) Fluorescence brightness per protein in the horn antenna as compared to the confocal reference for the three different proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The data points for the horn antenna correspond to the average 𝐶𝑅𝑀 determined by FCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To determine the number of molecules for the confocal data, we assume a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 fL confocal volume on our UV microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The UV-FCS average number of molecules is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2g as a function of the protein concentration for the different proteins used in this work (filled markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We find that the different datasets follow the same line whose slope is proportional to the size of the detection volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Experimentally, we determine a detection volume of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='07 attoliter (10-18 L), in good agreement with the numerical simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To further validate the UV-FCS data, the same 80 nm apertures in aluminum are probed with visible fluorescent dyes and 557 nm laser excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='60 We use a Cy3B label on LicT protein and the free fluorescent dye Alexa Fluor 546 to perform visible FCS experiments recording the number of fluorescent molecules as a function of the concentration (raw FCS data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The data for both LicT-Cy3B and Alexa Fluor 546 follow the same linear relationship with the concentration defining a similar detection volume inside the aperture (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Because of the longer illumination wavelength (557 instead of 266 nm) the penetration depth inside the nanoaperture 61 and hence the size of the detection volume are different between the UV and the visible experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This difference can be accounted for by the numerical simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2h & S12), providing a supplementary control of the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We compare the brightness between the three different proteins using the horn antenna and the confocal reference (the brightness for the confocal case is estimated from the measured concentration and the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 fL value of the confocal volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The horn antenna improves the brightness by about 9× for all the different proteins (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As the horn antenna is a weakly resonant structure (as compared to a dimer nanogap antenna62,63), its gain is essentially brought by the local excitation intensity increase and the improved collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='35 The quantum yield enhancement plays a minor role here,36 so similar net fluorescence enhancement are expected despite proteins with different Trp quantum yields are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The 9× enhancement stands also in good agreement with our calibration using the UV fluorescent dye p-terphenyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='35 If we compare between the best results found for the horn antenna (single Trp brightness 70 cts/s, background 600 cts/s) and the confocal setup (single Trp 8 cts/s, background 2000 cts/s), our combined solution improves the signal to background ratio by 30×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The 1000× lower detection volume with the antenna 10 efficiently eliminates the background intensity stemming from the solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S6), yet at the expense of a supplementary background from the antenna luminescence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 1i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In presence of background, the signal to noise ratio for determining 𝐺(0) is given by 𝑆𝑁𝑅 = 𝐶𝑅𝑀 ( 𝑆𝐵𝑅 1+𝑆𝐵𝑅) √𝑇𝑡𝑜𝑡 Δ𝜏 (see Supporting Information section S14), where 𝑆𝐵𝑅 = (𝑁 𝐶𝑅𝑀)/𝐵 is the signal to background ratio, 𝑇𝑡𝑜𝑡 is the total integration time and Δ𝜏 is the temporal width of the counting interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The 𝑆𝑁𝑅 provides a figure of merit to compare between experiments and discuss the feasibility of an FCS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For the horn antennas and the proteins used here, the 𝑆𝑁𝑅 ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7,S8), while for the confocal configuration and even with one hour integration time, the 𝑆𝑁𝑅 remains below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This consideration further highlights the key role played by the nanophotonic antenna to enable UV-FCS on proteins featuring a low number of Trp residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Besides, the FCS diffusion time influences the choice of the temporal width Δτ, as Δτ must remain significantly smaller than the FCS diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The detection of slower diffusing species enables the use of a longer counting interval Δ𝜏 which improves the signal to noise ratio and can partly compensate for a lower autofluorescence brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As most proteins bear only a few tryptophan residues, being able to detect a single tryptophan (instead of several tens) is a major breakthrough opening the possibility to apply the UV-FCS technique to a huge library of label-free proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This challenging task requires a careful optimization of the signal to background ratio combining approaches to maximize the signal (optical horn antenna, antioxidants) and reduce the background intensity (FIB milling depth, spectral filtering, time gating, buffer composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Our calculations provide useful guidelines to predict the feasibility of the experiments based on the correlation amplitude and the signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Altogether, the data presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2 demonstrate that a protein bearing a single Trp residue can be detected using UV-FCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We envision that the methods developed here to optimize the UV autofluorescence signal to background ratio will be useful to a wide range of future studies on label-free single protein spectroscopy,4,5,31–33 as well as the advancement of plasmonics into the UV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='21,22,25,26 UV- FCS can provide information about local concentration, diffusion properties, and autofluorescence brightness per molecule to shine new light on protein interaction dynamics with ligands or other molecular partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='46– 48 While in scattering microscopy the interference signal scales with the 3rd power of the nanoparticle diameter,4,33,64,65 UV-FCS is less sensitive to the protein size, relying more its tryptophan content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The TNase proteins detected here have a molecular weight lesser than 20 kDa, opening the possibility to detect label- free proteins with molecular weights in the single-digit kDa range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As supplementary advantage of the technique, the detection volume is in the attoliter range, three orders of magnitude below that of a diffraction-limited confocal microscope, so that single molecule detection and UV-FCS can operate at micromolar concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='66,67 Being able to work at high concentrations with single molecule resolution and/or FCS is essential to study a broad range of enzymatic reactions, protein-protein and protein-DNA/RNA interactions with Michaelis constants or dissociation constants in the micromolar range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='68,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='69 11 Supporting Information Tryptophan occurrence in human proteins,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein information and sequences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Calculations of FCS correlation amplitude in presence of background,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background intensity dependence with the FIB milling depth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background reduction using spectral filtering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Autofluorescence signal improvement using antioxidants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background from the buffer solution in the confocal configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation traces and fit results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation is observed in the absence of spectral filtering and time gating,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Validation of the experimental correlations with the calculations model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Control FCS experiments on visible fluorescent dyes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Numerical simulations of the detection volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein absorption and emission spectra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Signal to noise ratio in FCS in presence of background,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Supplementary methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Data availability All data are available from the corresponding author upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Note The authors declare no competing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Author contributions J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' designed and supervised research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' performed research and analyzed data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' built the microscope and contributed to preliminary experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' fabricated horn antennas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' performed electromagnetic simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' J.' metadata={'source': 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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Tenghamn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Sjösten, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Šípová-Jungová, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Albinsson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Lubart, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' van Leeuwen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Westerlund, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Volpe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Langhammer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Label-Free Nanofluidic Scattering Microscopy of Size and Mass of Single Diffusing Molecules and Nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Methods 2022, 19, 751–758.' metadata={'source': 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Waveguides to Enhance Single Molecule Fluorescence Detection and Fluorescence Correlation Spectroscopy toward Physiological Concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Wiley Interdiscip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Nanomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Nanobiotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2014, 6, 268–282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (68) Holzmeister, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Acuna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Grohmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Tinnefeld, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Breaking the Concentration Limit of Optical Single-Molecule Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2014, 43, 1014–1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (69) Patra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Claude, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Naubron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Wenger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Fast Interaction Dynamics of G-Quadruplex and RGG-Rich Peptides Unveiled in Zero-Mode Waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Nucleic Acids Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2021, 49, 12348–12357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 15 Supporting Information for Ultraviolet Nanophotonics Enables Autofluorescence Correlation Spectroscopy on Label-Free Proteins With a Single Tryptophan Prithu Roy,1 Jean-Benoît Claude,1 Sunny Tiwari,1 Aleksandr Barulin,1 Jérôme Wenger1,* 1 Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, AMUTech, 13013 Marseille, France Corresponding author: jerome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='wenger@fresnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='fr Contents: S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Tryptophan occurrence in human proteins S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein information and sequences S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Calculations of FCS correlation amplitude in presence of background S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background intensity dependence with the FIB milling depth S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background reduction using spectral filtering S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Autofluorescence signal improvement using antioxidants S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background from the buffer solution in the confocal configuration S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation traces and fit results S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation is observed in the absence of spectral filtering and time gating S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Validation of the experimental correlations with the calculations model S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Control FCS experiments on visible fluorescent dyes S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Numerical simulations of the detection volume S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein absorption and emission spectra S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Signal to noise ratio in FCS in presence of background S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Supplementary methods 16 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Tryptophan occurrence in human proteins Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Statistics of tryptophan distribution in human proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A total of 20 399 proteins was extracted from the UniProt database corresponding to human proteins whose entries were reviewed by Swiss-Prot 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' From the sequence information, the histograms of protein sequence length (a) and number of tryptophan residues (b) are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The scatter plot in (c) shows the correlation between the number of Trp residues and the total sequence length and enables a better view of the extreme values of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Out of the 20 399 human proteins with available data, 1353 do not bear a Trp residue (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6% of total), so 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4% of humans proteins have at least one Trp residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The Trp frequency measured from the entire distribution is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2%, which corresponds also to the ratio of the average values (on average, a human protein contains 558 amino acids among which 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2%) are tryptophans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' About half of the human proteins (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='9%) have between 1 and 5 tryptophans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Only 4% of the human proteins have more than 20 Trp residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2000 into 50 aa bins Median415aa Average 558 aa 1000 Occurrences residues 150 Number of tryptophan r 0 100 0 500 1000 1500 2000 2500 3000 Protein sequence length (aa) 50 Median5Trps Number of proteins 2000 Average6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8Trps 0 2000 4000 6000 8000 Protein sequence length (aa) 1000 0 0 10 20 30 40 Numberoftrvptophanresidues17 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein information and sequences Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Detailed information about the proteins used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Acronym TNase Strep LicT Name Thermonuclease Staphylococcal nuclease Streptavidin Transcription antiterminator protein LicT Organism Staphylococcus aureus Streptomyces avidinii Bacillus subtilis UniPROT reference P00644 P22629 P39805 RCSB PDB structure ref 1A2U 2RTR 6TWR Sigma Aldrich product number N 3755 S 4762 n/a Form Monomer Homotetramer Homodimer Monomer molecular weight (Da) 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='807 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='834 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='908 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='Monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='sequence length (aa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='149 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='183 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='Monomer sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='(tryptophan W and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='tyrosine Y highlighted) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='ATSTKKLHKEPATLIKAID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='GDTVKLMYKGPQMTFR ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='MKIAKVINNNVISVVNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='QGKELVVMGRGLAFQK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='KSGDDVDEARIEKVFTLD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='NKDVSEKFKTLLYDIPIEC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='MEVSEEIISYAKLQLGKKL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='NDSIYVSLTDHINFAIQR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='NQKGLDIKNALLWETKR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='LYKDEFAIGKEALVMVK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='NKTGVSLPEDEAGFIALHI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='VNAELNELQHHHHHH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='Tryptophan count per ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='Tyrosine count per ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='Extinction coefficient 𝜀 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='at 280 nm (M-1 cm-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='050 169,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='360 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='785 18 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Calculations results show that the background intensity plays a major role in determining the feasibility of UV-FCS experiments on label-free proteins Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Calculations of the FCS correlation amplitude 𝐺(0) = (1 − 𝐵 𝐹) 2 1 𝑁 , where 𝐵 is the background intensity, 𝑁 the number of detected fluorescent molecules, and 𝐹 = 𝐵 + 𝑁 ∗ 𝐶𝑅𝑀 is the total detected intensity in presence of the fluorescent molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 𝐶𝑅𝑀 denotes the average fluorescence brightness per molecule (count rate per molecule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a-c) Correlation amplitude as a function of the number of detected molecules 𝑁 for different values of the fluorescence brightness per molecule 𝐶𝑅𝑀 and background intensity 𝐵 (as indicated on the graphs, these values are typical for the experiments performed here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The shaded area indicates correlation amplitudes below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='005 which are highly challenging to detect with FCS due to the residual background correlation and the electronic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We define the threshold of G(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='005 as a minimum amplitude for possible detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This definition is an arbitrary choice, any value in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 would be realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As a consequence of the low signal to background ratio, the 𝐺(0) dependence with 𝑁 is more complex than just the classical 1/𝑁 rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For low 𝑁 values, the background term (1 − 𝐵 𝐹) 2 = (1 − 𝐵 𝐵+𝑁∗𝐶𝑅𝑀) 2 plays a major role and the correlation amplitude increases when the number of molecules grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For large 𝑁 values, the background term is not so influential, and we retrieve the 1/𝑁 dependence leading to a decrease of 𝐺(0) when 𝑁 grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' It can be shown that the correlation amplitude 𝐺(0) reaches its maximum when the number of molecules amounts to 𝑁 = 𝐵/𝐶𝑅𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In that case, the maximum amplitude is 𝐺𝑚𝑎𝑥(0) = 𝐶𝑅𝑀/4𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (d) Correlation amplitude as a function of the 𝐶𝑅𝑀 for different values of background 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For simplicity, the number of molecules is fixed to 𝑁 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (e) Maximum 𝐺𝑚𝑎𝑥(0) value at the optimum number of molecules 𝑁 = 𝐵/𝐶𝑅𝑀 corresponding to the highest correlation amplitude achievable for a given (𝐵, 𝐶𝑅𝑀) set of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The calculations results throughout (a-e) show that the background intensity plays a major role in determining the feasibility of an FCS experiment at a given brightness 𝐶𝑅𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For the range of brightness between 10 and 100 counts/s/molecule typically achievable in our experiments, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1F CRM = 140 cts/s B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6kcts/s b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 B = 2 kcts/s c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1F B=5kcts/s Correlation amplitudeG(0) 70 Correlation amplitude G(0) Correlation amplitude G(0) CRM=140cts/s 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01E 70 CRM=140cts/s 20 40 10 70 20 40 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 10 20 5 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0001 1 10 100 10 100 1 10 100 Numberofmolecules Numberofmolecules Numberofmolecules d N= 5 e N=B/CRM f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 kcts/s B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 kcts/s Correlation amplitude G(0) Correlation amplitude G(0) Correlation amplitude G(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 Horn antenna 2 B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 kcts/s 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01F 4 Gain = 12 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 Confocal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 B = 2 kcts/s Gain=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0001 10 100 1000 10 100 1000 10 100 Brightness(photons/second/molecule) Numberoftryptophanresiduesperprotein19 the background intensity must remain below 2-3 kcts/s to yield a detectable FCS correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (f) Simulated correlation amplitudes as a function of the number of tryptophan residues for the cases corresponding to the horn antenna (red trace) and the confocal reference (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The total number of protein is set to 𝑁 = 5 and we assume a constant brightness of 5 counts per second for each tryptophan residue, which is typical for our UV microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We also assume a gain of 12 brought by the presence of the horn antenna, which enhances the tryptophan brightness to 12*5 = 60 counts per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The graph in (f) shows that the detection of a protein with a single tryptophan is feasible with the horn antenna, while for the confocal case, at least 15 tryptophan residues per protein are needed to yield a detectable FCS correlation amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 20 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background intensity dependence with the FIB milling depth indicate that the gallium oxide implanted during FIB milling has a major contribution in the total background intensity Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a) Cross-cut scanning electron microscope image of nanoapertures milled with different depths Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The sample has been filled with platinum for a better side-view imaging, cut by FIB up to the half of the aperture, and tilted by 52° to enable a cross-cut view of the nanoaperture profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The scheme on the right is a guide to understand the SEM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Due to the FIB milling process, the region at the bottom of the nanoaperture is enriched in gallium, forming different alloys with the SiO2 quartz substrate, notably gallium oxide Ga2O3 which is UV-photoluminescent 2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The aperture on the middle corresponds to the configuration used for milling the central aperture of the horn antenna in this work, using a 60 nm deep undercut into the quartz substrate to maximize the signal enhancement 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The aperture on the left is not completely milled, leading to a degradation of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (b) Background intensity from a single nanoaperture as a function of the 266 nm laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The different data points correspond to different nanoapertures milled with the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The black lines are numerical fits using a fluorescence saturation model A*Plaser/(1+Plaser/Psat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The saturation indicates that the background stems mostly from photoluminescence, as scattering and backreflection scale linearly with the excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (c) Evolution of the background intensity with the undercut depth for single apertures milled with different Z parameter conditions (the Z parameter is the input used by our FEI DB235 focused ion beam system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (d) Assuming a uniform thickness of 20 nm for the FIB gallium oxide implant, we compute the total volume of gallium oxide implant and plot the background intensity as a function of this volume for 80 nm apertures and 200 nm apertures with different milling depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The linear relationship between the total background intensity and the volume of the gallium implanted region confirms that the gallium implantation during FIB is the main source of background in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We have tried to further reduce this background by annealing to 400°C or photobleaching with prolonged illumination with UV light but both were unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' a Platinumdepositedforcross-sectionimaging SiO,PECVDlayer Al layer2 Al layer1 Al oxide Undercut layer Diam80nm Galliumoxide 100nm DifferentFIBmillingdepthsZ implantduringFIB Quartzsubstrate b C d 10 (kcnts/s) Diam80nm 10 Diam80nm Diam200nm Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 S (kcnts/ 5 Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 Backgroundat10μw Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 8 Diam80nm 5 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 Y 4 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 B 0 10 20 30 50 100 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 266nmlaserpower(uw) Undercut (nm)21 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background reduction using spectral filtering Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a) Background intensity recorded on a single horn antenna with 10 µW average excitation power with and without spectral filtering and time gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (b) Time-resolved TCSPC photon arrival time histograms respective to the 80 MHz synchronization signal from the pulsed laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' All histograms are sampled into 8 ps time bins and are integrated over the same 20 s duration so that the intensities on the left axis can be directly compared across the different curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Dark blue traces correspond to the background without any spectral filter while light blue traces are with the 310-360 nm bandpass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The horn antenna background intensity significantly differs from the background recorded on the bare aluminum film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This shows that the backreflection from the laser and the residual luminescence from the quartz substrate and the aluminum layer are not the primary source of background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' On the contrary, the horn antenna background resembles very closely to the background recorded in the center of a 2 µm diameter aperture (no aluminum is illuminated in this case, only the photoluminescent gallium oxide implanted during FIB remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We used pure heavy water to reduce the background from the solution in this experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Without the spectral filter, the horn antenna background TCSPC histogram has a baseline at around 15 counts per 8 ps binning (corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 kcnts/s average intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This baseline is representative of a photoluminescence with a lifetime much larger than the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 ns period of the laser pulses, so that the arrival time of the photoluminescence photons appears uncorrelated with the laser pulses synchronization signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The spectral filter strongly reduces this baseline, indicating that most of the baseline photoluminescence falls outside the 310-360 nm region (as expected for gallium oxide photoluminescence2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A similar baseline is also found on the 2 µm wide FIB-processed region where the aluminum layer has been totally removed and gallium oxide has been implanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' a Background intensity (kcnts/s) Hornantennabackgroundwle proteir 6 No filter,wlo timegating Withfilter,w/otimegating Withfilter+timegating 0 10 20 30 40 Time (s) b 1000 Hornantennabackground 2μmmilledareabackground Aluminumfilmbackground 100 withoutspectralfilter per Counts 10 WIt spectralfilter 0 2 4 6 8 10 12 0 2 6 8 10 12 2 4 6 4 0 8 10 12 Time (ns) Time (ns) Time (ns)22 S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Autofluorescence signal improvement using antioxidants Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a) Total fluorescence intensity collected from the confocal volume as a function of the 266 nm laser power for a 8 µM TNase solution in presence of different chemical agents (MEA mercaptoethylamine or cysteamine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' GSH glutathione;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' AA ascorbic acid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' DABCO 1,4-diazabicyclo[2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2]octane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Oxygen removal using degassing did not modify the results for TNase significantly, therefore the oxygen dissolved in the buffer solution was not modified for the TNase experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (b,c) Evolution of the TNase fluorescence intensity for different GSH concentrations and the associated fluorescence lifetime decays (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Moderate GSH concentrations in the 5-25 mM range improve the signal intensity by increasing the quantum yield, the fluorescence lifetime and the excitation power leading to saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This indicates a reduction of the nonradiative decay rate as well as a reduction of the radical state buildup 5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' However, higher GSH concentrations lead to a decrease of the signal, which may indicate the formation of GSH-TNase complex and/or quenching of the tryptophan singlet excited state by concentrated GSH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (e) Simplified Jablonski diagram of the ground state S0, the singlet excited state S1 and the radical state R1 (the triplet state is omitted for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The presence of reactive oxygen species ROS produced by UV illumination 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 promote a b TNase8μM MEA10mM+GSH5mM TNase8μM GSH15mM 80 MEA10mM+GSH25mM S (kcnts/s AA10mM 60 GSH15mM GSH10mM 60 MEA10mM GSH25mM intensity MEA5mM+GSH15mM GSH35mM 40 40 GSH50mM Fluorescence 20 GSH5mM 20 DABCO10mM Hepesbuffer Hepesbuffer 0 0 0 10 20 30 40 0 10 20 30 40 266nmlaserpower (μW) 266 nm laser power (μW) d e +GSH TNase8μM TNase S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' ROS 60 lintensity GSH10mM +GSH 20μWexcitation R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 40 ROS GSH25mM GSH50mM Normalize GSH35mM Ket bleach 10uwexcitation 20 red Hepesbuffer +GSH ob S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 0 20 40 5 10 Time (ns) GSHconcentration(mM) f g h LicTdimer3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4uM LicT S ■GSH25mM Hepes Fluorescence intensity (kcnts/s 60 O,removed Hepesbuffer LicT l intensity MEA10mM-0, O2 removed O,removed O,removed ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='MEA10mM TNase 40 +GSH25mM AA10 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 O,removed GSH25mM MEA10 MEA10mM-0 Oremoved 20 MEA10mM MEA10mM GSH25 +GSH25mM +GSH25mM Hepesbuffer O,removed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 MEA 10 GSH 25 0 0 20 40 0 2 4 6 8 1 3 5 7 266nmlaserpower (μW) Time (ns) Signal gain23 nonradiative decay and oxidation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The addition of antioxidant GSH or MEA neutralize the negative effects of ROS and can also promote the reduction from R1 to S0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' However, high concentrations of GSH above 25 mM also tend to increase the nonradiative decay knr and quench fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (f,g) Same as (a,d) for LicT proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Here, the removal of oxygen using argon degassing plays a beneficial role, improving the signal linearity and delaying the occurrence of saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Oxygen removal also tends to increase the fluorescence lifetime and promote the quantum yield, yet the addition of reductants (MEA, GSH) is further reducing the fluorescence lifetime without significantly changing the total detected intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (h) Signal gain at 10 µW power for TNase and LicT in presence of different antifading compositions as compared to the hepes buffer reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For LicT, oxygen is removed by argon degassing while for TNase we keep the oxygen dissolved in the buffer as this did not modify significantly our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The FCS experiments with the horn antennas use 10 mM MEA and 25 mM GSH with and without oxygen for TNase and LicT respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Our data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S5 show that moderate MEA and GSH concentrations in the 5-25 mM range improve the linearity of the autofluorescence intensity with the excitation power and increase the autofluorescence intensity and the lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' These features correspond to a reduction of the nonradiative decay rate as well as a reduction of the radical state buildup, which we relate to a neutralization of ROS by the antioxidants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We have also found that the oxygen scavenger system GODCAT (glucose oxidase and catalase enzymes) was leading to a too high background for our goal and was prone to photopolymerization issues in the nanoaperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background from the buffer solution in the confocal configuration Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Background intensity for the confocal microscope (without horn antenna).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Ascorbic acid yields a high background intensity and is discarded as reducing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' MEA and GSH do not show a significantly higher background intensity than the normal buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Impurities present in sucrose and glycerol contribute to increase significantly the background, therefore we have discarded their use to increase the buffer viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Confocal background-noprotein Ascorbic acid 10 mM 80 (kcnts) DABCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='10mM Sucrose 50% 10 + MEA 10 mM 60 MEA10mM intensi Hepesbuffer Glycerol25% 40 + MEA 10 mM GSH25mM 5 MEA10mM+GSH25mM 20 0 20 40 0 10 20 30 266nmlaserpower(uW) 266 nm laserpower (uW)24 S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation traces and fit results Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation functions for different TNase concentrations in the horn antenna (orange dots) and their numerical fits (black line) with spectral filtering and time gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The fit parameters are indicated for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We also indicate the values for the signal to background (𝑆𝐵𝑅) and the signal to noise (𝑆𝑁𝑅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For definitions of these quantities, see section 14 page S16 of this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The gray data trace is the residual background correlation recorded in the absence of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 TNase4μM TNase8μM F=909cts/s F=1155cts/s B = 670 cts/s B = 680 cts/s P1 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6e-3 P, = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 P2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 P2= 6e-3 Correlation Correlation T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='45 ms T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='45 ms T2 = 55 ms T2=55ms N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM=85cts/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM = 82 cts/s SBR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='36 SBR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='70 SNR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='67 SNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Lag time (ms) Lag time (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 TNase12μM TNase15μM F=1136cts/s F = 1204 cts/s B=540cts/s B=530cts/s P1 = 34e-3 P, = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 P2 = 8e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 Correlation Correlation P2 = 7e-3 T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='45 ms T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='45 ms T2 = 55 ms T2= 50 ms N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 N= 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM = 74 cts/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM = 48 cts/s SBR= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='11 SBR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='26 SNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='17 SNR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='80 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Lag time (ms) Lag time (ms)25 Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation functions for streptavidin and LicT in the horn antenna (blue and green markers) and their numerical fits (black line) with spectral filtering and time gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The fit parameters are indicated for each case as well as the corresponding signal to background (𝑆𝐵𝑅) and signal to noise (𝑆𝑁𝑅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The gray data trace is the residual background correlation recorded in the absence of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 Streptavidin7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8μM LicT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 μM F=1944cts/s F=560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5cts/s B=1230cts/s B = 400 cts/s P, = 26e-3 P1 = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 P2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 P2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7e-3 Correlation Correlation T, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='25 ms T, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='15 ms T2 = 150 ms T2 = 50 ms N= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 N= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM=138cts/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM = 77 cts/s SBR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='58 SBR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='40 SNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='52 SNR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='66 0 0 L LL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Lag time (ms) Lag time (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 LicT 7 μM LicT14uM F= 803 cts/s F=989 cts/s B=480 cts/s B =440 cts/s P1 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2e-3 P1 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 P2 = 4e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 Correlation P2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3e-3 Correlation T, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='16 ms T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='21 ms T2 = 50 ms T2 = 65 ms N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 N= 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM=63cts/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 CRM=60 cts/s SBR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='67 SBR= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='25 SNR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='76 SNR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='00 0 0 l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Lagtime(ms) Lagtime (ms)26 S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' No FCS correlation is observed in the absence of spectral filtering and time gating Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Comparison of the FCS correlation with (a) and without (b) spectral filtering and time gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The horn antenna and TNase concentration are identical for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Due to the higher background level in the absence of spectral filtering and time gating, the correlation amplitude is strongly decreased and becomes undistinguishable from the background correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' From the calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S1d, for a background intensity of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 kcnts/s, 5 proteins and a brightness of 70 cnts/s, the estimated G(0) amplitude is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001, in good agreement with the experimental results in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Validation of the experimental correlations with the calculations model Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Comparison of the experimental correlation amplitudes for the first species 𝜌1 (markers, corresponding to proteins contribution separated from background) with the theoretical predictions (lines) from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S1 for TNase (a) and LicT (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For LicT, we used a specially selected sample with low background (presumably because of a slightly reduced milling depth during the FIB process, the background values were recorded for each structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Two lines are shown for LicT corresponding to slightly different parameters for the background intensity 𝐵 and the fluorescence brightness 𝐶𝑅𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The dispersion remains within the uncertainty margins on our experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The shaded region corresponds to the detection limit as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' a Withspectralfilterandtimegating b Withoutspectralfilterandtimegating 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='006 TNase 8 μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='004 Background lation TNase8μM lation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='002 Correla 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 Corre 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Background 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 100 Lagtime(ms) Lagtime(ms)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1F TNase b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1F a LicT B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='40kHz B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 kHz CRM=77cts/s CRM=70cts/s Correlation amplitude Correlation amplitude B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='46kHz CRM=61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5cts/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 10 100 1 10 100 Numberofmolecules Numberofmolecules27 S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Control FCS experiments on visible fluorescent dyes To confirm the UV FCS experiments, we have performed FCS measurements on a visible fluorescence microscope using Alexa 546 and Cy3B fluorescent dyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The experimental setup is described in details in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 9 and features a 557 nm laser focused by a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2NA water immersion objective and 570-620 nm confocal detection by an avalanche photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We use 80 nm apertures milled in similar conditions to the devices used for the UV experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Therefore the number of detected molecules and the size of the detection volume can be compared considering the difference in the illumination wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure S11 shows FCS correlation functions recorded with various Alexa Fluor 546 concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The FCS correlation data is fitted using a three dimensional Brownian diffusion model with an additional blinking term:10 𝐺(𝜏) = 1 𝑁 [1 + 𝑇 1−𝑇 exp (− 𝜏 𝜏𝑇)] (1 + 𝜏 𝜏𝑑) −1 (1 + 1 𝜅² 𝜏 𝜏𝑑) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 (S1) where N is the total number of molecules, T the fraction of dyes in the dark state, 𝜏𝑇 the dark state blinking time, 𝜏𝑑 the mean diffusion time and 𝜅 the aspect ratio of the axial to transversal dimensions of the nanohole volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' While the ZMW geometry obviously does not fulfill the assumption of free 3D diffusion, the above model equation was found to empirically describe well the FCS data inside ZMWs using an aspect ratio set to 𝜅 = 1 as found previously 9,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The dark state contribution remains quite small and is only needed to account for the fast dynamics below 10 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Owing to the larger statistical noise, this type of fluctuation is not currently detectable in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' FCS correlation (green) and numerical fits (black dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The fit parameters are indicated each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The 80 nm ZMW is milled with identical parameters as the one used for the UV experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Alexa5464μM Alexa5466μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='84 N=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='43 T = 73 μs Ta = 71 μs T =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='25 Correlation T = 4 μs Correlation T = 4 μs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 OL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 10 Lagtime(ms) Lagtime (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 Alexa5468μM 4 μM N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 Ta = 64 μs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='17 6μM Correlation T-=3μs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 8 μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 11 μM OF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 1 10 Lagtime (ms) Lag time (ms)28 S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Numerical simulations of the detection volume We use the wave optics module of COMSOL Multiphysics 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 to simulate the propagation of light inside a 80 nm diameter nanoaperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The excitation is a plane wave with 266 or 557 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The vertical profile of the nanoaperture takes into account the tapering due to FIB milling and the 12 nm thick silica layer deposited by PECVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To mimick our experiments, light is incoming from the bottom of the aperture where the diameter is the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The inside volume of the aperture and the upper medium are set to a refractive index of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A tetrahedral mesh is used with mesh size ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3 nm to 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Scattering boundary conditions were used to suppress reflections from the domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure S12 show the simulation results for the two excitation wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The decay profile of the excitation intensity along the vertical center axis of the aperture is also represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The light with the longer 557 nm wavelength has a shorter penetration inside the sub-wavelength aperture, explaining why the detection volume is smaller with 557 nm as compared to 266 nm excitation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To compute the FCS detection volume at each wavelength, we take into account the undercut into the quartz substrates, which add a constant volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='45 aL for both wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Then we sum the volume from the undercut to the volume from the aperture, assuming for simplicity a monoexponential decay inside the aperture with characteristic decay length 27 nm at λ = 557 nm and 35 nm at λ = 266 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For the UV illumination, a cavity-like mode is excited, which shifts the attenuation decay by an extra 50 nm inside the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' With these values, we obtain a detection volume of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 aL at 266 nm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7 aL at 557 nm, in very good agreement with the experimental volumes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2h) determined from the slope of the linear fits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a,b) Normalized intensity profiles at two different wavelengths 266 nm (a) and 557 nm (b) computed for a 80 nm diameter aperture milled in aluminum and covered by a 12 nm thick silica layer aiming at reproducing the experimental FIB-milled configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The peak enhancement value along the vertical center axis is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 for 266 nm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='7 for 557 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (c) Comparison of the normalized decay profiles of the excitation intensities along the vertical center axis of the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The origin (Z=0) is taken at the bottom quartz-aluminum interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Monoexponential fits of the evanescently decaying sections are shown in black dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' a b C 266 nm 557 nm Normalized intensity [E[2/|Ecentel 266nm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 557 nm 0 0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='nm 100 50nm 0 100 200 Z coordinate (nm)29 S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein absorption and emission spectra Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (a-c) Absorbance spectra corrected for background recorded for TNase, Streptavidin and LicT on a Tecan Spark 10M spectrofluorometer with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='05 cm path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (d) Normalized autofluorescence emission spectra for the proteins and pure tryptophan dissolved in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The emission spectrum of tryptophan depends on its local environment, with a tendency to red-shift when exposed to water 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='10 b 5mmAbsorbance TNase60μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 mm Absorbance Streptavin80μM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='0L 250 300 350 400 250 300 350 400 Wavelength (nm) Wavelength(nm) C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='03 Normalized fluorescence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 mm Absorbance Tryptophan inwater LicT18μM LicT Streptavidin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='02 TNase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='00 0 250 300 350 400 300 350 400 Wavelength(nm) Wavelength(nm)30 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Signal to noise ratio in FCS in presence of background The signal to noise ratio 𝑆𝑁𝑅 in a FCS experiment is commonly defined by 𝑆𝑁𝑅 = 𝐺(0) 𝜎(𝐺(0)) where 𝜎(𝐺(0)) is the standard deviation of the correlation amplitude 𝐺(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Following Koppel’s seminal work about statistical accuracy in FCS,13 the 𝑆𝑁𝑅 is generally derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (40) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='13 as 𝑆𝑁𝑅 = 𝐶𝑅𝑀√𝑇𝑡𝑜𝑡 Δ𝜏, where 𝐶𝑅𝑀 denotes the fluorescence brightness (detected photons per second and per molecule), 𝑇𝑡𝑜𝑡 the total integration time and Δ𝜏 the minimum lag time defining the time interval for computing the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Note that Koppel’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (40) contains a denominator taking the form (1 + 4 𝐶𝑅𝑀 Δ𝜏 + 2 𝐶𝑅𝑀2Δ𝜏 𝜏𝑑)1/2 where 𝜏𝑑 is the diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' However in practice the correction introduced by this denominator is small and can generally be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Note also that an extra term 1 √1+1/𝑁 can be introduced to account for possibly low average number of molecules in the detection volume 14 (Koppel’s formula assumes 𝑁 ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' It is important to stress out that the generally used formula 𝑆𝑁𝑅 = 𝐶𝑅𝑀√𝑇𝑡𝑜𝑡 Δ𝜏 does not consider the presence of background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='13 In presence of an extra background intensity 𝐵 (counts per second) on the detector, the 𝑆𝑁𝑅 expression must be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Combining Koppel’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (40), (63) and (64) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='13 and using our notations, the signal to noise ratio in FCS in presence of background can be expressed as: 𝑆𝑁𝑅 = 𝐶𝑅𝑀 ( 𝑆𝐵𝑅 1 + 𝑆𝐵𝑅) √𝑇𝑡𝑜𝑡 Δ𝜏 (1 + 4 𝐶𝑅𝑀 Δ𝜏 ( 𝑆𝐵𝑅 1 + 𝑆𝐵𝑅) + 2 𝐶𝑅𝑀2 ( 𝑆𝐵𝑅 1 + 𝑆𝐵𝑅) 2 Δ𝜏 𝜏𝑑)1/2 (S2) In presence of background, this equation includes an extra term ( 𝑆𝐵𝑅 1+𝑆𝐵𝑅) where 𝑆𝐵𝑅 = (𝑁 ∗ 𝐶𝑅𝑀)/𝐵 is the signal to background ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In practice, the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (S2) can often be neglected so that the simplified expression for the 𝑆𝑁𝑅 with background becomes 𝑆𝑁𝑅 = 𝐶𝑅𝑀 ( 𝑆𝐵𝑅 1 + 𝑆𝐵𝑅) √𝑇𝑡𝑜𝑡 Δ𝜏 = 𝐶𝑅𝑀 (1 − 𝐵 𝐵 + 𝑁 ∗ 𝐶𝑅𝑀) √𝑇𝑡𝑜𝑡 Δ𝜏 (S3) For large signal to background ratios, one retrieves Koppel’s established formula, while for very low 𝑆𝐵𝑅 values, the signal to background becomes a linear prefactor lowering the 𝑆𝑁𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In our UV-FCS experiments with the horn antennas, our typical values are 𝐶𝑅𝑀 = 70 cts/s, 𝐵 = 600 cts/s, 𝑁 = 3 to 14 molecules, 𝑇𝑡𝑜𝑡 = 90 s, Δ𝜏 = 10 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' This gives signal to background ratios between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='35 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='6, and signal to noise ratios between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In the confocal case, using 𝐶𝑅𝑀 = 8 cts/s, 𝐵 = 2000 cts/s, 𝑁 = 10 molecules, 𝑇𝑡𝑜𝑡 = 90 s, Δ𝜏 = 10 µs, the signal to background becomes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 and the signal to noise is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01, preventing any possible experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Increasing the integration time to 1 hour and the minimum lag time to 100 µs, improves the 𝑆𝑁𝑅 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='18, yet this low value shows that any confocal UV-FCS experiment on single Trp protein would remain highly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Lastly, we would like to point out that maximizing the 𝑆𝑁𝑅 should not be the only consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As already pointed out by Koppel, another constraint is that 𝐺(0) must remain much greater that the residual background correlation amplitude 𝜌𝐵: “If this isn’t much greater than [𝜌𝐵], the experiment is in trouble” (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='13 page 1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In our antenna experiments, we have 𝜌𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01, while our different signal correlation amplitudes range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='022 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='04 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7-S8), satisfying the condition 𝐺(0) > 𝜌𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 31 The 𝑆𝑁𝑅 formula can be used to predict the feasibility of a UV-FCS experiment and provide guidelines so as to set the experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Considering the case of free tryptophan molecules in solution (quantum yield 12%, expected brightness in the antenna 30 counts/s), the small molecular mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2 kDa implies a diffusion time below 10 µs imposing to set Δτ in the microsecond range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In the current conditions with 20 molecules in the antenna, the predicted 𝑆𝑁𝑅 for free tryptophan is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='2, which remains too low to yield relevant FCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' However, if with further work the background can be further reduced to 200 counts/s, and if the diffusion can be slowed down without introducing some photopolymerization artefacts, then UV- FCS on free tryptophans would become within experimental range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Supplementary Methods Optical horn antennas The fabrication of horn antennas builds on our recent protocol 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We first mill the microreflector into a NEGS1 quartz coverslip covered with 100 nm aluminum using FIB (FEI dual beam DB235 Strata, 30 kV acceleration voltage, 300 pA ion current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The cone angle is chosen around 30° following our previous work 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A second 100 nm thick aluminum layer is deposited on top of the sample by electron-beam evaporation (Bühler Syrus Pro 710).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Then a 80 nm diameter aperture is milled by FIB (10 pA current) in the center of the horn antenna unit with a 60 nm undercut chosen to optimize the signal to background ratio (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S3) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' To protect the aluminum surface against corrosion 8,17,18, the horn antennas are covered by a 12 nm-thick conformal silica layer using plasma-enhanced chemical vapor protection (PECVD, PlasmaPro NGP80 from Oxford Instruments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Protein samples Thermonuclease staphylococcal nuclease from Staphylococcus aureus and streptavidin from Streptomyces avidinii are purchased from Sigma-Aldrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Transcription antiterminator protein LicT from Bacillus subtilis is provided by Emmanuel Margeat, Nathalie Declerck and Caroline Clerté (CBS Montpellier, France) 19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' All details about the proteins used in this work are summarized in the Supporting Information Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The proteins are dissolved in a Hepes buffer (25 mM Hepes, 300 mM NaCl, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='1 v/v% Tween20, 1 mM DTT, and 1 mM EDTA 1 mM at pH 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The protein solutions are centrifuged for 12 min at 142,000 g using an air centrifuge (Airfuge 20 psi) and the supernatants are stored in small aliquots at -20°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The concentrations are assessed using a spectrofluorometer (Tecan Spark 10 M, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S13) using the extinction coefficients derived from the protein sequence and summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 10 mM of mercaptoethylamine MEA and 25 mM of glutathione GSH (both from Sigma Aldrich) are added to the buffer just before the experiments to improve the photostability and neutralize the reactive oxygen species (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Oxygen removal using degassing did not modify significantly the TNase autofluorescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Therefore, we decided to not undertake any special action to remove the oxygen dissolved in the buffer solution for the TNase and the streptavidin experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For LicT experiments, the oxygen dissolved in the solution is removed by bubbling the buffer with argon for at least 5 minutes, as it has a significant impact on the LicT autofluorescence signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The solution is then quickly placed on the UV microscope and the chamber is filled with argon and covered with a coverslip to prevent the oxygen from the air to enter into the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The LicT autofluorescence remained stable for about one hour, showing no sign of oxygen rediffusion during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 32 Experimental setup The laser source is a 266 nm picosecond laser (Picoquant LDH-P-FA-266, 70 ps pulse duration, 80 MHz repetition rate) with an average power of 10 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The laser beam is spatially filtered by a 50 µm pinhole to ensure a quasi-Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A dichroic mirror (Semrock FF310-Di01-25-D) reflects the laser beam towards the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The UV objective is a LOMO 58x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='8 NA with water immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The optical horn antenna is positioned at the laser focus with a 3-axis piezoelectric stage (Physik Instrumente P-517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='3CD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The fluorescence light is collected by the same microscope objective and focused onto a 80 µm pinhole by a quartz lens with 200 mm focal length (Thorlabs ACA254-200-UV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' A stack of three emission filters (Semrock FF01-300/LP-25, FF01-375/110-25 and FF01-334/40-25) selects the fluorescence photons within the 310-360 nm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The detection is performed by a photomultiplier tube (Picoquant PMA 175) connected to a time- correlated single photon counting TCSPC module (Picoquant Picoharp 300 with time-tagged time-resolved mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The integration time is 90 sec per antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Since we work at quasi-neutral pH, some photocorrosion of the aluminum can still occur after a longer UV exposure 17, this is why we decide to limit the illumination time to ensure the maximum data reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For each horn antenna used in this work, we record the background intensity 𝐵 by replacing the protein solution by the buffer only, all the other experimental conditions are kept identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Fluorescence correlation spectroscopy The fluorescence time traces data are analyzed with Symphotime 64 (Picoquant) and Igor Pro 7 (Wavemetrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' For the time gating, only the photons within a 3 ns window after the fluorescence peak are selected for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The rationale behind this choice is that for an exponential decay of characteristic time τ, 95% of the signal is collected within a time window of width 3τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' We thus consider a time window corresponding to 3× the fluorescence lifetime (which is about 1 ns for tryptophan in the horn antenna for the different proteins tested here) in order to provide a good trade-off between signal collection and noise rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The FCS correlations are computed using Symphotime 64 and fitted with a two species model:21 G(τ) = ρ1 (1 + τ τ1) −1 (1 + 1 κ² τ τ1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 + ρ2 (1 + τ τ2) −1 (1 + 1 κ² τ τ2) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='5 (S4) where 𝜌𝑖 and τi are the amplitude and diffusion time of each species and κ is the aspect ratio of the axial to transversal dimensions of the detection volume (set to κ = 1 for the horn antenna following our previous works 15,16,22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The rationale behind this two species model is that the first fast-diffusing species accounts for the protein while the second slow-diffusing species corresponds to the residual background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' In the absence of any protein sample (using only the same buffer solution), a correlation is still observed from the background with an amplitude below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content='01 and a characteristic time of 50 ms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7-S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' The origin of this background correlation remains unclear, the 50 ms time (20 Hz) suggests some remaining mechanical vibration or electric noise on our microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As the diffusion times of the proteins are below 500 µs, the background contribution (𝜌2, τ2) can be readily separated on the FCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' All the fit results are detailed in the Supporting Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' S7 & S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' From the correlation amplitude 𝜌1, the total fluorescence intensity 𝐹 and the background intensity 𝐵, we compute the background-corrected number of proteins 21 as 𝑁 = (1 − 𝐵/𝐹)² /𝜌1 and the fluorescence brightness per protein as 𝐶𝑅𝑀 = (𝐹 − 𝐵)/𝑁 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' As we consider lag times longer than 10 µs for the UV-FCS analysis, the afterpulsing from the photomultiplier and the repetition rate of the laser are not a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 33 Supplementary references (1) The UniProt Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' UniProt: The Universal Protein Knowledgebase in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Nucleic Acids Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2021, 49, D480–D489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' (2) Binet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Gourier, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} +page_content=' 2021, 54, 425101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfjf0-/content/2301.01516v1.pdf'} diff --git a/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf b/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..19d016b06290081eecbb13e4ab5e0bc7c0db93b0 --- /dev/null +++ b/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9dda044e2e417f968b8fb995634bd407096ba496088408b6dcf82148fee06dba +size 299825 diff --git a/KdFLT4oBgHgl3EQfLC8K/vector_store/index.faiss b/KdFLT4oBgHgl3EQfLC8K/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..caf85d3856a89eeafe3d4ae5525f28cf8e876001 --- /dev/null +++ b/KdFLT4oBgHgl3EQfLC8K/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5401dd020cd3bd6070da9ab229733eeab2b6ef516fc7cd4f7c021d132d7dc299 +size 6815789 diff --git a/MNAzT4oBgHgl3EQfV_xV/content/tmp_files/2301.01293v1.pdf.txt b/MNAzT4oBgHgl3EQfV_xV/content/tmp_files/2301.01293v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d717748610993979816c270254761e6dc9134f07 --- /dev/null +++ b/MNAzT4oBgHgl3EQfV_xV/content/tmp_files/2301.01293v1.pdf.txt @@ -0,0 +1,880 @@ + + + + + + + +Preprint +Linear chain conditional random fields, hidden Markov mod- +els, and related classifiers + +Elie Azeraf 1, Emmanuel Monfrini 2 and Wojciech Pieczynski 2,* +1 Watson Department, IBM France, Paris, France; elie.azeraf@ibm.com +2 Telecom SudParis, Institut Polytechnique de Paris, Evry, France ; {Emmanuel.Monfrini, Woj- +ciech.Pieczynski}@telecom-sudparis.eu +* Correspondence: Wojciech.Pieczynski@telecom-sudparis.eu +Abstract: Practitioners use Hidden Markov Models (HMMs) in different problems for about sixty +years. Besides, Conditional Random Fields (CRFs) are an alternative to HMMs and appear in the +literature as different and somewhat concurrent models. We propose two contributions. First, we +show that basic Linear-Chain CRFs (LC-CRFs), considered as different from the HMMs, are in fact +equivalent to them in the sense that for each LC-CRF there exists a HMM – that we specify – whom +posterior distribution is identical to the given LC-CRF. Second, we show that it is possible to refor- +mulate the generative Bayesian classifiers Maximum Posterior Mode (MPM) and Maximum a Pos- +teriori (MAP) used in HMMs, as discriminative ones. The last point is of importance in many fields, +especially in Natural Language Processing (NLP), as it shows that in some situations dropping +HMMs in favor of CRFs was not necessary. +Keywords: Hidden Markov Model; Linear Chain Conditional Random Field; Bayesian Classifier; +Discriminative Classifier; Maximum Posterior Mode; Maximum A Posteriori + +1. Introduction +Let 𝑍�:� = (𝑍�, … , 𝑍�) be a stochastic sequence, with 𝑍� = (𝑋�, 𝑌�). Random varia- +bles 𝑋�, … , 𝑋� take their values in a finite set Λ, while 𝑌�, … , 𝑌� take their values either +in a discrete or continuous set Ω. Realizations of 𝑋�:� = (𝑋�, … , 𝑋�) are hidden while re- +alizations of 𝑌�:� = (𝑌�, … , 𝑌�) are observed, and the problem we deal with is to estimate +𝑋�:� = 𝑥�:� from 𝑌�:� = 𝑦�:�. We deal with Bayesian methods of estimation, which need +some probabilistic model. Probabilistic model is a distribution – or a family of distribu- +tions – which will be denoted with 𝑝(𝑧�:�), or 𝑝(𝑥�:�, 𝑦�:�). We are interested in the case +of dependent 𝑍�, … , 𝑍�. The simplest model taking into account this dependence is the +well-known hidden Markov model (HMM) [1, 2, 3, 4, 5], whose distribution is given with +𝑝(𝑥�:�, 𝑦�:�) = 𝑝(𝑥�) 𝑝(𝑦�|𝑥�) ∏ +𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���) +��� +��� +. + + +(1.1) +HMMs allow recursive fast computation of Bayesian estimators called “classifiers” in this +paper and recalled below. In spite of their simplicity, HMMs are very robust and provide +quite satisfactory results in many applications. +Beside, conditional random fields (CRFs) [6, 7] also allow estimating 𝑋�:� = 𝑥�:� +from 𝑌�:� = 𝑦�:�. Their definition is different from the definition of HMMs in that in CRFs, +one directly considers 𝑝(𝑥�:�|𝑦�:�), and neither 𝑝(𝑥�:�, 𝑦�:�) nor 𝑝(𝑦�:�|𝑥�:�) are needed +to perform the estimation. In some areas, like Natural Language Processing (NLP), CRFs +are preferred over HMCs because 𝑝(𝑥�:�, 𝑦�:�) and 𝑝(𝑦�:�|𝑥�:�) are difficult to model. +General CRFs are written + + + +2 of 8 + +𝑝(𝑥�:�|𝑦�:�) = 𝑝(𝑥�|𝑦�:�) ∏ +𝑝(𝑥���|𝑥�:�, 𝑦�:�) +��� +��� +, + + +(1.2) +In this paper we will consider the following basic “linear-chain” CRF (LC-CRF): +𝑝(𝑥�:�|𝑦�:�) = +� +�(��:�) 𝑒𝑥𝑝[∑ +𝑉�(𝑥�, 𝑥���) +��� +��� ++ ∑ +𝑈�(𝑥�, 𝑦�) +� +��� +], + + +(1.3) +with 𝜅(𝑦�:�) the normalizing constant. +Authors usually consider the two families HMMs and CRFs as different [6, 7, 8, 9, 10, +11, 12, 13]. They classify the former in the category of “generative models”, while they +classify the latter in the category of “discriminative” models. +Considering the simples cases (1.1) and (1.3), we propose two contributions: +(a) We establish an equivalence between HMMs (1.1) and basic linear-chain CRFs +(1.3), which completes the results presented in [14]. +Let us notice that wanting to compare the two models directly is somewhat mislead- +ing. Indeed, HMMs and CRFs are defined with distributions on different spaces. To be +precise, we adopt the following definition +Definition 1 +Let 𝑋�, … , 𝑋�, 𝑌�, … , 𝑌� be two stochastic sequences defined above. +(i) we will call “model” a distribution 𝑝(𝑥�:�, 𝑦�:�); +(ii) we will call “conditional model” a distribution 𝑝(𝑥�:�|𝑦�:�); +(iii) we will say that a model 𝑝(𝑥�:�, 𝑦�:�) is “equivalent” to a conditional model +𝑞(𝑥�:�|𝑦�:�) if +there +exists +a +distribution +𝑟(𝑦�:�) such +that +𝑝(𝑥�:�, 𝑦�:�) = +𝑞(𝑥�:�|𝑦�:�)𝑟(𝑦�:�); +(iv) we will say that a family of models Α is “equivalent” to a family of conditional +models Β if for each model 𝑝(𝑥�:�, 𝑦�:�) in Α there exists an equivalent conditional +model 𝑞(𝑥�:�|𝑦�:�) in Β. +According to Definition 1 HMMs are particular “models”, while CRFs are particular +“conditional models”. Then a particular model HMM cannot be equal to a particular con- +ditional model CRF, but it can be equivalent to it. +Our aim is to show that the family of LC-CRFs (1.3) is equivalent to the family of +HMMs (1.1). In addition we specify, for each LC-CRF 𝑞(𝑥�:�|𝑦�:�), a particular HMM +𝑝(𝑥�:�, 𝑦�:�) such that 𝑝(𝑥�:�|𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�). +Let 𝑝(𝑥�:�, 𝑦�:�, 𝜃) be an HMM (1.1), with parameters 𝜃. Taking 𝑟(𝑦�:�) = 𝑝(𝑦�:�, 𝜃), +it is immediate to see that 𝑝(𝑥�:�|𝑦�:�, 𝜃) is an equivalent CRF. The converse is not imme- +diate. Is a given CRF 𝑝(𝑥�:�|𝑦�:�, 𝜃) equivalent to a HMM? If yes, can we find 𝑟(𝑦�:�) +such that 𝑝(𝑥�:�|𝑦�:�, 𝜃)𝑟(𝑦�:�) is a HMM? Besides, can we give its (1.1) form? Respond- +ing these questions in a simple linear-chain CRF case is our first contribution. More pre- +cisely, we show that the family of LC-CRFs (1.3) is equivalent to the family of HMMs (1.1), +and we specify, for each LC-CRF 𝑞(𝑥�:�|𝑦�:�), a particular HMM 𝑝(𝑥�:�, 𝑦�:�) given in the +form (1.1), such that 𝑝(𝑥�:�|𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�). +Note that numerous papers addressed similarities between HMMs and linear-chain +CRFs [7, 15]; however, to the best of our knowledge, results proposed here, which are +mathematically rigorous in the framework of Definition 1, are new. +(b) We show that “generative” estimators MPM and MAP in HMM are computable +in a “discriminative” manner, exactly as in LC-CRF. +One of interests of HMMs and CRFs is that in both of them there exist Bayesian clas- +sifiers, which allow estimating 𝑥�:� from 𝑦�:� in a reasonable computer time. As exam- +ples, let us consider the “maximum of posterior margins” (MPM) defined with: +[𝑔(𝑦�:�) = 𝑥��:� = (𝑥��, … , 𝑥��)] ⟺ [∀𝑛 = 1, … , 𝑁, 𝑝( 𝑥��|𝑦�:�) = 𝑠𝑢𝑝 +�� +(𝑝(𝑥�|𝑦�:�))], (1.4) +and the “maximum a posteriori” (MAP) is defined with +[𝑔(𝑦�:�) = 𝑥��:�] ⟺ [𝑝( 𝑥��:�|𝑦�:�) = 𝑠𝑢𝑝 +��:� +(𝑝(𝑥�:�|𝑦�:�))], + + +(1.5) + + +3 of 8 + +Note that likely to any other Bayesian classifier, MPM and MAP are independent +from 𝑝(𝑦�:�). This means that in any generative model 𝑝(𝑥�:�, 𝑦�:�), related Bayesian clas- +sifier is strictly the same as that related to the equivalent (in the meaning of Definition 1) +CRF model 𝑝(𝑥�:�|𝑦�:�). We see that the distinction between “generative” and “discrimi- +native” classifiers is not justified: all Bayesian classifiers are discriminative. However, in +HMM the related MPM and MAP classifiers are computed calling on 𝑝(𝑦�|𝑥�), while this +is not the case in LC-CRF. We show that both MPM and MAP in HMM can also be com- +puted in a “discriminative” way, without calling on 𝑝(𝑦�|𝑥�). Thus, the use of MPM and +MAP in HMM is strictly the same as its use in LC-CRF, which is our second contribution. +One of the consequences is that the use of MPMs and MAPs in the two families HMMs +and LC-CRFs presents exactly the same interest, in particular in NLP. This shows that +abandoning HMMs in favor of LC-CRFs in NLP because of their “generative” nature [6, +7, 8, 9, 16, 17, 18, 19] of related Bayesian classifiers was not justified. +2. Equivalence between HMMs and simple linear-chain CRFs +We will use the following Lemma: +Lemma +Let 𝑊�:� = (𝑊�, … , 𝑊�) be random sequence, taking its values in a finite set Δ. Then +(i) 𝑊�:� is a Markov chain iff there exist 𝑁 − 1 functions 𝜑�, … , 𝜑��� from Δ� to +R�such that +𝑝(𝑤�, … , 𝑤�) ∝ 𝜑�(𝑤�, 𝑤�) … 𝜑���(𝑤���, 𝑤�), + + + +(2.1) +where “∝” means “proportional to”; +(ii) for HMM defined with 𝜑�, …, 𝜑��� verifying (2.1), 𝑝(𝑤�) and 𝑝(𝑤���|𝑤�) are +given with +𝑝(𝑤�) = +��(��) +∑ +��(��) +�� + ; 𝑝(𝑤���|𝑤�) = +��(��,����)����(����) +��(��) +, + + +(2.2) +where 𝛽�(𝑤�), …, 𝛽�(𝑤�) are defined with the following backward recursion: +𝛽�(𝑤�)=1, 𝛽�(𝑤�) = ∑ +𝜑�(𝑤�, 𝑤���)𝛽���(𝑤���) +���� + + + +(2.3) +Proof of Lemma. +1. Let 𝑊�:� be Markov : 𝑝(𝑤�, … , 𝑤�) = 𝑝(𝑤�)𝑝(𝑤�| 𝑤�)𝑝(𝑤�| 𝑤�) … 𝑝(𝑤�| 𝑤���). Then +(2.1) is verified by +𝜑�(𝑤�, 𝑤�) = 𝑝(𝑤�)𝑝(𝑤�| 𝑤�) , +𝜑�(𝑤�, 𝑤�) = 𝑝(𝑤�| 𝑤�) , …, +𝜑���(𝑤���, 𝑤�) = 𝑝(𝑤�| 𝑤���). +2. +Conversely, +let +𝑝(𝑤�, … , 𝑤�) + verifies +(2.1). +Thus +𝑝(𝑤�, … , 𝑤�) = +𝐾𝜑�(𝑤�, 𝑤�) … 𝜑���(𝑤���, 𝑤�) with 𝐾 constant. This implies that for each 𝑛 = 1, …, +𝑁 − 1 we have +𝑝(𝑤���|𝑤�, … , 𝑤�) = +�(��,…,��,����) +�(��,…,��) += +∑ +��(��,��) …��(��,����)����(����,����)…����(����,��) +�����,…,��,� +∑ +��(��,��) …��(��,����)����(����,����)…����(����,��) +�����,����,…,��,� += + + +(2.4) +��(��,����) ∑ +����(����,����)…����(����,��) +�����,…,��,� +∑ +��(��,����)����(����,����)…����(����,��) +�����,����,…,��,� += 𝑝(𝑤���| 𝑤�), +which shows that 𝑝(𝑤�, … , 𝑤�) is Markov. +Besides, let us set 𝛽�(𝑤�) = ∑ +𝜑�(𝑤�, 𝑤���) … 𝜑���(𝑤���, 𝑤�) +(����,����,…,��) + for 𝑛 = 1, +…, 𝑁 − 1. On the one hand, we see that 𝛽�(𝑤�) = ∑ +𝜑�(𝑤�, 𝑤���)𝛽���(𝑤���) +���� +. On the +other hand, according to (2.4) we have 𝑝(𝑤���|𝑤�) = +��(��,����)����(����) +��(��) +. As 𝑝(𝑤�) = +��(��) +∑ +��(��) +�� +, (2.2) and (2.3) are verified, which ends the proof □ + + +4 of 8 + +Proposition 1 below shows that “linear-chain” CRF defined with (1.3) is equivalent +to a HMM defined with (1.1). In addition, 𝑝(𝑥�), 𝑝(𝑥���|𝑥�), and 𝑝(𝑦�|𝑥�) in (1.1) de- +fining an equivalent HMM are computed from 𝑉�(𝑥�, 𝑥���) and 𝑈�(𝑥�, 𝑦�). To the best of +our knowledge, except some first weaker results in [15], these results are new. +Proposition 1. Let 𝑍�:� = (𝑍�, … , 𝑍�) be stochastic sequence, with 𝑍� = (𝑋�, 𝑌�). Each +(𝑋�, 𝑌�) takes its values in 𝛬 × 𝛺, with 𝛬 and 𝛺 finite. If 𝑍�:� is a linear chain conditional ran- +dom field (LC-CRF) with the distribution 𝑝(𝑥�:�|𝑦�:�) defined by + 𝑝(𝑥�:�|𝑦�:�) = +� +�(��:�) 𝑒𝑥𝑝[∑ +𝑉�(𝑥�, 𝑥���) +��� +��� ++ ∑ +𝑈�(𝑥�, 𝑦�) +� +��� +]. + +(2.5) +then (2.5) is the posterior distribution of the HMM +𝑞(𝑥�:�, 𝑦�:�) = 𝑞�(𝑥�)𝑞(𝑦�|𝑥�) ∏ +𝑞���(𝑥���|𝑥�)𝑞(𝑦���|𝑥���) +��� +��� +, + +(2.6) +with +𝑞(𝑥�) = +��(��) +∑ +��(��) +�� +; + + + + + + + + + +(2.7) +𝑞(𝑥���|𝑥�) = +��� [��(��,����)] +∑ +�(����)��� [��(��,����)] +���� +; + + + + + +(2.8) +𝑞(𝑦�|𝑥�) = +��� [����(����,����)] ����(����,����) +�(����) +, + + + + + +(2.9) +where +𝜓(𝑥���) = ∑ +𝑒𝑥𝑝 [𝑈(𝑥���, 𝑦���)]𝛾���(𝑥���, 𝑦���) +���� +, + + + +(2.10) +and 𝛾�(𝑥�, 𝑦�), …, 𝛾�(𝑥�, 𝑦�) are given by the backward recursion +𝛾�(𝑥�, 𝑦�) = 1, 𝛾�(𝑥�, 𝑦�) = ∑ +𝜑�(𝑥�, 𝑦�, 𝑥���, 𝑦���)𝛾���(𝑥���, 𝑦���) +(����,����) + +(2.11) +Proof of Proposition 1. According to the Lemma, functions 𝜑�, .., 𝜑� defined on + Δ�, with Δ = Λ × Ω, by +𝜑�(𝑥�, 𝑦�, 𝑥�, 𝑦�) = 𝑒𝑥𝑝[𝑉�(𝑥�, 𝑥�) + 𝑈�(𝑥�, 𝑦�) + 𝑈�(𝑥�, 𝑦�)]; + + +(2.12) +for 𝑛 = 2, … , 𝑁 − 1, 𝜑�(𝑥�, 𝑦�, 𝑥���, 𝑦���) = exp [𝑉�(𝑥�, 𝑥���) + 𝑈���(𝑥���, 𝑦���)], (2.13) +define a Markov chain 𝑍�:� = (𝑍�, … , 𝑍�) , with 𝑍� = (𝑋�, 𝑌�) . Let us denote with +𝑟(𝑧�:�) = 𝑟(𝑥�:�, 𝑦�:�) + its +distribution. +As +𝑟(𝑥�:�, 𝑦�:�) = 𝐾𝑒𝑥𝑝[∑ +𝑉�(𝑥�, 𝑥���) +��� +��� ++ +∑ +𝑈�(𝑥�, 𝑦�) +� +��� +] with 𝐾 constant, we have 𝑟(𝑥�:�|𝑦�:�) = 𝑝(𝑥�:�|𝑦�:�). Thus the problem +is to show that 𝑟(𝑥�:�|𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�). This is implied by 𝛽�(𝑥�) = 𝛾�(𝑥�), for 𝑛 = 𝑁, +…, 1. We have 𝛽�(𝑥�) = 𝛾�(𝑥�) = 1. Let us show that 𝛽���(𝑥���) = 𝛾���(𝑥���) implies +𝛽�(𝑥�) = 𝛾�(𝑥�). According to the lemma +𝑟(𝑥���, 𝑦���|𝑥�, 𝑦�) = +��(��,��,����,����)����(����,����) +∑ +��(��,��,����,����)����(����,����) +(����,����) += +��� [��(��,����)�����(����,����)] ����(����,����) +∑ +��� [��(��,����)�����(����,����)]����(����,����) +(����,����) + + + +(2.14) +Let +𝜓(𝑥���) = ∑ +exp [𝑈(𝑥���, 𝑦���)]𝛾���(𝑥���, 𝑦���) +���� + + + + +(2.15) +According to (2.13) +𝑟(𝑥���|𝑥�, 𝑦�) = 𝑟(𝑥���|𝑥�) = +�(����)��� [��(��,����)] +∑ +�(����)��� [��(��,����)] +���� + , + + +(2.16) +𝑟(𝑦���|𝑥���, 𝑥�, 𝑦�) = 𝑟(𝑦���|𝑥���) = +��� [��(��,����)�����(����,����)] ����(����,����) +�(����)��� [��(��,����)] += + + + + +(2.17) +��� [����(����,����)] ����(����,����) +�(����) +. + + + + + +5 of 8 + +Finally +𝑟(𝑥���, 𝑦���|𝑥�, 𝑦�) = 𝑟(𝑥���|𝑥�)𝑟(𝑦���|𝑥���) = +[ +��� [��(��,����)] +∑ +�(����)��� [��(��,����)] +���� +][ +��� [����(����,����)] ����(����,����) +�(����) +], +which ends the proof □ +3. Discriminative classifiers in generative HMMs +One of interests of HMMs and some CRFs with hidden discrete finite data lies in +possibilities of analytic fast computation of Bayesian classifiers. As examples of classic +Bayesian classifiers, let us consider the MPM (1.4) and the MAP (1.5). However, in some +domains like NLP, CRFs are preferred to HMMs for the following reasons. +As HMM is a generative model, MPM and MAP used in HMM are also called “gen- +erative”, and people consider that HMM based MPM and MAP need the knowledge of +𝑝(𝑦�|𝑥�). Then people consider it as improper to use them in situations where these dis- +tributions are hard to handle. We show that this reason is not valid. More precisely, we +show two points: +(i) First, we notice that whatever distribution 𝑝(𝑥�:�, 𝑦�:�), all Bayesian classifiers are +independent from 𝑝(𝑦�:�), so that the distinction between « generative » and « discrimi- +native » classifiers is misleading: they are all discriminative; +(ii) Second, “discriminative” computation of MPM and MAP in HMMs is not intrin- +sic to HMMs but is due to its particular classic parameterization (1.1). In other words, +changing the parametrization, it is possible to compute the HMM based MPM and MAP +calling neither on 𝑝(𝑦�:�|𝑥�:�) nor of 𝑝(𝑦�:�). +The first point is rather immediate: we note that Bayesian classifier 𝑔� is defined by +a loss function 𝐿: Ω� → ℝ� through +[𝑔�(𝑦�:�) = 𝑥��:�] ⟺ [𝐸[𝐿(𝑔�(𝑦�:�), 𝑋�:�)|𝑦�:�] = 𝑖𝑛𝑓 +��:� +𝐸[𝐿(𝑥�:�, 𝑋�:�)|𝑦�:�]; +(3.1) +it is thus immediate to notice that 𝑔�(𝑦�:�) only depends on 𝑝(𝑥�:�|𝑦�:�). This implies +that it is the same in a generative model or its equivalent (within the meaning of Definition +1.1) discriminative model. +We show (ii) by separately considering MPM and MAP cases. +3.1. Discriminative computing of HMM based MPM +To show (ii) let us consider (1.1) with 𝑝(𝑦�|𝑥�) = +�(��)�(��|��) +�(��) +. It becomes +𝑝(𝑥�:�, 𝑦�:�) = 𝑝(𝑥�|𝑦�) ∏ +𝑝(𝑥�|𝑥���) �(��|��) +�(��) +� +��� +∏ +𝑝(𝑦�) +� +��� +, + + +(3.2) +We see that (3.2) is of the form 𝑝(𝑥�:�, 𝑦�:�) = ℎ(𝑥�:�, 𝑦�:�) ∏ +𝑝(𝑦�) +� +��� +, where +ℎ(𝑥�:�, 𝑦�:�) does not depend on 𝑝(𝑦�) , …, 𝑝(𝑦�) . This implies that ℎ(𝑥�, 𝑦�:�) = +∑ +ℎ(𝑥�:�, 𝑦�:�) +(��:���,����:�) + does not depend on 𝑝(𝑦�) , …, 𝑝(𝑦�) either. Then +𝑝(𝑥�|𝑦�:�) = +�(��,��:�) +�(��:�) = +�(��,��:�) ∏ +�(��) +� +��� +[∑ +�(��,��:�)] +�� +∏ +�(��) +� +��� += +�(��,��:�) +[∑ +�(��,��:�)] +�� +, so that + +𝑝(𝑥�|𝑦�:�) = +∑ +��𝑥��𝑦�� ∏ +��𝑥��𝑥�����(��|��) +�(��) +� +��� +(��:���,����:�) +∑ +��𝑥��𝑦�� ∏ +��𝑥��𝑥�����(��|��) +�(��) +� +��� +(��:�) + + + + +(3.3) +neither depends on 𝑝(𝑦�), …, 𝑝(𝑦�). Thus, HMM based classifier MPM also verifies the +“discriminative classifier” definition. +How to compute 𝑝(𝑥�|𝑦�:�) ? It is classically computable using “forward” probabili- +ties 𝛼�(𝑥�) and “backward” ones 𝛽�(𝑥�) defined with +𝛼�(𝑥�) = 𝑝(𝑥�, 𝑦�:�), + + + + + +(3.4) +𝛽�(𝑥�) = 𝑝(𝑦���:�|𝑥�). + + + + + +(3.5) + + +6 of 8 + +Then +𝑝(𝑥�|𝑦�:�) = +��(��)��(��) +∑ +��(��)��(��) +�� +, + + + + +(3.6) +with all 𝛼�(𝑥�) and 𝛽�(𝑥�) computed using the following forward and backward recur- +sions [20]: +𝛼�(𝑥�) = 𝑝(𝑥�)𝑝(𝑦�|𝑥�); 𝛼���(𝑥���) = ∑ +𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���) +�� +𝛼�(𝑥�), (3.7) +𝛽�(𝑥�) = 1; 𝛽�(𝑥�) = ∑ +𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���) +���� +𝛽���(𝑥���). + +(3.8) +Setting 𝑝(𝑦�|𝑥�) = +�(��)�(��|��) +�(��) + and recalling that 𝑝(𝑥�|𝑦�:�) does not depend on 𝑝(𝑦�), +…, 𝑝(𝑦�), we can arbitrarily modify them. Let us consider the uniform distribution over +Ω, so that 𝑝(𝑦�) = ⋯ = 𝑝(𝑦�) = +� +#� = 𝑐. Then (3.7), and (3.8) become + 𝛼��� +∗ +(𝑥���) = ∑ +𝑝(𝑥���|𝑥�) ��(����|����) +�(����) +�� +𝛼� +∗(𝑥�) +; + + + +(3.9) + 𝛽� +∗(𝑥�) = ∑ +𝑝(𝑥���|𝑥�) ��(����|����) +�(����) +���� +𝛽��� +∗ +(𝑥���), + + + +(3.10) +and we still have +𝑝(𝑥�|𝑦�:�) = +��∗ (��)��∗ (��) +∑ +��∗ ��∗ (��) +�� +. + + + + + +(3.11) +Finally, we see that 𝑝(𝑥�|𝑦�:�) is independent from 𝑐, so that we can take 𝑐 = 1. Then we +can state +Proposition 2 Let 𝑋�, … , 𝑋�, 𝑌�, … , 𝑌� be a HMM (1.1). Let define “forward discriminative” +quantities 𝛼� +�(𝑥�), …, 𝛼� +�(𝑥�), and “backward discriminative” ones 𝛽� +�(𝑥�), …, 𝛽� +�(𝑥�) by the +following forward and backward recursions: +𝛼� +�(𝑥�) = 𝑝(𝑥�|𝑦�); 𝛼��� +� +(𝑥���) = ∑ +𝑝(𝑥���|𝑥�) �(����|����) +�(����) +�� +𝛼� +�(𝑥�), + +(3.12) +𝛽� +�(𝑥�) = 1; 𝛽� +�(𝑥�) = ∑ +𝑝(𝑥���|𝑥�) �(����|����) +�(����) +���� +𝛽��� +� +(𝑥���). + +(3.13) +Then +𝑝(𝑥�|𝑦�:�) = +���(��)���(��) +∑ +���(��)���(��) +�� +. + + + + +(3.14) +Consequently, we can compute MPM classifier in a discriminative manner, only using 𝑝(𝑥�), …, +𝑝(𝑥�), 𝑝(𝑥�|𝑥�), …, 𝑝(𝑥�|𝑥���), and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�). + +Then for 𝑛 = 1, …, 𝑁, 𝑝(𝑥�|𝑦�:�) can be computed par the only use of 𝑝(𝑥�), …, 𝑝(𝑥�), +𝑝(𝑥�|𝑥�), …, 𝑝(𝑥�|𝑥���), and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�). This shows that 𝑝(𝑥�|𝑦�:�) can be +computed with the only use of 𝑝(𝑥�|𝑦�) and 𝑝(𝑥�), exactly as in CRF case. For example, +in supervised NLP problems 𝑝(𝑥�|𝑦�) and 𝑝(𝑥�) can be easily estimated and thus the +interest of HMMs is equivalent to that of CRFs. This result is similar to [21], but with a +more concise proof. +Remark 1 +Let us notice that according to (3.12), (3.13), it is possible to compute 𝛼� +�(𝑥�) and +𝛽� +�(𝑥�) by a very slight adaptation of classic computing programs giving classic 𝛼�(𝑥�) = +𝑝(𝑥�, 𝑦�:�) and 𝛽�(𝑥�) = 𝑝(𝑦���:�|𝑥�) with recursions (3.7), (3.8). All we have to do is to +replace 𝑝(𝑦���|𝑥���) with +�(����|����) +�(����) +. Of course, 𝛼� +�(𝑥�) ≠ 𝑝(𝑥�, 𝑦�:�) and 𝛽� +�(𝑥�) ≠ +𝑝(𝑦���:�|𝑥�), but (3.14) holds. This is the core point: one uses a wrong model to find +𝑝(𝑥�|𝑦�:�) of the true HMC, that is the final goal. + + + + +7 of 8 + +Remark 2 +We see that we can compute MPM in HMM only using 𝑝(𝑥�), …, 𝑝(𝑥�) and +𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�). This means that in supervised classification, where we have a learn +sample, we can use any parametrization to estimate them. For example, we can model +them with logistic regression, as currently do in CRFs. It of importance to note that such +a parametrization is unusual; however, important is that the model remains the same. +3.2. Discriminative computing of HMM based MAP: discriminative Viterbi +Let 𝑋�, … , 𝑋�, 𝑌�, … , 𝑌� be a HMM (1.1). Bayesian MAP classifier (1.5) based on is +computed with the following Viterbi algorithm [22]. For each 𝑛 = 1, … , 𝑁, and each 𝑥�, +let 𝑥�:��� +��� (𝑥�) = (𝑥� +���, … , 𝑥��� +���)(𝑥�) be the path 𝑥� +���, …, 𝑥��� +��� verifying +𝑝(𝑥�:��� +��� (𝑥�), 𝑥�, 𝑦�:�) = 𝑠𝑢𝑝 +��:��� +𝑝(𝑥�:���, 𝑥�, 𝑦�:�); + + + + +(3.15) +We see that 𝑥�:��� +��� (𝑥�) is a path maximizing 𝑝(𝑥�:���, 𝑥�|𝑦�:�) over all paths ending in +𝑥�. Then having the paths 𝑥�:��� +��� (𝑥�) and the probabilities 𝑝(𝑥�:��� +��� (𝑥�), 𝑥�, 𝑦�:�) for each +𝑥� , one determines, for each 𝑥��� , the paths 𝑥�:� +���(𝑥���) and the probabilities +𝑝(𝑥�:� +���(𝑥���), 𝑥���, 𝑦�:���) = 𝑝(𝑥�:��� +��� (𝑥� +���), 𝑥� +���, 𝑥���, 𝑦�:���), searching 𝑥� +��� with +𝑝(𝑥�:��� +��� (𝑥� +���), 𝑥� +���, 𝑥���, 𝑦�:���)) = + + + + + +(3.16) +𝑠𝑢𝑝 +�� +[𝑝(𝑥�:��� +��� (𝑥�), 𝑥�, 𝑦�:�)𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���)]. +Setting in (3.16) 𝑝(𝑦���|𝑥���) = +�(����)�(����|����) +�(����) +, we see that 𝑥� +��� which verifies (3.16) +is the same that 𝑥� +��� which maximizes 𝑝(𝑥�:��� +��� (𝑥�), 𝑥�, 𝑦�:�)𝑝(𝑥���|𝑥�) +�(����|����) +�(����) +, so +that we can suppress 𝑝(𝑦���). In other words we can replace (3.16) with + + +𝑝(𝑥�:��� +��� (𝑥� +���), 𝑥� +���, 𝑥���, 𝑦�:���)) = + + + + + +(3.17) +𝑠𝑢𝑝 +�� +[𝑝(𝑥�:��� +��� (𝑥�), 𝑥�, 𝑦�:�)𝑝(𝑥���|𝑥�) +�(����|����) +�(����) +] +Finally, we propose the following discriminative version of the Viterbi algorithm: + - set 𝑥� +��� = 𝑎𝑟𝑔𝑚𝑎𝑥 +�� +[𝑝(𝑥�|𝑦�)]; +- for each 𝑛 = 1, …, 𝑁 − 1, and each 𝑥���, apply (3.17) to find a path 𝑥�:� +���(𝑥���) from +the paths 𝑥�:��� +��� (𝑥�) (for all 𝑥�), and the probabilities 𝑝(𝑥�:� +���(𝑥���), 𝑥���, 𝑦�:���) (for all +𝑥���); +- end setting 𝑥�:� +��� = 𝑎𝑟𝑔𝑚𝑎𝑥 +�� +[𝑝(𝑥�:��� +��� (𝑥�), 𝑥�, 𝑦�:�)]. +Likely for MPM above, we see that we can find 𝑥�:� +��� with the only use of 𝑝(𝑥�), …, +𝑝(𝑥�), 𝑝(𝑥�|𝑥�), …, 𝑝(𝑥�|𝑥���), and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�), exactly as in CRF case. As +above, it appears that dropping HMMs in some NLP tasks on the grounds that MAP is a +“generative” classifier, is not justified. In particular, in supervised stationary framework +distributions 𝑝(𝑥�), 𝑝(𝑥���|𝑥�), and 𝑝(𝑥�|𝑦�) can be estimated in the same way as in LC- +CRFs case. +4. Discussion +We proposed two results. We showed that the basic LC-CRF (1.3) is equivalent to a +classic HMM (1.1) in that one can find a HMM whose posterior distribution is exactly the +given LC-CRF. More precisely, we specified the way of computing 𝑝(𝑥�) and 𝑝(𝑥���|𝑥�), +𝑝(𝑦�|𝑥�) defining (1.1) from 𝑉�(𝑥�, 𝑥���), 𝑈�(𝑥�, 𝑦�) defining (1.3). Then, noticing that all +Bayesian classifiers are discriminative in that they do not depend on the observations dis- +tribution, we showed that HMMs based classifiers, usually considered as “generative” + + +8 of 8 + +ones, can also be considered as discriminative classifiers. More precisely, we proposed +discriminative ways of computing the classic HMMs based Maximum Posterior Mode +(MPM) and Maximum a Posteriori (MAP) classifiers. +We considered basic LC-CRFs and HMMs; extending the proposed results to more +sophisticated models like Pairwise Markov chains” (PMCs [23]) is an interesting perspec- +tive for further investigations. +Author Contributions: Conceptualization, E.A., E.M. and W.P.; methodology, E.A., E.M. and W.P.; +validation, E.A., E.M. and W.P.; investigation, E.A., E.M. and W.P.; writing—original draft prepara- +tion, W.P.; writing—review and editing, E.A., E.M. and W.P; supervision, W.P.; project administra- +tion, W.P. All authors have read and agreed to the published version of the manuscript. +Funding: This research was partly funded by the French Government Agency ASSOCIATION NA- +TIONALE RECHERCHE TECHNOLOGIES (ANRT). +Conflicts of Interest: The authors declare no conflict of interest. +References +1. +Stratonovich, R. L. Conditional Markov Processes. Non-linear Transformations of Stochastic Processes, Pergamon, 1965, pp. 427-453. +2. +Baum, L. E.; Petrie, T. Statistical Inference for Probabilistic Functions of Finite state Markov Chains. The Annals of Mathematical +Statistics, 1966, 37(6), pp. 1554-1563. +3. +Rabiner, L.; Juang, B. An Introduction to Hidden Markov Models. IEEE ASSP Magazine, 3(1), 1986, pp. 4-16. +4. +Ephraim Y. Hidden Markov Processes, IEEE Trans. on Information Theory, 2002, vol. 48, no. 6, pp.1518–1569. +5. +Cappé O.; Moulines E.; Ryden T. Inference in Hidden Markov Models, Springer, Series in Statistics, 2005. +6. +Lafferty, J.; McCallum, A.; Pereira, F. C. Conditional Random Fields: Probabilistic Models for Segmenting and Labeling Se- +quence Data, Proceedings of the International Conference on Machine Learning, 2001. +7. +Sutton, C.; McCallum, A. An Introduction to Conditional Random Fields. Foundations and Trends® in Machine Learning, 4(4), +2012, pp. 267-373. +8. +Jurafsky D.; Martin J.H. Speech and Language Processing, Copyright © 2020. All rights reserved. Draft of December 30, 2020. +9. +Ng, A.; Jordan, M. On Discriminative vs. Generative Classifiers: A Comparison of Logistic Regression and Naive Bayes. Ad- +vances in Neural Information Processing Systems, 2001, pp. 841-848. +10. +He, H.; Liu, Z.; Jiao, R.; Yan, G. A Novel Nonintrusive Load Monitoring Approach based on Linear-Chain Conditional Random +Fields, Energies, 12(9), 2019. +11. +Condori, G. C.; Castro-Gutierrez, E.; Casas, L.A. Virtual Rehabilitation Using Sequential Learning Algorithms, International +Journal of Advanced Computer science and Applications, vol. 9, no. 11, 2018, pp. 639-645. +12. +Fang, M.; Kodamana, H.; Huang, B.; Sammaknejad, N. A Novel Approach to Process Operating Mode Diagnosis using Condi- +tional Random Fields in the Presence of Missing Data, Computers and Chemical Engineering 111, 2018, pp. 149–163. +13. +Saa, J. F. D.; Cetin, M. A Latent Discriminative model-based Approach for Classification of Imaginary Motor tasks from EEG +data, Journal of Neural Engineering, vol. 9, no. 2, 2012. +14. +Azeraf, E.; Monfrini, E.; and Pieczynski, W. On Equivalence between Linear-Chain Conditional Random Fields and Hidden +Markov Chains, International Conference on Agents and Artificial Intelligence, 2022. +15. +Heigold, G.; Ney, H.; Lehnen, P.; Gass, T.; Schluter, R. Equivalence of Generative and Log-Linear Models. IEEE Transactions on +Audio, Speech, and Language Processing, 19(5), 2010, pp. 1138-1148. +16. +Liliana, D. Y.; Basaruddin C. A Review on Conditional Random Fields as a Sequential Classifier in Machine Learning, Interna- +tional Conference on Electrical Engineering and Computer Science (ICECOS), 2017, pp.143-148. +17. +Ayogu, I. I.; Adetunmbi, A. O.; Ojokoh, B. A.; Oluwadare, S. A. A Comparative Study of Hidden Markov Model and Conditional +Random Fields on a Yorùba Part-of-Speech Tagging task. IEEE International Conference on Computing Networking and Informatics +(ICCNI), 2017, pp. 1-6. +18. +McCallum, A.; Freitag, D.; Pereira, F. C. Maximum Entropy Markov Models for Information Extraction and Segmentation. +ICML, 2000, pp. 591-598. +19. +Song, S. L.; Zhang, N.; Huang, H. T. Named Entity Recognition based on Conditional Random Fields, Cluster Computing – the +Journal of Networks Software Tools and Applications, 22, 2019, pp. S5195-S5206. +20. +Baum, L.E.; Petrie, T.; Soules, G.; Weiss N. A Maximization Technique Occurring in the Statistical Analysis of Probabilistic +Functions of Markov Chains, Annals of Mathematical Statistics, vol. 41, no. 1, 1970, pp. 164-171. +21. +Azeraf, E.; Monfrini, E.; Vignon, E.; Pieczynski, W. Hidden Markov Chains, Entropic Forward-Backward, and Part-Of-Speech +Tagging. arXiv preprint arXiv:2005.10629, 2020. +22. +Viterbi, A. Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm, IEEE transactions on +Information Theory, 13(2), 1967, pp. 260-269. +23. +Pieczynski, W. Pairwise Markov Chains, IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 25, No. 5, 2003, pp. 634- +639. + diff --git a/MNAzT4oBgHgl3EQfV_xV/content/tmp_files/load_file.txt b/MNAzT4oBgHgl3EQfV_xV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85788173b844681e17140c2168889a7e1c7fa011 --- /dev/null +++ b/MNAzT4oBgHgl3EQfV_xV/content/tmp_files/load_file.txt @@ -0,0 +1,509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf,len=508 +page_content='Preprint Linear chain conditional random fields, hidden Markov mod- els, and related classifiers Elie Azeraf 1, Emmanuel Monfrini 2 and Wojciech Pieczynski 2,* 1 Watson Department, IBM France, Paris, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' elie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='azeraf@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='com 2 Telecom SudParis, Institut Polytechnique de Paris, Evry, France ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' {Emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='Monfrini, Woj- ciech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='Pieczynski}@telecom-sudparis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='eu * Correspondence: Wojciech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='Pieczynski@telecom-sudparis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='eu Abstract: Practitioners use Hidden Markov Models (HMMs) in different problems for about sixty years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Besides, Conditional Random Fields (CRFs) are an alternative to HMMs and appear in the literature as different and somewhat concurrent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We propose two contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' First, we show that basic Linear-Chain CRFs (LC-CRFs), considered as different from the HMMs, are in fact equivalent to them in the sense that for each LC-CRF there exists a HMM – that we specify – whom posterior distribution is identical to the given LC-CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Second, we show that it is possible to refor- mulate the generative Bayesian classifiers Maximum Posterior Mode (MPM) and Maximum a Pos- teriori (MAP) used in HMMs, as discriminative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' The last point is of importance in many fields, especially in Natural Language Processing (NLP), as it shows that in some situations dropping HMMs in favor of CRFs was not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Keywords: Hidden Markov Model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Linear Chain Conditional Random Field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Bayesian Classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Discriminative Classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Maximum Posterior Mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Maximum A Posteriori 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Introduction Let 𝑍�:� = (𝑍�, … , 𝑍�) be a stochastic sequence, with 𝑍� = (𝑋�, 𝑌�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Random varia- bles 𝑋�, … , 𝑋� take their values in a finite set Λ, while 𝑌�, … , 𝑌� take their values either in a discrete or continuous set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Realizations of 𝑋�:� = (𝑋�, … , 𝑋�) are hidden while re- alizations of 𝑌�:� = (𝑌�, … , 𝑌�) are observed, and the problem we deal with is to estimate 𝑋�:� = 𝑥�:� from 𝑌�:� = 𝑦�:�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We deal with Bayesian methods of estimation, which need some probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Probabilistic model is a distribution – or a family of distribu- tions – which will be denoted with 𝑝(𝑧�:�), or 𝑝(𝑥�:�, 𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We are interested in the case of dependent 𝑍�, … , 𝑍�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' The simplest model taking into account this dependence is the well-known hidden Markov model (HMM) [1, 2, 3, 4, 5], whose distribution is given with 𝑝(𝑥�:�, 𝑦�:�) = 𝑝(𝑥�) 𝑝(𝑦�|𝑥�) ∏ 𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���) ��� ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) HMMs allow recursive fast computation of Bayesian estimators called “classifiers” in this paper and recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In spite of their simplicity, HMMs are very robust and provide quite satisfactory results in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Beside, conditional random fields (CRFs) [6, 7] also allow estimating 𝑋�:� = 𝑥�:� from 𝑌�:� = 𝑦�:�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Their definition is different from the definition of HMMs in that in CRFs, one directly considers 𝑝(𝑥�:�|𝑦�:�), and neither 𝑝(𝑥�:�, 𝑦�:�) nor 𝑝(𝑦�:�|𝑥�:�) are needed to perform the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In some areas, like Natural Language Processing (NLP), CRFs are preferred over HMCs because 𝑝(𝑥�:�, 𝑦�:�) and 𝑝(𝑦�:�|𝑥�:�) are difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' General CRFs are written 2 of 8 𝑝(𝑥�:�|𝑦�:�) = 𝑝(𝑥�|𝑦�:�) ∏ 𝑝(𝑥���|𝑥�:�, 𝑦�:�) ��� ��� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='2) In this paper we will consider the following basic “linear-chain” CRF (LC-CRF): 𝑝(𝑥�:�|𝑦�:�) = � �(��:�) 𝑒𝑥𝑝[∑ 𝑉�(𝑥�, 𝑥���) ��� ��� + ∑ 𝑈�(𝑥�, 𝑦�) � ��� ], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) with 𝜅(𝑦�:�) the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Authors usually consider the two families HMMs and CRFs as different [6, 7, 8, 9, 10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' They classify the former in the category of “generative models”, while they classify the latter in the category of “discriminative” models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Considering the simples cases (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3), we propose two contributions: (a) We establish an equivalence between HMMs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) and basic linear-chain CRFs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3), which completes the results presented in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let us notice that wanting to compare the two models directly is somewhat mislead- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Indeed, HMMs and CRFs are defined with distributions on different spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' To be precise, we adopt the following definition Definition 1 Let 𝑋�, … , 𝑋�, 𝑌�, … , 𝑌� be two stochastic sequences defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (i) we will call “model” a distribution 𝑝(𝑥�:�, 𝑦�:�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (ii) we will call “conditional model” a distribution 𝑝(𝑥�:�|𝑦�:�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (iii) we will say that a model 𝑝(𝑥�:�, 𝑦�:�) is “equivalent” to a conditional model 𝑞(𝑥�:�|𝑦�:�) if there exists a distribution 𝑟(𝑦�:�) such that 𝑝(𝑥�:�, 𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�)𝑟(𝑦�:�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (iv) we will say that a family of models Α is “equivalent” to a family of conditional models Β if for each model 𝑝(𝑥�:�, 𝑦�:�) in Α there exists an equivalent conditional model 𝑞(𝑥�:�|𝑦�:�) in Β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' According to Definition 1 HMMs are particular “models”, while CRFs are particular “conditional models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then a particular model HMM cannot be equal to a particular con- ditional model CRF, but it can be equivalent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Our aim is to show that the family of LC-CRFs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) is equivalent to the family of HMMs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In addition we specify, for each LC-CRF 𝑞(𝑥�:�|𝑦�:�), a particular HMM 𝑝(𝑥�:�, 𝑦�:�) such that 𝑝(𝑥�:�|𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let 𝑝(𝑥�:�, 𝑦�:�, 𝜃) be an HMM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1), with parameters 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Taking 𝑟(𝑦�:�) = 𝑝(𝑦�:�, 𝜃), it is immediate to see that 𝑝(𝑥�:�|𝑦�:�, 𝜃) is an equivalent CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' The converse is not imme- diate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Is a given CRF 𝑝(𝑥�:�|𝑦�:�, 𝜃) equivalent to a HMM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' If yes, can we find 𝑟(𝑦�:�) such that 𝑝(𝑥�:�|𝑦�:�, 𝜃)𝑟(𝑦�:�) is a HMM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Besides, can we give its (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Respond- ing these questions in a simple linear-chain CRF case is our first contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' More pre- cisely, we show that the family of LC-CRFs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) is equivalent to the family of HMMs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1), and we specify, for each LC-CRF 𝑞(𝑥�:�|𝑦�:�), a particular HMM 𝑝(𝑥�:�, 𝑦�:�) given in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1), such that 𝑝(𝑥�:�|𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Note that numerous papers addressed similarities between HMMs and linear-chain CRFs [7, 15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' however, to the best of our knowledge, results proposed here, which are mathematically rigorous in the framework of Definition 1, are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (b) We show that “generative” estimators MPM and MAP in HMM are computable in a “discriminative” manner, exactly as in LC-CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' One of interests of HMMs and CRFs is that in both of them there exist Bayesian clas- sifiers, which allow estimating 𝑥�:� from 𝑦�:� in a reasonable computer time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' As exam- ples, let us consider the “maximum of posterior margins” (MPM) defined with: [𝑔(𝑦�:�) = 𝑥��:� = (𝑥��, … , 𝑥��)] ⟺ [∀𝑛 = 1, … , 𝑁, 𝑝( 𝑥��|𝑦�:�) = 𝑠𝑢𝑝 �� (𝑝(𝑥�|𝑦�:�))], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='4) and the “maximum a posteriori” (MAP) is defined with [𝑔(𝑦�:�) = 𝑥��:�] ⟺ [𝑝( 𝑥��:�|𝑦�:�) = 𝑠𝑢𝑝 ��:� (𝑝(𝑥�:�|𝑦�:�))], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='5) 3 of 8 Note that likely to any other Bayesian classifier, MPM and MAP are independent from 𝑝(𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This means that in any generative model 𝑝(𝑥�:�, 𝑦�:�), related Bayesian clas- sifier is strictly the same as that related to the equivalent (in the meaning of Definition 1) CRF model 𝑝(𝑥�:�|𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We see that the distinction between “generative” and “discrimi- native” classifiers is not justified: all Bayesian classifiers are discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' However, in HMM the related MPM and MAP classifiers are computed calling on 𝑝(𝑦�|𝑥�), while this is not the case in LC-CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We show that both MPM and MAP in HMM can also be com- puted in a “discriminative” way, without calling on 𝑝(𝑦�|𝑥�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Thus, the use of MPM and MAP in HMM is strictly the same as its use in LC-CRF, which is our second contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' One of the consequences is that the use of MPMs and MAPs in the two families HMMs and LC-CRFs presents exactly the same interest, in particular in NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This shows that abandoning HMMs in favor of LC-CRFs in NLP because of their “generative” nature [6, 7, 8, 9, 16, 17, 18, 19] of related Bayesian classifiers was not justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Equivalence between HMMs and simple linear-chain CRFs We will use the following Lemma: Lemma Let 𝑊�:� = (𝑊�, … , 𝑊�) be random sequence, taking its values in a finite set Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then (i) 𝑊�:� is a Markov chain iff there exist 𝑁 − 1 functions 𝜑�, … , 𝜑��� from Δ� to R�such that 𝑝(𝑤�, … , 𝑤�) ∝ 𝜑�(𝑤�, 𝑤�) … 𝜑���(𝑤���, 𝑤�), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) where “∝” means “proportional to”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (ii) for HMM defined with 𝜑�, …, 𝜑��� verifying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1), 𝑝(𝑤�) and 𝑝(𝑤���|𝑤�) are given with 𝑝(𝑤�) = ��(��) ∑ ��(��) �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 𝑝(𝑤���|𝑤�) = ��(��,����)����(����) ��(��) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='2) where 𝛽�(𝑤�), …, 𝛽�(𝑤�) are defined with the following backward recursion: 𝛽�(𝑤�)=1, 𝛽�(𝑤�) = ∑ 𝜑�(𝑤�, 𝑤���)𝛽���(𝑤���) ���� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) Proof of Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let 𝑊�:� be Markov : 𝑝(𝑤�, … , 𝑤�) = 𝑝(𝑤�)𝑝(𝑤�| 𝑤�)𝑝(𝑤�| 𝑤�) … 𝑝(𝑤�| 𝑤���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) is verified by 𝜑�(𝑤�, 𝑤�) = 𝑝(𝑤�)𝑝(𝑤�| 𝑤�) , 𝜑�(𝑤�, 𝑤�) = 𝑝(𝑤�| 𝑤�) , …, 𝜑���(𝑤���, 𝑤�) = 𝑝(𝑤�| 𝑤���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Conversely, let 𝑝(𝑤�, … , 𝑤�) verifies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Thus 𝑝(𝑤�, … , 𝑤�) = 𝐾𝜑�(𝑤�, 𝑤�) … 𝜑���(𝑤���, 𝑤�) with 𝐾 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This implies that for each 𝑛 = 1, …, 𝑁 − 1 we have 𝑝(𝑤���|𝑤�, … , 𝑤�) = �(��,…,��,����) �(��,…,��) = ∑ ��(��,��) …��(��,����)����(����,����)…����(����,��) �����,…,��,� ∑ ��(��,��) …��(��,����)����(����,����)…����(����,��) �����,����,…,��,� = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='4) ��(��,����) ∑ ����(����,����)…����(����,��) �����,…,��,� ∑ ��(��,����)����(����,����)…����(����,��) �����,����,…,��,� = 𝑝(𝑤���| 𝑤�), which shows that 𝑝(𝑤�, … , 𝑤�) is Markov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Besides, let us set 𝛽�(𝑤�) = ∑ 𝜑�(𝑤�, 𝑤���) … 𝜑���(𝑤���, 𝑤�) (����,����,…,��) for 𝑛 = 1, …, 𝑁 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' On the one hand, we see that 𝛽�(𝑤�) = ∑ 𝜑�(𝑤�, 𝑤���)𝛽���(𝑤���) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' On the other hand, according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='4) we have 𝑝(𝑤���|𝑤�) = ��(��,����)����(����) ��(��) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' As 𝑝(𝑤�) = ��(��) ∑ ��(��) �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) are verified, which ends the proof □ 4 of 8 Proposition 1 below shows that “linear-chain” CRF defined with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) is equivalent to a HMM defined with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In addition, 𝑝(𝑥�), 𝑝(𝑥���|𝑥�), and 𝑝(𝑦�|𝑥�) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) de- fining an equivalent HMM are computed from 𝑉�(𝑥�, 𝑥���) and 𝑈�(𝑥�, 𝑦�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' To the best of our knowledge, except some first weaker results in [15], these results are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let 𝑍�:� = (𝑍�, … , 𝑍�) be stochastic sequence, with 𝑍� = (𝑋�, 𝑌�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Each (𝑋�, 𝑌�) takes its values in 𝛬 × 𝛺, with 𝛬 and 𝛺 finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' If 𝑍�:� is a linear chain conditional ran- dom field (LC-CRF) with the distribution 𝑝(𝑥�:�|𝑦�:�) defined by 𝑝(𝑥�:�|𝑦�:�) = � �(��:�) 𝑒𝑥𝑝[∑ 𝑉�(𝑥�, 𝑥���) ��� ��� + ∑ 𝑈�(𝑥�, 𝑦�) � ��� ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='5) then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='5) is the posterior distribution of the HMM 𝑞(𝑥�:�, 𝑦�:�) = 𝑞�(𝑥�)𝑞(𝑦�|𝑥�) ∏ 𝑞���(𝑥���|𝑥�)𝑞(𝑦���|𝑥���) ��� ��� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='6) with 𝑞(𝑥�) = ��(��) ∑ ��(��) �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='7) 𝑞(𝑥���|𝑥�) = ��� [��(��,����)] ∑ �(����)��� [��(��,����)] ���� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='8) 𝑞(𝑦�|𝑥�) = ��� [����(����,����)] ����(����,����) �(����) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='9) where 𝜓(𝑥���) = ∑ 𝑒𝑥𝑝 [𝑈(𝑥���, 𝑦���)]𝛾���(𝑥���, 𝑦���) ���� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='10) and 𝛾�(𝑥�, 𝑦�), …, 𝛾�(𝑥�, 𝑦�) are given by the backward recursion 𝛾�(𝑥�, 𝑦�) = 1, 𝛾�(𝑥�, 𝑦�) = ∑ 𝜑�(𝑥�, 𝑦�, 𝑥���, 𝑦���)𝛾���(𝑥���, 𝑦���) (����,����) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='11) Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' According to the Lemma, functions 𝜑�, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='., 𝜑� defined on Δ�, with Δ = Λ × Ω, by 𝜑�(𝑥�, 𝑦�, 𝑥�, 𝑦�) = 𝑒𝑥𝑝[𝑉�(𝑥�, 𝑥�) + 𝑈�(𝑥�, 𝑦�) + 𝑈�(𝑥�, 𝑦�)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='12) for 𝑛 = 2, … , 𝑁 − 1, 𝜑�(𝑥�, 𝑦�, 𝑥���, 𝑦���) = exp [𝑉�(𝑥�, 𝑥���) + 𝑈���(𝑥���, 𝑦���)], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='13) define a Markov chain 𝑍�:� = (𝑍�, … , 𝑍�) , with 𝑍� = (𝑋�, 𝑌�) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let us denote with 𝑟(𝑧�:�) = 𝑟(𝑥�:�, 𝑦�:�) its distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' As 𝑟(𝑥�:�, 𝑦�:�) = 𝐾𝑒𝑥𝑝[∑ 𝑉�(𝑥�, 𝑥���) ��� ��� + ∑ 𝑈�(𝑥�, 𝑦�) � ��� ] with 𝐾 constant, we have 𝑟(𝑥�:�|𝑦�:�) = 𝑝(𝑥�:�|𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Thus the problem is to show that 𝑟(𝑥�:�|𝑦�:�) = 𝑞(𝑥�:�|𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This is implied by 𝛽�(𝑥�) = 𝛾�(𝑥�), for 𝑛 = 𝑁, …, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We have 𝛽�(𝑥�) = 𝛾�(𝑥�) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let us show that 𝛽���(𝑥���) = 𝛾���(𝑥���) implies 𝛽�(𝑥�) = 𝛾�(𝑥�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' According to the lemma 𝑟(𝑥���, 𝑦���|𝑥�, 𝑦�) = ��(��,��,����,����)����(����,����) ∑ ��(��,��,����,����)����(����,����) (����,����) = ��� [��(��,����)�����(����,����)] ����(����,����) ∑ ��� [��(��,����)�����(����,����)]����(����,����) (����,����) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='14) Let 𝜓(𝑥���) = ∑ exp [𝑈(𝑥���, 𝑦���)]𝛾���(𝑥���, 𝑦���) ���� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='15) According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='13) 𝑟(𝑥���|𝑥�, 𝑦�) = 𝑟(𝑥���|𝑥�) = �(����)��� [��(��,����)] ∑ �(����)��� [��(��,����)] ���� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='16) 𝑟(𝑦���|𝑥���, 𝑥�, 𝑦�) = 𝑟(𝑦���|𝑥���) = ��� [��(��,����)�����(����,����)] ����(����,����) �(����)��� [��(��,����)] = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='17) ��� [����(����,����)] ����(����,����) �(����) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 5 of 8 Finally 𝑟(𝑥���, 𝑦���|𝑥�, 𝑦�) = 𝑟(𝑥���|𝑥�)𝑟(𝑦���|𝑥���) = [ ��� [��(��,����)] ∑ �(����)��� [��(��,����)] ���� ][ ��� [����(����,����)] ����(����,����) �(����) ], which ends the proof □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Discriminative classifiers in generative HMMs One of interests of HMMs and some CRFs with hidden discrete finite data lies in possibilities of analytic fast computation of Bayesian classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' As examples of classic Bayesian classifiers, let us consider the MPM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='4) and the MAP (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' However, in some domains like NLP, CRFs are preferred to HMMs for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' As HMM is a generative model, MPM and MAP used in HMM are also called “gen- erative”, and people consider that HMM based MPM and MAP need the knowledge of 𝑝(𝑦�|𝑥�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then people consider it as improper to use them in situations where these dis- tributions are hard to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We show that this reason is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' More precisely, we show two points: (i) First, we notice that whatever distribution 𝑝(𝑥�:�, 𝑦�:�), all Bayesian classifiers are independent from 𝑝(𝑦�:�), so that the distinction between « generative » and « discrimi- native » classifiers is misleading: they are all discriminative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (ii) Second, “discriminative” computation of MPM and MAP in HMMs is not intrin- sic to HMMs but is due to its particular classic parameterization (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In other words, changing the parametrization, it is possible to compute the HMM based MPM and MAP calling neither on 𝑝(𝑦�:�|𝑥�:�) nor of 𝑝(𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' The first point is rather immediate: we note that Bayesian classifier 𝑔� is defined by a loss function 𝐿: Ω� → ℝ� through [𝑔�(𝑦�:�) = 𝑥��:�] ⟺ [𝐸[𝐿(𝑔�(𝑦�:�), 𝑋�:�)|𝑦�:�] = 𝑖𝑛𝑓 ��:� 𝐸[𝐿(𝑥�:�, 𝑋�:�)|𝑦�:�];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) it is thus immediate to notice that 𝑔�(𝑦�:�) only depends on 𝑝(𝑥�:�|𝑦�:�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This implies that it is the same in a generative model or its equivalent (within the meaning of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) discriminative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We show (ii) by separately considering MPM and MAP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Discriminative computing of HMM based MPM To show (ii) let us consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) with 𝑝(𝑦�|𝑥�) = �(��)�(��|��) �(��) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' It becomes 𝑝(𝑥�:�, 𝑦�:�) = 𝑝(𝑥�|𝑦�) ∏ 𝑝(𝑥�|𝑥���) �(��|��) �(��) � ��� ∏ 𝑝(𝑦�) � ��� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='2) We see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='2) is of the form 𝑝(𝑥�:�, 𝑦�:�) = ℎ(𝑥�:�, 𝑦�:�) ∏ 𝑝(𝑦�) � ��� , where ℎ(𝑥�:�, 𝑦�:�) does not depend on 𝑝(𝑦�) , …, 𝑝(𝑦�) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This implies that ℎ(𝑥�, 𝑦�:�) = ∑ ℎ(𝑥�:�, 𝑦�:�) (��:���,����:�) does not depend on 𝑝(𝑦�) , …, 𝑝(𝑦�) either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then 𝑝(𝑥�|𝑦�:�) = �(��,��:�) �(��:�) = �(��,��:�) ∏ �(��) � ��� [∑ �(��,��:�)] �� ∏ �(��) � ��� = �(��,��:�) [∑ �(��,��:�)] �� , so that 𝑝(𝑥�|𝑦�:�) = ∑ ��𝑥��𝑦�� ∏ ��𝑥��𝑥�����(��|��) �(��) � ��� (��:���,����:�) ∑ ��𝑥��𝑦�� ∏ ��𝑥��𝑥�����(��|��) �(��) � ��� (��:�) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) neither depends on 𝑝(𝑦�), …, 𝑝(𝑦�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Thus, HMM based classifier MPM also verifies the “discriminative classifier” definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' How to compute 𝑝(𝑥�|𝑦�:�) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' It is classically computable using “forward” probabili- ties 𝛼�(𝑥�) and “backward” ones 𝛽�(𝑥�) defined with 𝛼�(𝑥�) = 𝑝(𝑥�, 𝑦�:�), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='4) 𝛽�(𝑥�) = 𝑝(𝑦���:�|𝑥�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='5) 6 of 8 Then 𝑝(𝑥�|𝑦�:�) = ��(��)��(��) ∑ ��(��)��(��) �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='6) with all 𝛼�(𝑥�) and 𝛽�(𝑥�) computed using the following forward and backward recur- sions [20]: 𝛼�(𝑥�) = 𝑝(𝑥�)𝑝(𝑦�|𝑥�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 𝛼���(𝑥���) = ∑ 𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���) �� 𝛼�(𝑥�), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='7) 𝛽�(𝑥�) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 𝛽�(𝑥�) = ∑ 𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���) ���� 𝛽���(𝑥���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='8) Setting 𝑝(𝑦�|𝑥�) = �(��)�(��|��) �(��) and recalling that 𝑝(𝑥�|𝑦�:�) does not depend on 𝑝(𝑦�), …, 𝑝(𝑦�), we can arbitrarily modify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let us consider the uniform distribution over Ω, so that 𝑝(𝑦�) = ⋯ = 𝑝(𝑦�) = � #� = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='7), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='8) become 𝛼��� ∗ (𝑥���) = ∑ 𝑝(𝑥���|𝑥�) ��(����|����) �(����) �� 𝛼� ∗(𝑥�) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='9) 𝛽� ∗(𝑥�) = ∑ 𝑝(𝑥���|𝑥�) ��(����|����) �(����) ���� 𝛽��� ∗ (𝑥���), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='10) and we still have 𝑝(𝑥�|𝑦�:�) = ��∗ (��)��∗ (��) ∑ ��∗ ��∗ (��) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='11) Finally, we see that 𝑝(𝑥�|𝑦�:�) is independent from 𝑐, so that we can take 𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then we can state Proposition 2 Let 𝑋�, … , 𝑋�, 𝑌�, … , 𝑌� be a HMM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Let define “forward discriminative” quantities 𝛼� �(𝑥�), …, 𝛼� �(𝑥�), and “backward discriminative” ones 𝛽� �(𝑥�), …, 𝛽� �(𝑥�) by the following forward and backward recursions: 𝛼� �(𝑥�) = 𝑝(𝑥�|𝑦�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 𝛼��� � (𝑥���) = ∑ 𝑝(𝑥���|𝑥�) �(����|����) �(����) �� 𝛼� �(𝑥�), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='12) 𝛽� �(𝑥�) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 𝛽� �(𝑥�) = ∑ 𝑝(𝑥���|𝑥�) �(����|����) �(����) ���� 𝛽��� � (𝑥���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='13) Then 𝑝(𝑥�|𝑦�:�) = ���(��)���(��) ∑ ���(��)���(��) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='14) Consequently, we can compute MPM classifier in a discriminative manner, only using 𝑝(𝑥�), …, 𝑝(𝑥�), 𝑝(𝑥�|𝑥�), …, 𝑝(𝑥�|𝑥���), and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then for 𝑛 = 1, …, 𝑁, 𝑝(𝑥�|𝑦�:�) can be computed par the only use of 𝑝(𝑥�), …, 𝑝(𝑥�), 𝑝(𝑥�|𝑥�), …, 𝑝(𝑥�|𝑥���), and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This shows that 𝑝(𝑥�|𝑦�:�) can be computed with the only use of 𝑝(𝑥�|𝑦�) and 𝑝(𝑥�), exactly as in CRF case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' For example, in supervised NLP problems 𝑝(𝑥�|𝑦�) and 𝑝(𝑥�) can be easily estimated and thus the interest of HMMs is equivalent to that of CRFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This result is similar to [21], but with a more concise proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Remark 1 Let us notice that according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='13), it is possible to compute 𝛼� �(𝑥�) and 𝛽� �(𝑥�) by a very slight adaptation of classic computing programs giving classic 𝛼�(𝑥�) = 𝑝(𝑥�, 𝑦�:�) and 𝛽�(𝑥�) = 𝑝(𝑦���:�|𝑥�) with recursions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' All we have to do is to replace 𝑝(𝑦���|𝑥���) with �(����|����) �(����) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Of course, 𝛼� �(𝑥�) ≠ 𝑝(𝑥�, 𝑦�:�) and 𝛽� �(𝑥�) ≠ 𝑝(𝑦���:�|𝑥�), but (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This is the core point: one uses a wrong model to find 𝑝(𝑥�|𝑦�:�) of the true HMC, that is the final goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 7 of 8 Remark 2 We see that we can compute MPM in HMM only using 𝑝(𝑥�), …, 𝑝(𝑥�) and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' This means that in supervised classification, where we have a learn sample, we can use any parametrization to estimate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' For example, we can model them with logistic regression, as currently do in CRFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' It of importance to note that such a parametrization is unusual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' however, important is that the model remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Discriminative computing of HMM based MAP: discriminative Viterbi Let 𝑋�, … , 𝑋�, 𝑌�, … , 𝑌� be a HMM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Bayesian MAP classifier (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='5) based on is computed with the following Viterbi algorithm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' For each 𝑛 = 1, … , 𝑁, and each 𝑥�, let 𝑥�:��� ��� (𝑥�) = (𝑥� ���, … , 𝑥��� ���)(𝑥�) be the path 𝑥� ���, …, 𝑥��� ��� verifying 𝑝(𝑥�:��� ��� (𝑥�), 𝑥�, 𝑦�:�) = 𝑠𝑢𝑝 ��:��� 𝑝(𝑥�:���, 𝑥�, 𝑦�:�);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='15) We see that 𝑥�:��� ��� (𝑥�) is a path maximizing 𝑝(𝑥�:���, 𝑥�|𝑦�:�) over all paths ending in 𝑥�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then having the paths 𝑥�:��� ��� (𝑥�) and the probabilities 𝑝(𝑥�:��� ��� (𝑥�), 𝑥�, 𝑦�:�) for each 𝑥� , one determines, for each 𝑥��� , the paths 𝑥�:� ���(𝑥���) and the probabilities 𝑝(𝑥�:� ���(𝑥���), 𝑥���, 𝑦�:���) = 𝑝(𝑥�:��� ��� (𝑥� ���), 𝑥� ���, 𝑥���, 𝑦�:���), searching 𝑥� ��� with 𝑝(𝑥�:��� ��� (𝑥� ���), 𝑥� ���, 𝑥���, 𝑦�:���)) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='16) 𝑠𝑢𝑝 �� [𝑝(𝑥�:��� ��� (𝑥�), 𝑥�, 𝑦�:�)𝑝(𝑥���|𝑥�)𝑝(𝑦���|𝑥���)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Setting in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='16) 𝑝(𝑦���|𝑥���) = �(����)�(����|����) �(����) , we see that 𝑥� ��� which verifies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='16) is the same that 𝑥� ��� which maximizes 𝑝(𝑥�:��� ��� (𝑥�), 𝑥�, 𝑦�:�)𝑝(𝑥���|𝑥�) �(����|����) �(����) , so that we can suppress 𝑝(𝑦���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In other words we can replace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='16) with 𝑝(𝑥�:��� ��� (𝑥� ���), 𝑥� ���, 𝑥���, 𝑦�:���)) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='17) 𝑠𝑢𝑝 �� [𝑝(𝑥�:��� ��� (𝑥�), 𝑥�, 𝑦�:�)𝑝(𝑥���|𝑥�) �(����|����) �(����) ] Finally, we propose the following discriminative version of the Viterbi algorithm: - set 𝑥� ��� = 𝑎𝑟𝑔𝑚𝑎𝑥 �� [𝑝(𝑥�|𝑦�)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' - for each 𝑛 = 1, …, 𝑁 − 1, and each 𝑥���, apply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='17) to find a path 𝑥�:� ���(𝑥���) from the paths 𝑥�:��� ��� (𝑥�) (for all 𝑥�), and the probabilities 𝑝(𝑥�:� ���(𝑥���), 𝑥���, 𝑦�:���) (for all 𝑥���);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' - end setting 𝑥�:� ��� = 𝑎𝑟𝑔𝑚𝑎𝑥 �� [𝑝(𝑥�:��� ��� (𝑥�), 𝑥�, 𝑦�:�)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Likely for MPM above, we see that we can find 𝑥�:� ��� with the only use of 𝑝(𝑥�), …, 𝑝(𝑥�), 𝑝(𝑥�|𝑥�), …, 𝑝(𝑥�|𝑥���), and 𝑝(𝑥�|𝑦�), …, 𝑝(𝑥�|𝑦�), exactly as in CRF case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' As above, it appears that dropping HMMs in some NLP tasks on the grounds that MAP is a “generative” classifier, is not justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' In particular, in supervised stationary framework distributions 𝑝(𝑥�), 𝑝(𝑥���|𝑥�), and 𝑝(𝑥�|𝑦�) can be estimated in the same way as in LC- CRFs case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Discussion We proposed two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We showed that the basic LC-CRF (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3) is equivalent to a classic HMM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) in that one can find a HMM whose posterior distribution is exactly the given LC-CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' More precisely, we specified the way of computing 𝑝(𝑥�) and 𝑝(𝑥���|𝑥�), 𝑝(𝑦�|𝑥�) defining (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='1) from 𝑉�(𝑥�, 𝑥���), 𝑈�(𝑥�, 𝑦�) defining (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Then, noticing that all Bayesian classifiers are discriminative in that they do not depend on the observations dis- tribution, we showed that HMMs based classifiers, usually considered as “generative” 8 of 8 ones, can also be considered as discriminative classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' More precisely, we proposed discriminative ways of computing the classic HMMs based Maximum Posterior Mode (MPM) and Maximum a Posteriori (MAP) classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' We considered basic LC-CRFs and HMMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' extending the proposed results to more sophisticated models like Pairwise Markov chains” (PMCs [23]) is an interesting perspec- tive for further investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Author Contributions: Conceptualization, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' methodology, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' validation, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' investigation, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' writing—original draft prepara- tion, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' writing—review and editing, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' supervision, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' project administra- tion, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Funding: This research was partly funded by the French Government Agency ASSOCIATION NA- TIONALE RECHERCHE TECHNOLOGIES (ANRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Stratonovich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Conditional Markov Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' Non-linear Transformations of Stochastic Processes, Pergamon, 1965, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 427-453.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} +page_content=' 634- 639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf'} diff --git a/MtE0T4oBgHgl3EQfjQEE/content/tmp_files/2301.02455v1.pdf.txt b/MtE0T4oBgHgl3EQfjQEE/content/tmp_files/2301.02455v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4d07f362537357b61beff7d4c62cff872549674 --- /dev/null +++ b/MtE0T4oBgHgl3EQfjQEE/content/tmp_files/2301.02455v1.pdf.txt @@ -0,0 +1,1363 @@ +KCL-PH-TH/2023-01 +NEC violation: Tunnelling versus the Casimir effect +Jean Alexandre1 and Drew Backhouse1 +1Theoretical Particle Physics and Cosmology, King’s College London, WC2R 2LS, UK +We show that tunnelling between two degenerate minima, as allowed in a finite volume, leads +to a non-extensive symmetric ground state. This results in Null Energy Condition violation for +sufficiently low temperatures, when a continuous set of momenta in the box containing the field is +assumed. Taking into account discrete momenta can modify this picture and is achieved via the +addition of the Casimir energy to the tunnelling-induced ground state energy. Focusing on zero- +temperature, these non-trivial effects are found to compete, depending on the typical length scales +involved. +I. +INTRODUCTION +Spontaneous Symmetry Breaking (SSB) is strictly +speaking valid for infinite volumes only, where tunnelling +between degenerate vacua is completely suppressed. On +the other hand, for a field confined in a box of finite vol- +ume, tunnelling between degenerate vacua is allowed and +we study here the energetic consequences. +Involving tunnelling in the quantisation of a system +automatically takes into account the different vacua and +is known to lead to a convex effective action [1]. This is +not the case in the situation of SSB, where the different +vacua are decoupled and quantisation over a single vac- +uum does not necessarily lead to convexity. Taking into +account several degenerate vacua in the partition func- +tion comes with a remarkable energetic feature, generated +dynamically: the effective action is non-extensive, as was +shown in [2, 3] with a semi-classical approximation for +the partition function. +The latter works were done in an O(4)-symmetric Eu- +clidean spacetime though, and to account for a full de- +scription of tunnelling one needs a finite spatial volume +V and an independent large Euclidean time β. The natu- +ral context for these studies is therefore equilibrium field +theory at a finite-temperature T = 1/β. The correspond- +ing Quantum Mechanics study was done in [4], involving +a gas of instantons/anti-instantons which dominates the +partition function in the limit of small temperature. It is +shown there that the Null Energy Condition (NEC - see +[5] for reviews) is violated, as a consequence of a non- +extensive effective action induced by tunnelling. +The +present article extends this study to full 4-dimensional +quantum fluctuations, and we find that NEC violation +occurs in any finite volume for sufficiently low tempera- +tures. +Our +study +does +not +however +deal +with +high- +temperature symmetry restoration, as seen in the Kibble- +Zureck mechanism [6]. +We are instead interested in +the low-temperature regime, where tunnelling dominates +over thermal fluctuations providing an opportunity to vi- +olate the NEC, which the Kibble-Zurek mechanism does +not. +We first evaluate quantum corrections with continuous +momentum for fluctuations above each saddle point, to +describe the fundamental dynamical mechanism induced +by tunnelling. We then take into account the modifica- +tion arising from discrete momentum in a finite volume, +using results known from studies of the Casimir effect +(see [7] for a review). The latter is known to be either +attractive or repulsive, depending on the geometry of the +box containing the field, as well as the boundary condi- +tions the field satisfies on the walls of the box. +As a +consequence, as far as NEC violation is concerned, the +difference between discrete and continuous momentum +can play an important role. +In Section II we describe the semi-classical approx- +imation in which the partition function is derived, to +take into account the different saddle points which are +relevant to tunnelling: +static saddle points and the +instanton/anti-instanton dilute gas. Details of the calcu- +lations with continuous momentum are given in Appen- +dices A and B. Section III focuses on the ground state of +the effective action, with a non-extensive energy density +providing the origin of NEC violation. The maximum +effect occurs at zero temperature and is the regime in +which we introduce corrections arising from discrete mo- +mentum in Section IV via the Casimir energy. We find +that tunnelling and the Casimir effect compete when the +typical size of the box containing the field is of the order +of the Compton wavelength of the corresponding parti- +cle. +For a larger box, the Casimir effect seems to be +dominant. +To summarise our results: +in the low-temperature +regime, the sum of energy density ρ and pressure p can +be written in the form +ρ + p ≃ Afinite-T + Btunnelling + CCasimir , +where +• the finite-temperature contribution A is always +positive (and vanishes exponentially for T → 0); +• the tunnelling contribution B, calculated with con- +tinuous momentum, is always negative (and van- +ishes exponentially for V → ∞); +• the discrete momentum correction C has a sign +which depends on the geometry and topology of +arXiv:2301.02455v1 [hep-th] 6 Jan 2023 + +2 +the finite box containing the field (and vanishes for +V → ∞). +As expected, the NEC is satisfied at zero temperature +and for infinite volume, where ρ + p = 0 for a homoge- +neous vacuum. +II. +SEMI-CLASSICAL APPROXIMATION +A. +Model +Consider a single real scalar field φ(t, x) in Euclidean +space, at finite-temperature T = 1/β and in a three- +dimensional spacial volume V , described by the Eu- +clidean action +� β +0 +dt +� +V +d3x +�1 +2(∂φ)2 + λ +24(φ2 − v2)2 + jφ +� +. +(1) +The finite volume is represented by a physical box con- +taining the scalar field, in which we assume continu- +ous momenta to calculate quantum corrections. Section +IV discusses corrections arising from discrete momen- +tum and the boundary conditions the field satisfies at +the walls of the box. Finite temperature requires field +configurations to have periodic boundary conditions in +Euclidean time and, as later discussed, has an impact on +the saddle point configurations which are allowed in the +partition function. +Introducing the dimensionless variables τ ≡ ωt and +ω ≡ v +� +λ +6 +, +ϕ ≡ +� +λ +6 +φ +ω +, +k ≡ +� +λ +6 +j +ω3 , +(2) +leads to the bare action +S[ϕ] = λv4 +12ω +� ωβ +0 +dτ +� +V +d3x +� +(ϕ′)2 + 1 +ω2 (∇ϕ)2 +(3) ++ 1 +2(ϕ2 − 1)2 + 2kϕ +� +, +where a prime represents a derivative with respect to the +dimensionless Euclidean time τ. As shown further in this +article, the effective action is convex and we thus focus +on the true vacuum, which occurs for vanishing source +j = 0 = k. As a consequence, no bubbles of true/false +vacuum can form as they would have an infinite radius +[8]. +We are therefore interested in time-dependent in- +stantons only, beyond the static and homogeneous saddle +points. The corresponding equation of motion is then +ϕ′′ − ϕ3 + ϕ − k = 0 , +(4) +where the solutions to this equation, ϕi(τ, x), are the +saddle points of the partition function to be introduced +below. +B. +Static saddle points +Introducing the critical dimensionless source +kc ≡ 2/(3 +√ +3) , +(5) +allows us to distinguish two cases. +For |k| > kc, there is only one static and homogeneous +(real) solution to the equation of motion (4), and quanti- +sation of the theory can therefore be based on one saddle +point only, leading to the usual 1PI effective potential. +For |k| < kc, the regime we focus on, there are two +such solutions +ϕL(k) = +2 +√ +3 cos +� +π/3 − (1/3) arccos(k/kc) +� +(6) +ϕR(k) = +2 +√ +3 cos +� +π − (1/3) arccos(k/kc) +� += −ϕL(−k) . +The actions for these configurations are +SL ≡ S[ϕL(k)] = Bωβ +� +4k − k2 + O(k3) +� +(7) +SR ≡ S[ϕR(k)] = S[ϕL(−k)] , +where +B ≡ λv4V +24ω +. +(8) +C. +Instanton/anti-instanton gas +In Euclidean time, and with the absence of a source, +the motion described by equation (4) corresponds to +the motion in real-time with the upside-down potential +V (ϕ) ≡ −(ϕ2 −1)/2, for which the minimum action Sinst +is obtained by the known solution +ϕinst(τ) = ± tanh +�τ − τ0 +√ +2 +� +, +(9) +where 0 ≤ τ0 ≤ ωβ, and +Sinst ≡ S[ϕinst] = 8 +√ +2 +3 B . +(10) +Because of finite temperature though, field configurations +should be periodic in Euclidean time, such that one needs +to consider an instanton/anti-instanton pair as the basic +building block. For two distant “jumps” at τ1 and τ2 such +that |τ1−τ2| ≫ 1, the configuration can be approximated +by [4] +ϕpair(τ) ≃ − tanh +�τ − τ1 +√ +2 +� +tanh +�τ − τ2 +√ +2 +� +, +(11) +with an action exponentially close to 2Sinst. +In the presence of a source, the basic building block is +in principle either a bounce or a shot (see [9] for reviews). +However, since we are interested in the limit of vanishing + +3 +source and periodic boundary conditions, the fundamen- +tal saddle point we consider behaves as the function (11). +Assuming the jumps occur over a short time in compar- +ison to β, the instanton/anti-instanton pair spends the +same time β/2 exponentially close to each static saddle +point, resulting in an action for such a pair ϕpair of +Spair ≃ 1 +2SL + 1 +2SR + 2Sinst . +(12) +Revisiting the analogy of classical mechanics in the +upside-down potential V (ϕ), the other possible sad- +dle points consist of periodic oscillations made of n +instanton/anti-instanton pairs, where the value of n de- +pends on how “exponentially close” the oscillations from +a static saddle point begins. An example of an exact sad- +dle point is given in Fig.1a. Assuming the total Euclidean +time β is large enough to leave the structure of pairs in- +tact, the time spent close to one static saddle point is the +same as the time spent close to the other and the total +action for n pairs is +Sn pairs ≃ 1 +2SL + 1 +2SR + 2nSinst . +(13) +The latter “crystalline” structure, with n periodic os- +cillations, corresponds to an exact solution of the equa- +tion of motion. +For large β, where the average dis- +tance between instantons and anti-instantons remains +large compared to their width, a translation of each jump +leaves the action Sinst invariant and the resulting highly +degenerate “gas” of instanton/anti-instanton pairs dom- +inates the partition function. An example of an approxi- +mate saddle point is given in Fig.1b. In this “dilute gas” +approximation, the n instanton/anti-instanton pair con- +figurations spend on average an equal time β/2 close to +each static saddle point, with the same total action (13) +as for an exact n-pair configuration as a result of the +translational invariance of jumps. +D. +Partition function +The partition function is evaluated in the semi-classical +approximation via a sum over the two static saddle +points, ϕL and ϕR, and the dilute gas of n instanton/anti- +instanton pairs for all possible values of n. +Together, +with the corresponding one-loop fluctuation factors FL,R +and Fn, the semi-classical approximation of the partition +function reads +Z[k] ≃ FL(β) exp(−SL) + FR(β) exp(−SR) +(14) ++ +∞ +� +n=1 +� 2n +� +i=1 +� ωRβ +τi−1 +τi +� +Fn exp(−Sn pairs) . +In the latter expression, the product of integrals over +the times τi where the jumps occur corresponds to the +zero-mode of the fluctuation factor for the saddle point +made of n instanton/anti-instanton pairs. +Indeed, the +10 +20 +30 +40 +50 +60 +70 +-1.0 +-0.5 +0.5 +1.0 +(a) An exact saddle point configuration with 3 +instanton/anti-instanton pairs and action S3 pairs: the +oscillations are periodic. +10 +20 +30 +40 +50 +60 +70 +-1.0 +-0.5 +0.5 +1.0 +(b) An approximate saddle point configuration with 3 +instanton/anti-instanton pairs: the jumps are randomly +distributed, but the average distance between them is larger +than their width, such that they keep their shape and the +action of the configuration is also S3 pairs. +FIG. 1: Examples of exact and approximate saddle +points. In the dilute gas approximation, the difference +between the corresponding actions is of order +Bωβ exp(−ωβ) ≪ 1, and the partition function is +dominated by the whole set of approximate saddle +points. +translational invariance of the action means that the ith +jump can happen at any time τi ∈ [τi−1, β]. +For fi- +nite temperature though, there is a maximum number of +instanton/anti-instanton pairs, however, the error made +in the summation for n → ∞ is negligible since each +term is suppressed by exp(−nSinst). With the fluctua- +tion factors derived in Appendix A and Appendix B, we +can write +Z[k] = exp +� +− ΣL(β) +� ++ exp +� +− ΣR(β) +� ++ exp +� +− Σgas(β) +� +, +(15) +where ΣL, ΣR and Σgas are the connected graphs gen- +erating functionals for the static saddle points and the +gas of instanton/anti-instanton pairs respectively. +We + +4 +note that an instanton or anti-instanton does not lead +to any imaginary part in the partition function, unlike a +bounce, since the former are monotonous functions of the +Euclidean time, such that the fluctuation operator does +not have negative eigenvalues [10]. +1. +Static saddle points +One-loop quantum corrections can be split into two +contributions: +the zero-temperature corrections, con- +taining all the divergences, and the divergence free +finite-temperature dependent corrections. +The zero- +temperature contribution is calculated in [3] and is ex- +pressed in terms of the renormalised parameters. +It +is mentioned here that, in the case of several saddle +points and in order to avoid confusion between loop +orders, renormalisation should be done at the level of +the individual connected graphs generating functionals +before performing the Legendre transform. The finite- +temperature contribution can be calculated using the +Schwinger proper time representation - see Appendix A +- and the overall contribution is +ΣL,R(β) +(16) += Brωrβ +� +(ϕ2 +L,R − 1)2 + 4kϕL,R ++ λr +96π2 (3ϕ2 +L,R − 1)2 ln +�3 +2ϕ2 +L,R − 1 +2 +� +− +λr(3ϕ2 +L,R − 1) +3π2 +∞ +� +l=1 +K2 +� +lωrβ +� +3ϕ2 +L,R − 1 +� +(lωrβ)2 +� +. +In the previous expression, the renormalised parameters +are +λr ≡ λ − 3λ2 +32π2 log +� Λ2 +λv2 +� +, +(17) +v2 +r ≡ v2 − 3Λ2 +16π2 + λv2 +16π2 log +� Λ2 +λv2 +� +, +Br ≡ λrv4 +rV +24ωr +, +ωr ≡ vr +� +λr +6 , +and K2(z) is a modified Bessel function of the second +kind with asymptotic behaviour +K2(z → ∞) ≃ e−z +� π +2z . +(18) +We note that l does not correspond to Matsubara modes. +Also, the temperature-independent part of the expression +(16) reproduces the zero-temperature result derived in +[3]. +2. +Gas of instanton/anti-instanton pairs +The evaluation of Σgas involves the fluctuation factor +above each jump and includes a summation over the al- +lowed jump positions in the interval τ ∈ [0, β] [10]. The +additional contribution of quantum fluctuations arises +from the “flat” parts of the instanton/anti-instanton con- +figurations, which are exponentially close to each static +saddle point for the approximate average time of β/2 +when neglecting the width of each jump compared to +β. +Performing the resummation over instantons/anti- +instantons, we show in Appendix B that the correspond- +ing connected graphs generating functional is then +Σgas(β) ≃ ΣL(β/2) + ΣR(β/2) +(19) +− ln +� +cosh( ¯N) − 1 +� +, +where +¯N ≡ ωrβ +� +6 +π Sinst e−Sinst , +(20) +corresponding to the average number of instanton/anti- +instanton pairs at temperature T = 1/β. In this article +we are interested in the limit ωrβ ≫ 1 for a fixed volume +- and thus fixed action Sinst - such that we consider the +situation where ¯N ≫ 1, corresponding to the full tun- +nelling regime. In the situation where β is fixed and V +becomes large we have ¯N ≪ 1, where tunnelling is sup- +pressed and the system is better approximated by SSB +[4]. +III. +NON-EXTENSIVE GROUND SATE +A. +One-particle-irreducible effective action +From the partition function evaluated for a constant +source j, the classical field is obtained as +φc ≡ − 1 +Z +δZ +δj +→ +− +1 +V βZ +∂Z +∂j , +(21) +which, in terms of the dimensionless quantities previously +introduced, can be written as +ϕc = − +1 +4BrωrβZ +∂Z[k] +∂k +. +(22) +From the expression (15) for the partition function, to- +gether with the expressions (16) and (19), the classical +field is expanded in powers of the source k +ϕc = +� +−f0 + +λr +128π2 f1 +� +k + O(k3) , +(23) +where +f0 ≡ 1 + 16Brωrβ + cosh( ¯N) +2 +� +1 + cosh( ¯N) +� +(24) +f1 ≡ 7 + 32Brωrβ + 7 cosh( ¯N) +1 + cosh( ¯N) +. + +5 +Consistently with the symmetry of the bare potential, the +classical field φc is an odd function of k: the even powers +of k cancel out in the expression for ϕc after adding the +contribution of the different saddle points, leading to the +mapping k = 0 ⇔ ϕc = 0. +We then perform the Legendre transform, after ex- +pressing the source as a function of the classical field +k(ϕc) = −1 +2 +� +g0 + +λ +16π2 g1 +� +ϕc + O(ϕ3 +c) , +(25) +where +g0 ≡ +4 +� +1 + cosh( ¯N) +� +1 + 16Brωrβ + cosh( ¯N) +(26) +g1 ≡ +� +1 + cosh( ¯N) +�� +7 + 32Brωrβ + 7 cosh( ¯N) +� +� +1 + 16Brωrβ + cosh( ¯N) +�2 +. +The effective action for a constant configuration is finally +Γ(ϕc) = − ln Z +� +k(ϕc) +� +− 4Brωrβ +� +k(ϕc) dϕc +(27) += Γ(0) + Brωrβ +� +g0 + +λ +16π2 g1 +� +ϕ2 +c + O(ϕ4 +c) , +where +Γ(0) = − ln Z(0) +(28) += − ln +� +2e−Σ0(β) + e−2Σ0(β/2)� +cosh( ¯N) − 1 +�� +, +and Σ0 ≡ ΣL|k=0 = ΣR|k=0. The effective potential Ueff +is finally given by +Γ(φc) = V βUeff(φc) , +(29) +and, as expected, it satisfies the following properties: +• it is a convex function of φc, since the mass term is +positive; +• the ground state is at ϕc = 0, or equivalently k = 0; +• it has a non-trivial volume-dependence and is thus +non-extensive. +For the following studies of NEC violation we focus on +the ground state ϕc = 0. +B. +NEC violation +The ground state density ρ and pressure p are obtained +from the free energy +F = 1 +β Γ(0) = − 1 +β ln Z(0) , +(30) +and their sum can be written as [4] +ρ + p = 1 +V +� +F − T ∂F +∂T +� +− ∂F +∂V += −T ∂Ueff(0) +∂T +− V ∂Ueff(0) +∂V +. +(31) +From the expression (28), we obtain for ωrβ ≫ 1 +ρ + p ≃ +4ω5/2 +R +( +√ +2πβ)3/2 e−ωRβ/ +√ +2 +(32) +−ωR +V +� +Sinst + 1 +2 +� � +6 +π Sinst e−Sinst . +On the right-hand side, the first term corresponds to +thermal fluctuations and the second term corresponds to +tunnelling. These terms compete for the overall sign of +ρ + p leading to the following cases: +• Infinite volume: ρ + p ≥ 0 +In the limit of infinite volume tunnelling is sup- +pressed, as seen via the vanishing of the average +number (20) of instanton/anti-instanton pairs for +any fixed temperature: limV →∞ ¯N = 0 for fixed β. +Hence only thermal fluctuations contribute and +ρ + p = +4ω5/2 +R +( +√ +2πβ)3/2 e−ωRβ/ +√ +2 , +(33) +with ρ + p → 0 as the temperature goes to 0 or +equivalently β → ∞. This result is not surprising: +the limit of infinite volume corresponds to SSB and, +as expected, the NEC is satisfied; +• Finite volume and zero temperature: ρ + p < 0 +In this situation, only the tunnelling term con- +tributes and +ρ + p = −ωR +V +� +Sinst + 1 +2 +� � +6 +π Sinst e−Sinst . +(34) +The NEC is violated as a consequence of the ex- +plicit volume-dependence of the effective potential; +• Boundary ρ + p = 0 +We sketch in Fig.2 the boundary V (T) between the +region where the NEC is satisfied and the region +where the NEC is violated. +Finally, we note that NEC violation is suppressed ex- +ponentially with the volume, unlike the power law sup- +pression which is found with O(4)-symmetric Euclidean +spacetime coordinates [3, 19]. +IV. +DISCRETE MOMENTUM CORRECTIONS +We focus here on the ground state obtained for k = 0 +in the case of zero temperature, where NEC violation +arising from tunnelling is maximum. + +6 +FIG. 2: The boundary between the regions where the +NEC is satisfied and where it is violated due to the +competition of tunnelling and thermal fluctuations. The +plot shows the curve V (T) in terms of the dimensionless +variables used in this article. +The previous sections ignore quantisation of momen- +tum when calculating the connected graphs generating +functional for each static saddle point in a finite volume. +As we explain below, the evaluation of ΣL,R with discrete +momentum consists of taking into account the relevant +Casimir energy. There is no such contribution from the +jumps in the instantons/anti-instantons since the corre- +sponding one-loop corrections do not depend on momen- +tum. +A. +Vacuum energy +The Casimir contribution to the connected graphs gen- +erating functional is defined as +ΣCas ≡ ΣL,R|discrete − ΣL,R|continuum , +(35) +where the ultraviolet divergences cancel out since they +are identical in the discrete and continuum cases. For +zero temperature and vanishing source, the expression +(16) gives +ΣL,R(k = 0, T = 0)|continuum = lim +β→∞ Σ0(β) = 0 , +(36) +such that, instead of eq.(28), one-loop corrections ob- +tained with discrete momentum lead to +Γ(0) = − ln +� +2e−ΣCas + e−ΣCas� +cosh( ¯N) − 1 +�� += ΣCas − ln +� +cosh( ¯N) + 1 +� +. +(37) +The above expression takes advantage of the proportinal- +ity between Σ0 and β in the limit of vanishing tempera- +ture, such that +2Σ0(β/2) → Σ0(β), +(38) +as β → ∞. +In the situation of one saddle point, and +therefore no tunnelling, Γ(0) = ΣCas = βECas where +ECas is the Casimir energy corresponding to quantum +fluctuations about a single vacua ±v (where one has +approximately quadratic fluctuations with mass m = +√ +2ωr). Hence +Ueff(0) = ECas +V +− +1 +V β ln +� +cosh( ¯N) + 1 +� +, +(39) +and we see the additive nature of the Casimir effect +and tunnelling contributions, similarly to the finite- +temperature contribution. The sum of density and pres- +sure reads finally +ρ + p = ECas +V +− ∂ECas +∂V +(40) +−ωR +V +� +Sinst + 1 +2 +� � +6Sinst +π +e−Sinst . +B. +Casimir contribution to the NEC +The Casimir energy is highly sensitive to the geometry +of the box containing the field, as well as the boundary +conditions used on the corresponding surfaces [7]. For a +scalar field ϕ(t, x) in the interval x ∈ [0, L] for example, +the possible choices of boundary conditions are defined +as follows +Dirichlet:ϕ(t, 0) = ϕ(t, L) = 0 +(41) +Neumann:∂xϕ(t, 0) = ∂xϕ(t, L) = 0 +Periodic:ϕ(t, 0) = ϕ(t, L) . +(42) +For the cases we consider, the asymptotic form of +the Casimir effect is identical for both Dirichlet and +Neumann boundary conditions. We thus consider mixed +boundary conditions, where different subsets of the +boundary can possess either Dirichlet or Neumann +conditions. For the case of mixed boundary conditions, +the Casimir energy is dependent on the size/curvature +of the material boundaries, and for the case of periodic +boundary conditions, it is dependent on the period +length/curvature of the non-trivial spacetime. A ‘gen- +eral rule’ states that flat geometries lead to exponential +suppression of the Casimir energy for mL ≫ 1, where L +is the length scale of the relevant boundaries, and that +curved geometries lead to power-law suppression of the +Casimir energy for mR ≫ 1, where R is the radius of +curvature of the relevant surfaces. There are exceptions +to this general rule though, which are highlighted in the +following examples. +• Dirichlet boundary conditions, flat boundaries +The original Casimir configuration consists of a scalar +field constrained between two parallel, flat mirrors with +surface area A and separation a, with the scalar field +satisfying Dirichlet conditions on the boundaries. The + +5 +4 +3 +2 +p+p>0 +Od+d +1 +1 +2 +3 +4 +5 +9 +Br7 +corresponding Casimir energy is [7] +ECas ≃ +� +− +Aπ2 +1440a3 +for +am ≪ 1 +− A +8 +√ +2 +� m +πa +�3/2 e−2ma +for +am ≫ 1 +(43) +and is always negative. +• Dirichlet boundary conditions, curved boundaries +For dimensional reasons, the Casimir energy for a scalar +field confined within the curved boundary of a 2-sphere +of radius R with Dirichlet boundary conditions is given +in terms of the dimensionless function +ECas = 1 +Rf(mR) , +(44) +and is found to obey power law suppression in mR, for +mR ≫ 1 [12]. +• Periodic boundary conditions, flat spacetime +For a scalar field confined to the surface of a 3-torus (a +rectangular box with periodic boundary conditions), the +sign of the Casimir energy depends on the ratio of the +lengths of the box and we have [13] +ECas ≃ −(mL)3/2 +L +exp(−mL) +for +mL ≫ 1 , +(45) +where L is the typical size of the period length. +• Periodic boundary conditions, curved spacetime +For a scalar field confined to the surface of a 3-sphere +with radius R, we would expect the asymptotic form to +be a power law in R. However, this special case is an +exception to the general rule as a consequence of the +accidental vanishing of the heat-kernel coefficients (see +Sec. 3 of [7] for details). The resulting Casimir energy +has instead an exponential asymptotic form, as in the +case of flat geometries [14] +ECas ≃ +(mR)5/2 +R +exp(−2πmR) +for +mR ≫ 1 . (46) +The above examples display how the Casimir effect for +a massive scalar field is at most suppressed by the expo- +nential e−mL, where L is a typical size of the boundary +containing the field. On the other hand, the tunnelling +contribution to the NEC, calculated with continuous mo- +mentum, is proportional to +e−Sinst ∼ exp +� +−(mL)3 +λ +� +, +(47) +and is therefore negligible compared to the Casimir con- +tribution in the regime mL ≫ +√ +λ. +For mL ∼ +√ +λ +though, tunnelling competes with the Casimir effect and +can change the sign of ρ + p in the situation where the +Casimir energy is positive. As an example, we sketch in +FIG. 3: The boundary between the regions where the +NEC is satisfied and where it is violated, due to the +competition of the Casimir energy and tunnelling at +zero temperature on a 3-sphere. The plot shows the +curve R(λ) in terms of the dimensionless variables used +in this article. +Fig.3 the boundary R(λ) between the region where the +NEC is satisfied and the region where it is violated, due +to the competition between tunnelling and the Casimir +effect on a 3-sphere. +We note however two important points regarding the +Casimir examples cited here: (i) they are valid for ideal +surfaces only, and a realistic confining mechanism for the +scalar field would lead to a modification of the Casimir +vacuum energies, especially if the field is confined by an +external potential instead of a physical box [15]; (ii) they +assume free scalar fields and ignore its self-interactions. +On the other hand, the tunnelling mechanism described +here: (i) necessitates the field to be self-interacting; (ii) +is not sensitive to the geometry/topology of the box con- +taining the field. Hence the conclusions regarding which +effect dominates could be modified by a more thorough +study, depending on the situation which is considered. +Finally, the Average Null Energy Condition is not vio- +lated by the present mechanism. Indeed, if we take into +account the energy necessary to maintain the confining +mechanism the overall ground state of the system does +not violate the NEC [16], consistently with what is ex- +pected from causality [17]. +V. +CONCLUSIONS +Tunnelling between degenerate vacua is exponentially +suppressed with the volume of the box containing the +field, but nevertheless allows the possibility of NEC vio- +lation at low temperatures. Taking into account discrete +momentum of fluctuations in a finite volume implies this +effect is mainly relevant for situations where the typical +size of the box is not too large compared to the Compton +wave length of the particle, and where tunnelling can lead +to an overall NEC violation. A potential application lies + +1 +0.8 +Ovd+d +0.6 +V +p+p>0 +0.4 +0.2 +0.5 +1 +1.5 +2 +WrR8 +in axion physics, where the de Broglie wavelength can be +of order 1 kpc [18] with the confinement provided by a +gravitational well. +Exponential suppression in the volume could poten- +tially be avoided by a consideration of non-degenerate +vacua, +where +other +saddle points +with +a +volume- +independent action become relevant, as in the original +study of false vacuum decay [8]. The resulting effective +action would be non-extensive in a certain regime of the +classical field, but more studies need to be done for the +status of NEC violation in the corresponding vacuum. +Finally, NEC violation could play an important role in +Early Universe Cosmology, where tunnelling could pro- +vide a dynamical mechanism for a cosmological bounce, +as explained in [19]: +as the Universe contracts, tun- +nelling switches on and violates the NEC, which induces +a bounce after which tunnelling is suppressed as the +Universe expands. This scenario necessitates the study +of tunnelling in a Friedman-Lemaitre-Robertson-Walker +background though, and is left for future work. +ACKNOWLEDGEMENTS +The authors would like to thank Klaus Kirsten for valu- +able correspondence regarding the Casimir effect, and +JA would like to thank Janos Polonyi for enlightening +discussions. This work is supported by the Leverhulme +Trust (grant RPG-2021-299) and the Science and Tech- +nology Facilities Council (grant STFC-ST/T000759/1). +For the purpose of Open Access, the authors have ap- +plied a CC BY public copyright licence to any Author +Accepted Manuscript version arising from this submis- +sion. +Appendix A: Fluctuation factor for a static saddle +point +The fluctuation factors for the static saddle points are +calculated with continuous 3-dimensional momenta, in- +troducing the cut-off Λ in the Schwinger proper time +representation of the propagator. Introducing the dimen- +sionless Matsubara frequency νn ≡ 2πn/ωβ, we have +Tr +� +ln +� +δ2S[ϕi] +�� +(A1) += V +� +d3p +(2π)3 +∞ +� +n=−∞ +� ∞ +1/Λ2 +ds +s e−4Bωβs(p2/ω2+ν2 +n+3ϕi−1) += V ω3 +2π2 +∞ +� +n=−∞ +� ∞ +1/X2 +dx +x +� ∞ +0 +dq q2e−x(q2+ν2 +n+3ϕi−1) += V ω3 +8π3/2 +∞ +� +n=−∞ +� ∞ +1/X2 +dx +x5/2 e−x(ν2 +n+3ϕi−1) += V ω3 +8π3/2 +� ∞ +1/X2 +dx +x5/2 e−x(3ϕi−1)ϑ0 +� 4πx +ω2β2 +� +, +where the dimensionless variables are +q ≡ p +ω +, +x ≡ 4Bωβs +, +X2 ≡ +Λ2 +4Bωβ , +(A2) +and ϑ0(y) is the Jacobi function +ϑ0(y) ≡ +∞ +� +n=−∞ +e−πyn2 . +(A3) +Making use of the following property +ϑ0(y) = y−1/2ϑ0(1/y) , +(A4) +the above becomes +Tr +� +ln +� +δ2S[ϕi] +�� +(A5) += V ω4β +16π2 +� ∞ +1/X2 +dx +x3 e−x(3ϕi−1)ϑ0 +� ωβ +4πx +� += V ω4β +16π2 +� ∞ +1/X2 +dx +x3 e−x(3ϕi−1) +∞ +� +n=−∞ +e−ω2β2n2/4x += λBωβ +24π2 +� +IΛ(ϕi) + IT (ϕi) +� +, +where +IΛ(ϕi) ≡ +� ∞ +1/Λ2 +dx +x3 e−x(3ϕi−1) +(A6) +IT (ϕi) ≡ 2 +∞ +� +n=1 +� ∞ +0 +dx +x3 e−x(3ϕi−1)−ω2β2n2/4x . +The first integral IΛ is the temperature-independent di- +vergent integral which, after renormalisation, produces +the same results as in the zero-temperature case [3]. The +second integral IT is the temperature-dependent contri- +bution corresponding to the finite-temperature correc- +tions. It is finite, which is why the cut-off is taken to in- +finity in this specific term. This temperature-dependent +integral can be written in terms of the modified Bessel +functions of the second kind K2(z) as +IT (φi) = +∞ +� +n=1 +16(3ϕi − 1) +(nωβ)2 +K2(nωβ +� +3ϕi − 1) . +(A7) +Together with the integral IΛ, the connected graphs gen- +erating functional for homogeneous saddle points is given +by eq.(16). +Appendix B: Fluctuation factor for the +instantons/anti-instantons gas +We calculate here the contribution exp(−Σgas) to the +partition function (15), following the known approach in +studies of tunnelling effects [10]. +The invariance of the action for n instanton/anti- +instanton pairs under the translation of the jumps leads +to the degeneracy factor in the partition function +� 2n +� +i=1 +� ωβ +τi−1 +τi +� += (ωβ)2n +(2n)! +, +(B1) + +9 +where τi ∈ [τi−1, ωβ] and τ0 = 0, since successive in- +stanton jumps can only occur after previous ones. Each +jump has an associated fluctuation factor +� +6Sinst/π and +thus the total fluctuation factor is given by the product +of the contributions of the “flat” parts of the n-pairs of +instanton/anti-instantons and the n pairs of jumps. On +average, each configuration of n instanton/anti-instanton +pairs spends the same time ≃ β/2 close to each static +saddle point, such that the expression for Fn is finally +Fn = FL(β/2)FR(β/2) +�6Sint +π +�n +. +(B2) +Substituting the above results into the partition function +(14), along with the total action (13) for n pairs, yields +the total contribution to the partition function due to +instanton/anti-instanton pairs +exp(−Σgas) +(B3) += e−ΣL[β/2]e−ΣR[β/2] +∞ +� +n=1 +(ωβ)2n +(2n)! +�6Sint +π +�n +e−2nSint += exp +� +− ΣL[β/2] − ΣR[β/2] +� +× +� +cosh +� +ωβ +� +6Sint +π +e−Sinst +� +− 1 +� +, +This leads to the expression (19), where the parameters +can be replaced by their renormalised version, since the +overall expression is already at one-loop. +[1] K. Symanzik, Commun. Math. Phys. 16, 48 (1970); +S. R. Coleman, R. Jackiw and H. D. Politzer, Phys. Rev. +D 10, 2491 (1974); +J. Iliopoulos, C. Itzykson and A. Martin, Rev. Mod. +Phys. 47, 165 (1975); +R. W. Haymaker and J. 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D 103 (2021) +no.10, 105020 [arXiv:2101.08640 [hep-th]]. + diff --git a/MtE0T4oBgHgl3EQfjQEE/content/tmp_files/load_file.txt b/MtE0T4oBgHgl3EQfjQEE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6574683e757a2867bcb54ff5bb6ed19aad196859 --- /dev/null +++ b/MtE0T4oBgHgl3EQfjQEE/content/tmp_files/load_file.txt @@ -0,0 +1,413 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf,len=412 +page_content='KCL-PH-TH/2023-01 NEC violation: Tunnelling versus the Casimir effect Jean Alexandre1 and Drew Backhouse1 1Theoretical Particle Physics and Cosmology, King’s College London, WC2R 2LS, UK We show that tunnelling between two degenerate minima, as allowed in a finite volume, leads to a non-extensive symmetric ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' This results in Null Energy Condition violation for sufficiently low temperatures, when a continuous set of momenta in the box containing the field is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Taking into account discrete momenta can modify this picture and is achieved via the addition of the Casimir energy to the tunnelling-induced ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Focusing on zero- temperature, these non-trivial effects are found to compete, depending on the typical length scales involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' INTRODUCTION Spontaneous Symmetry Breaking (SSB) is strictly speaking valid for infinite volumes only, where tunnelling between degenerate vacua is completely suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' On the other hand, for a field confined in a box of finite vol- ume, tunnelling between degenerate vacua is allowed and we study here the energetic consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Involving tunnelling in the quantisation of a system automatically takes into account the different vacua and is known to lead to a convex effective action [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' This is not the case in the situation of SSB, where the different vacua are decoupled and quantisation over a single vac- uum does not necessarily lead to convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Taking into account several degenerate vacua in the partition func- tion comes with a remarkable energetic feature, generated dynamically: the effective action is non-extensive, as was shown in [2, 3] with a semi-classical approximation for the partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The latter works were done in an O(4)-symmetric Eu- clidean spacetime though, and to account for a full de- scription of tunnelling one needs a finite spatial volume V and an independent large Euclidean time β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The natu- ral context for these studies is therefore equilibrium field theory at a finite-temperature T = 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The correspond- ing Quantum Mechanics study was done in [4], involving a gas of instantons/anti-instantons which dominates the partition function in the limit of small temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' It is shown there that the Null Energy Condition (NEC - see [5] for reviews) is violated, as a consequence of a non- extensive effective action induced by tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The present article extends this study to full 4-dimensional quantum fluctuations, and we find that NEC violation occurs in any finite volume for sufficiently low tempera- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Our study does not however deal with high- temperature symmetry restoration, as seen in the Kibble- Zureck mechanism [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We are instead interested in the low-temperature regime, where tunnelling dominates over thermal fluctuations providing an opportunity to vi- olate the NEC, which the Kibble-Zurek mechanism does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We first evaluate quantum corrections with continuous momentum for fluctuations above each saddle point, to describe the fundamental dynamical mechanism induced by tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We then take into account the modifica- tion arising from discrete momentum in a finite volume, using results known from studies of the Casimir effect (see [7] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The latter is known to be either attractive or repulsive, depending on the geometry of the box containing the field, as well as the boundary condi- tions the field satisfies on the walls of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' As a consequence, as far as NEC violation is concerned, the difference between discrete and continuous momentum can play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In Section II we describe the semi-classical approx- imation in which the partition function is derived, to take into account the different saddle points which are relevant to tunnelling: static saddle points and the instanton/anti-instanton dilute gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Details of the calcu- lations with continuous momentum are given in Appen- dices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Section III focuses on the ground state of the effective action, with a non-extensive energy density providing the origin of NEC violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The maximum effect occurs at zero temperature and is the regime in which we introduce corrections arising from discrete mo- mentum in Section IV via the Casimir energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We find that tunnelling and the Casimir effect compete when the typical size of the box containing the field is of the order of the Compton wavelength of the corresponding parti- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For a larger box, the Casimir effect seems to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' To summarise our results: in the low-temperature regime, the sum of energy density ρ and pressure p can be written in the form ρ + p ≃ Afinite-T + Btunnelling + CCasimir , where the finite-temperature contribution A is always positive (and vanishes exponentially for T → 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' the tunnelling contribution B, calculated with con- tinuous momentum, is always negative (and van- ishes exponentially for V → ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' the discrete momentum correction C has a sign which depends on the geometry and topology of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='02455v1 [hep-th] 6 Jan 2023 2 the finite box containing the field (and vanishes for V → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' As expected, the NEC is satisfied at zero temperature and for infinite volume, where ρ + p = 0 for a homoge- neous vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' SEMI-CLASSICAL APPROXIMATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Model Consider a single real scalar field φ(t, x) in Euclidean space, at finite-temperature T = 1/β and in a three- dimensional spacial volume V , described by the Eu- clidean action � β 0 dt � V d3x �1 2(∂φ)2 + λ 24(φ2 − v2)2 + jφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (1) The finite volume is represented by a physical box con- taining the scalar field, in which we assume continu- ous momenta to calculate quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Section IV discusses corrections arising from discrete momen- tum and the boundary conditions the field satisfies at the walls of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Finite temperature requires field configurations to have periodic boundary conditions in Euclidean time and, as later discussed, has an impact on the saddle point configurations which are allowed in the partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Introducing the dimensionless variables τ ≡ ωt and ω ≡ v � λ 6 , ϕ ≡ � λ 6 φ ω , k ≡ � λ 6 j ω3 , (2) leads to the bare action S[ϕ] = λv4 12ω � ωβ 0 dτ � V d3x � (ϕ′)2 + 1 ω2 (∇ϕ)2 (3) + 1 2(ϕ2 − 1)2 + 2kϕ � , where a prime represents a derivative with respect to the dimensionless Euclidean time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' As shown further in this article, the effective action is convex and we thus focus on the true vacuum, which occurs for vanishing source j = 0 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' As a consequence, no bubbles of true/false vacuum can form as they would have an infinite radius [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We are therefore interested in time-dependent in- stantons only, beyond the static and homogeneous saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The corresponding equation of motion is then ϕ′′ − ϕ3 + ϕ − k = 0 , (4) where the solutions to this equation, ϕi(τ, x), are the saddle points of the partition function to be introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Static saddle points Introducing the critical dimensionless source kc ≡ 2/(3 √ 3) , (5) allows us to distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For |k| > kc, there is only one static and homogeneous (real) solution to the equation of motion (4), and quanti- sation of the theory can therefore be based on one saddle point only, leading to the usual 1PI effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For |k| < kc, the regime we focus on, there are two such solutions ϕL(k) = 2 √ 3 cos � π/3 − (1/3) arccos(k/kc) � (6) ϕR(k) = 2 √ 3 cos � π − (1/3) arccos(k/kc) � = −ϕL(−k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The actions for these configurations are SL ≡ S[ϕL(k)] = Bωβ � 4k − k2 + O(k3) � (7) SR ≡ S[ϕR(k)] = S[ϕL(−k)] , where B ≡ λv4V 24ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (8) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Instanton/anti-instanton gas In Euclidean time, and with the absence of a source, the motion described by equation (4) corresponds to the motion in real-time with the upside-down potential V (ϕ) ≡ −(ϕ2 −1)/2, for which the minimum action Sinst is obtained by the known solution ϕinst(τ) = ± tanh �τ − τ0 √ 2 � , (9) where 0 ≤ τ0 ≤ ωβ, and Sinst ≡ S[ϕinst] = 8 √ 2 3 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (10) Because of finite temperature though, field configurations should be periodic in Euclidean time, such that one needs to consider an instanton/anti-instanton pair as the basic building block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For two distant “jumps” at τ1 and τ2 such that |τ1−τ2| ≫ 1, the configuration can be approximated by [4] ϕpair(τ) ≃ − tanh �τ − τ1 √ 2 � tanh �τ − τ2 √ 2 � , (11) with an action exponentially close to 2Sinst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In the presence of a source, the basic building block is in principle either a bounce or a shot (see [9] for reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' However, since we are interested in the limit of vanishing 3 source and periodic boundary conditions, the fundamen- tal saddle point we consider behaves as the function (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Assuming the jumps occur over a short time in compar- ison to β, the instanton/anti-instanton pair spends the same time β/2 exponentially close to each static saddle point, resulting in an action for such a pair ϕpair of Spair ≃ 1 2SL + 1 2SR + 2Sinst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (12) Revisiting the analogy of classical mechanics in the upside-down potential V (ϕ), the other possible sad- dle points consist of periodic oscillations made of n instanton/anti-instanton pairs, where the value of n de- pends on how “exponentially close” the oscillations from a static saddle point begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' An example of an exact sad- dle point is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Assuming the total Euclidean time β is large enough to leave the structure of pairs in- tact, the time spent close to one static saddle point is the same as the time spent close to the other and the total action for n pairs is Sn pairs ≃ 1 2SL + 1 2SR + 2nSinst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (13) The latter “crystalline” structure, with n periodic os- cillations, corresponds to an exact solution of the equa- tion of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For large β, where the average dis- tance between instantons and anti-instantons remains large compared to their width, a translation of each jump leaves the action Sinst invariant and the resulting highly degenerate “gas” of instanton/anti-instanton pairs dom- inates the partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' An example of an approxi- mate saddle point is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In this “dilute gas” approximation, the n instanton/anti-instanton pair con- figurations spend on average an equal time β/2 close to each static saddle point, with the same total action (13) as for an exact n-pair configuration as a result of the translational invariance of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Partition function The partition function is evaluated in the semi-classical approximation via a sum over the two static saddle points, ϕL and ϕR, and the dilute gas of n instanton/anti- instanton pairs for all possible values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Together, with the corresponding one-loop fluctuation factors FL,R and Fn, the semi-classical approximation of the partition function reads Z[k] ≃ FL(β) exp(−SL) + FR(β) exp(−SR) (14) + ∞ � n=1 � 2n � i=1 � ωRβ τi−1 τi � Fn exp(−Sn pairs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In the latter expression, the product of integrals over the times τi where the jumps occur corresponds to the zero-mode of the fluctuation factor for the saddle point made of n instanton/anti-instanton pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Indeed, the 10 20 30 40 50 60 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='0 (a) An exact saddle point configuration with 3 instanton/anti-instanton pairs and action S3 pairs: the oscillations are periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 10 20 30 40 50 60 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='0 (b) An approximate saddle point configuration with 3 instanton/anti-instanton pairs: the jumps are randomly distributed, but the average distance between them is larger than their width, such that they keep their shape and the action of the configuration is also S3 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 1: Examples of exact and approximate saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In the dilute gas approximation, the difference between the corresponding actions is of order Bωβ exp(−ωβ) ≪ 1, and the partition function is dominated by the whole set of approximate saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' translational invariance of the action means that the ith jump can happen at any time τi ∈ [τi−1, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For fi- nite temperature though, there is a maximum number of instanton/anti-instanton pairs, however, the error made in the summation for n → ∞ is negligible since each term is suppressed by exp(−nSinst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' With the fluctua- tion factors derived in Appendix A and Appendix B, we can write Z[k] = exp � − ΣL(β) � + exp � − ΣR(β) � + exp � − Σgas(β) � , (15) where ΣL, ΣR and Σgas are the connected graphs gen- erating functionals for the static saddle points and the gas of instanton/anti-instanton pairs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We 4 note that an instanton or anti-instanton does not lead to any imaginary part in the partition function, unlike a bounce, since the former are monotonous functions of the Euclidean time, such that the fluctuation operator does not have negative eigenvalues [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Static saddle points One-loop quantum corrections can be split into two contributions: the zero-temperature corrections, con- taining all the divergences, and the divergence free finite-temperature dependent corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The zero- temperature contribution is calculated in [3] and is ex- pressed in terms of the renormalised parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' It is mentioned here that, in the case of several saddle points and in order to avoid confusion between loop orders, renormalisation should be done at the level of the individual connected graphs generating functionals before performing the Legendre transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The finite- temperature contribution can be calculated using the Schwinger proper time representation - see Appendix A and the overall contribution is ΣL,R(β) (16) = Brωrβ � (ϕ2 L,R − 1)2 + 4kϕL,R + λr 96π2 (3ϕ2 L,R − 1)2 ln �3 2ϕ2 L,R − 1 2 � − λr(3ϕ2 L,R − 1) 3π2 ∞ � l=1 K2 � lωrβ � 3ϕ2 L,R − 1 � (lωrβ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In the previous expression, the renormalised parameters are λr ≡ λ − 3λ2 32π2 log � Λ2 λv2 � , (17) v2 r ≡ v2 − 3Λ2 16π2 + λv2 16π2 log � Λ2 λv2 � , Br ≡ λrv4 rV 24ωr , ωr ≡ vr � λr 6 , and K2(z) is a modified Bessel function of the second kind with asymptotic behaviour K2(z → ∞) ≃ e−z � π 2z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (18) We note that l does not correspond to Matsubara modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Also, the temperature-independent part of the expression (16) reproduces the zero-temperature result derived in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Gas of instanton/anti-instanton pairs The evaluation of Σgas involves the fluctuation factor above each jump and includes a summation over the al- lowed jump positions in the interval τ ∈ [0, β] [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The additional contribution of quantum fluctuations arises from the “flat” parts of the instanton/anti-instanton con- figurations, which are exponentially close to each static saddle point for the approximate average time of β/2 when neglecting the width of each jump compared to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Performing the resummation over instantons/anti- instantons, we show in Appendix B that the correspond- ing connected graphs generating functional is then Σgas(β) ≃ ΣL(β/2) + ΣR(β/2) (19) − ln � cosh( ¯N) − 1 � , where ¯N ≡ ωrβ � 6 π Sinst e−Sinst , (20) corresponding to the average number of instanton/anti- instanton pairs at temperature T = 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In this article we are interested in the limit ωrβ ≫ 1 for a fixed volume and thus fixed action Sinst - such that we consider the situation where ¯N ≫ 1, corresponding to the full tun- nelling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In the situation where β is fixed and V becomes large we have ¯N ≪ 1, where tunnelling is sup- pressed and the system is better approximated by SSB [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' NON-EXTENSIVE GROUND SATE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' One-particle-irreducible effective action From the partition function evaluated for a constant source j, the classical field is obtained as φc ≡ − 1 Z δZ δj → − 1 V βZ ∂Z ∂j , (21) which, in terms of the dimensionless quantities previously introduced, can be written as ϕc = − 1 4BrωrβZ ∂Z[k] ∂k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (22) From the expression (15) for the partition function, to- gether with the expressions (16) and (19), the classical field is expanded in powers of the source k ϕc = � −f0 + λr 128π2 f1 � k + O(k3) , (23) where f0 ≡ 1 + 16Brωrβ + cosh( ¯N) 2 � 1 + cosh( ¯N) � (24) f1 ≡ 7 + 32Brωrβ + 7 cosh( ¯N) 1 + cosh( ¯N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 5 Consistently with the symmetry of the bare potential, the classical field φc is an odd function of k: the even powers of k cancel out in the expression for ϕc after adding the contribution of the different saddle points, leading to the mapping k = 0 ⇔ ϕc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We then perform the Legendre transform, after ex- pressing the source as a function of the classical field k(ϕc) = −1 2 � g0 + λ 16π2 g1 � ϕc + O(ϕ3 c) , (25) where g0 ≡ 4 � 1 + cosh( ¯N) � 1 + 16Brωrβ + cosh( ¯N) (26) g1 ≡ � 1 + cosh( ¯N) �� 7 + 32Brωrβ + 7 cosh( ¯N) � � 1 + 16Brωrβ + cosh( ¯N) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The effective action for a constant configuration is finally Γ(ϕc) = − ln Z � k(ϕc) � − 4Brωrβ � k(ϕc) dϕc (27) = Γ(0) + Brωrβ � g0 + λ 16π2 g1 � ϕ2 c + O(ϕ4 c) , where Γ(0) = − ln Z(0) (28) = − ln � 2e−Σ0(β) + e−2Σ0(β/2)� cosh( ¯N) − 1 �� , and Σ0 ≡ ΣL|k=0 = ΣR|k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The effective potential Ueff is finally given by Γ(φc) = V βUeff(φc) , (29) and, as expected, it satisfies the following properties: it is a convex function of φc, since the mass term is positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' the ground state is at ϕc = 0, or equivalently k = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' it has a non-trivial volume-dependence and is thus non-extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For the following studies of NEC violation we focus on the ground state ϕc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' NEC violation The ground state density ρ and pressure p are obtained from the free energy F = 1 β Γ(0) = − 1 β ln Z(0) , (30) and their sum can be written as [4] ρ + p = 1 V � F − T ∂F ∂T � − ∂F ∂V = −T ∂Ueff(0) ∂T − V ∂Ueff(0) ∂V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (31) From the expression (28), we obtain for ωrβ ≫ 1 ρ + p ≃ 4ω5/2 R ( √ 2πβ)3/2 e−ωRβ/ √ 2 (32) −ωR V � Sinst + 1 2 � � 6 π Sinst e−Sinst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' On the right-hand side, the first term corresponds to thermal fluctuations and the second term corresponds to tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' These terms compete for the overall sign of ρ + p leading to the following cases: Infinite volume: ρ + p ≥ 0 In the limit of infinite volume tunnelling is sup- pressed, as seen via the vanishing of the average number (20) of instanton/anti-instanton pairs for any fixed temperature: limV →∞ ¯N = 0 for fixed β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Hence only thermal fluctuations contribute and ρ + p = 4ω5/2 R ( √ 2πβ)3/2 e−ωRβ/ √ 2 , (33) with ρ + p → 0 as the temperature goes to 0 or equivalently β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' This result is not surprising: the limit of infinite volume corresponds to SSB and, as expected, the NEC is satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Finite volume and zero temperature: ρ + p < 0 In this situation, only the tunnelling term con- tributes and ρ + p = −ωR V � Sinst + 1 2 � � 6 π Sinst e−Sinst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (34) The NEC is violated as a consequence of the ex- plicit volume-dependence of the effective potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Boundary ρ + p = 0 We sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='2 the boundary V (T) between the region where the NEC is satisfied and the region where the NEC is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Finally, we note that NEC violation is suppressed ex- ponentially with the volume, unlike the power law sup- pression which is found with O(4)-symmetric Euclidean spacetime coordinates [3, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' DISCRETE MOMENTUM CORRECTIONS We focus here on the ground state obtained for k = 0 in the case of zero temperature, where NEC violation arising from tunnelling is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 2: The boundary between the regions where the NEC is satisfied and where it is violated due to the competition of tunnelling and thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The plot shows the curve V (T) in terms of the dimensionless variables used in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The previous sections ignore quantisation of momen- tum when calculating the connected graphs generating functional for each static saddle point in a finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' As we explain below, the evaluation of ΣL,R with discrete momentum consists of taking into account the relevant Casimir energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' There is no such contribution from the jumps in the instantons/anti-instantons since the corre- sponding one-loop corrections do not depend on momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Vacuum energy The Casimir contribution to the connected graphs gen- erating functional is defined as ΣCas ≡ ΣL,R|discrete − ΣL,R|continuum , (35) where the ultraviolet divergences cancel out since they are identical in the discrete and continuum cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For zero temperature and vanishing source, the expression (16) gives ΣL,R(k = 0, T = 0)|continuum = lim β→∞ Σ0(β) = 0 , (36) such that, instead of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (28), one-loop corrections ob- tained with discrete momentum lead to Γ(0) = − ln � 2e−ΣCas + e−ΣCas� cosh( ¯N) − 1 �� = ΣCas − ln � cosh( ¯N) + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (37) The above expression takes advantage of the proportinal- ity between Σ0 and β in the limit of vanishing tempera- ture, such that 2Σ0(β/2) → Σ0(β), (38) as β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' In the situation of one saddle point, and therefore no tunnelling, Γ(0) = ΣCas = βECas where ECas is the Casimir energy corresponding to quantum fluctuations about a single vacua ±v (where one has approximately quadratic fluctuations with mass m = √ 2ωr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Hence Ueff(0) = ECas V − 1 V β ln � cosh( ¯N) + 1 � , (39) and we see the additive nature of the Casimir effect and tunnelling contributions, similarly to the finite- temperature contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The sum of density and pres- sure reads finally ρ + p = ECas V − ∂ECas ∂V (40) −ωR V � Sinst + 1 2 � � 6Sinst π e−Sinst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Casimir contribution to the NEC The Casimir energy is highly sensitive to the geometry of the box containing the field, as well as the boundary conditions used on the corresponding surfaces [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For a scalar field ϕ(t, x) in the interval x ∈ [0, L] for example, the possible choices of boundary conditions are defined as follows Dirichlet:ϕ(t, 0) = ϕ(t, L) = 0 (41) Neumann:∂xϕ(t, 0) = ∂xϕ(t, L) = 0 Periodic:ϕ(t, 0) = ϕ(t, L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (42) For the cases we consider, the asymptotic form of the Casimir effect is identical for both Dirichlet and Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We thus consider mixed boundary conditions, where different subsets of the boundary can possess either Dirichlet or Neumann conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For the case of mixed boundary conditions, the Casimir energy is dependent on the size/curvature of the material boundaries, and for the case of periodic boundary conditions, it is dependent on the period length/curvature of the non-trivial spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' A ‘gen- eral rule’ states that flat geometries lead to exponential suppression of the Casimir energy for mL ≫ 1, where L is the length scale of the relevant boundaries, and that curved geometries lead to power-law suppression of the Casimir energy for mR ≫ 1, where R is the radius of curvature of the relevant surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' There are exceptions to this general rule though, which are highlighted in the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Dirichlet boundary conditions, flat boundaries The original Casimir configuration consists of a scalar field constrained between two parallel, flat mirrors with surface area A and separation a, with the scalar field satisfying Dirichlet conditions on the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The 5 4 3 2 p+p>0 Od+d 1 1 2 3 4 5 9 Br7 corresponding Casimir energy is [7] ECas ≃ � − Aπ2 1440a3 for am ≪ 1 − A 8 √ 2 � m πa �3/2 e−2ma for am ≫ 1 (43) and is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Dirichlet boundary conditions, curved boundaries For dimensional reasons, the Casimir energy for a scalar field confined within the curved boundary of a 2-sphere of radius R with Dirichlet boundary conditions is given in terms of the dimensionless function ECas = 1 Rf(mR) , (44) and is found to obey power law suppression in mR, for mR ≫ 1 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Periodic boundary conditions, flat spacetime For a scalar field confined to the surface of a 3-torus (a rectangular box with periodic boundary conditions), the sign of the Casimir energy depends on the ratio of the lengths of the box and we have [13] ECas ≃ −(mL)3/2 L exp(−mL) for mL ≫ 1 , (45) where L is the typical size of the period length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Periodic boundary conditions, curved spacetime For a scalar field confined to the surface of a 3-sphere with radius R, we would expect the asymptotic form to be a power law in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' However, this special case is an exception to the general rule as a consequence of the accidental vanishing of the heat-kernel coefficients (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 3 of [7] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The resulting Casimir energy has instead an exponential asymptotic form, as in the case of flat geometries [14] ECas ≃ +(mR)5/2 R exp(−2πmR) for mR ≫ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (46) The above examples display how the Casimir effect for a massive scalar field is at most suppressed by the expo- nential e−mL, where L is a typical size of the boundary containing the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' On the other hand, the tunnelling contribution to the NEC, calculated with continuous mo- mentum, is proportional to e−Sinst ∼ exp � −(mL)3 λ � , (47) and is therefore negligible compared to the Casimir con- tribution in the regime mL ≫ √ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For mL ∼ √ λ though, tunnelling competes with the Casimir effect and can change the sign of ρ + p in the situation where the Casimir energy is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' As an example, we sketch in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' 3: The boundary between the regions where the NEC is satisfied and where it is violated, due to the competition of the Casimir energy and tunnelling at zero temperature on a 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The plot shows the curve R(λ) in terms of the dimensionless variables used in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='3 the boundary R(λ) between the region where the NEC is satisfied and the region where it is violated, due to the competition between tunnelling and the Casimir effect on a 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' We note however two important points regarding the Casimir examples cited here: (i) they are valid for ideal surfaces only, and a realistic confining mechanism for the scalar field would lead to a modification of the Casimir vacuum energies, especially if the field is confined by an external potential instead of a physical box [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (ii) they assume free scalar fields and ignore its self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' On the other hand, the tunnelling mechanism described here: (i) necessitates the field to be self-interacting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (ii) is not sensitive to the geometry/topology of the box con- taining the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Hence the conclusions regarding which effect dominates could be modified by a more thorough study, depending on the situation which is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Finally, the Average Null Energy Condition is not vio- lated by the present mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Indeed, if we take into account the energy necessary to maintain the confining mechanism the overall ground state of the system does not violate the NEC [16], consistently with what is ex- pected from causality [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' CONCLUSIONS Tunnelling between degenerate vacua is exponentially suppressed with the volume of the box containing the field, but nevertheless allows the possibility of NEC vio- lation at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Taking into account discrete momentum of fluctuations in a finite volume implies this effect is mainly relevant for situations where the typical size of the box is not too large compared to the Compton wave length of the particle, and where tunnelling can lead to an overall NEC violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' A potential application lies 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='8 Ovd+d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='6 V p+p>0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content='5 2 WrR8 in axion physics, where the de Broglie wavelength can be of order 1 kpc [18] with the confinement provided by a gravitational well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Exponential suppression in the volume could poten- tially be avoided by a consideration of non-degenerate vacua, where other saddle points with a volume- independent action become relevant, as in the original study of false vacuum decay [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The resulting effective action would be non-extensive in a certain regime of the classical field, but more studies need to be done for the status of NEC violation in the corresponding vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Finally, NEC violation could play an important role in Early Universe Cosmology, where tunnelling could pro- vide a dynamical mechanism for a cosmological bounce, as explained in [19]: as the Universe contracts, tun- nelling switches on and violates the NEC, which induces a bounce after which tunnelling is suppressed as the Universe expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' This scenario necessitates the study of tunnelling in a Friedman-Lemaitre-Robertson-Walker background though, and is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors would like to thank Klaus Kirsten for valu- able correspondence regarding the Casimir effect, and JA would like to thank Janos Polonyi for enlightening discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' This work is supported by the Leverhulme Trust (grant RPG-2021-299) and the Science and Tech- nology Facilities Council (grant STFC-ST/T000759/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' For the purpose of Open Access, the authors have ap- plied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Appendix A: Fluctuation factor for a static saddle point The fluctuation factors for the static saddle points are calculated with continuous 3-dimensional momenta, in- troducing the cut-off Λ in the Schwinger proper time representation of the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Introducing the dimen- sionless Matsubara frequency νn ≡ 2πn/ωβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' we have Tr � ln � δ2S[ϕi] �� (A1) = V � d3p (2π)3 ∞ � n=−∞ � ∞ 1/Λ2 ds s e−4Bωβs(p2/ω2+ν2 n+3ϕi−1) = V ω3 2π2 ∞ � n=−∞ � ∞ 1/X2 dx x � ∞ 0 dq q2e−x(q2+ν2 n+3ϕi−1) = V ω3 8π3/2 ∞ � n=−∞ � ∞ 1/X2 dx x5/2 e−x(ν2 n+3ϕi−1) = V ω3 8π3/2 � ∞ 1/X2 dx x5/2 e−x(3ϕi−1)ϑ0 � 4πx ω2β2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' where the dimensionless variables are q ≡ p ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' x ≡ 4Bωβs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' X2 ≡ Λ2 4Bωβ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (A2) and ϑ0(y) is the Jacobi function ϑ0(y) ≡ ∞ � n=−∞ e−πyn2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (A3) Making use of the following property ϑ0(y) = y−1/2ϑ0(1/y) , (A4) the above becomes Tr � ln � δ2S[ϕi] �� (A5) = V ω4β 16π2 � ∞ 1/X2 dx x3 e−x(3ϕi−1)ϑ0 � ωβ 4πx � = V ω4β 16π2 � ∞ 1/X2 dx x3 e−x(3ϕi−1) ∞ � n=−∞ e−ω2β2n2/4x = λBωβ 24π2 � IΛ(ϕi) + IT (ϕi) � , where IΛ(ϕi) ≡ � ∞ 1/Λ2 dx x3 e−x(3ϕi−1) (A6) IT (ϕi) ≡ 2 ∞ � n=1 � ∞ 0 dx x3 e−x(3ϕi−1)−ω2β2n2/4x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The first integral IΛ is the temperature-independent di- vergent integral which, after renormalisation, produces the same results as in the zero-temperature case [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The second integral IT is the temperature-dependent contri- bution corresponding to the finite-temperature correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' It is finite, which is why the cut-off is taken to in- finity in this specific term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' This temperature-dependent integral can be written in terms of the modified Bessel functions of the second kind K2(z) as IT (φi) = ∞ � n=1 16(3ϕi − 1) (nωβ)2 K2(nωβ � 3ϕi − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (A7) Together with the integral IΛ, the connected graphs gen- erating functional for homogeneous saddle points is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Appendix B: Fluctuation factor for the instantons/anti-instantons gas We calculate here the contribution exp(−Σgas) to the partition function (15), following the known approach in studies of tunnelling effects [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' The invariance of the action for n instanton/anti- instanton pairs under the translation of the jumps leads to the degeneracy factor in the partition function � 2n � i=1 � ωβ τi−1 τi � = (ωβ)2n (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' , (B1) 9 where τi ∈ [τi−1, ωβ] and τ0 = 0, since successive in- stanton jumps can only occur after previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Each jump has an associated fluctuation factor � 6Sinst/π and thus the total fluctuation factor is given by the product of the contributions of the “flat” parts of the n-pairs of instanton/anti-instantons and the n pairs of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' On average, each configuration of n instanton/anti-instanton pairs spends the same time ≃ β/2 close to each static saddle point, such that the expression for Fn is finally Fn = FL(β/2)FR(β/2) �6Sint π �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' (B2) Substituting the above results into the partition function (14), along with the total action (13) for n pairs, yields the total contribution to the partition function due to instanton/anti-instanton pairs exp(−Σgas) (B3) = e−ΣL[β/2]e−ΣR[β/2] ∞ � n=1 (ωβ)2n (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' �6Sint π �n e−2nSint = exp � − ΣL[β/2] − ΣR[β/2] � × � cosh � ωβ � 6Sint π e−Sinst � − 1 � , This leads to the expression (19), where the parameters can be replaced by their renormalised version, since the overall expression is already at one-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Symanzik, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf'} +page_content=' Math.' metadata={'source': 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0000000000000000000000000000000000000000..0bd90492c842a5f09c487e639cab82f83f143bd5 --- /dev/null +++ b/NNAyT4oBgHgl3EQfgvgA/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ceeaabd7abbe251bed6812270e71d0fac7b8af38a981a0961a10b1c50d72c389 +size 120917 diff --git a/NNE3T4oBgHgl3EQfBglT/content/tmp_files/2301.04267v1.pdf.txt b/NNE3T4oBgHgl3EQfBglT/content/tmp_files/2301.04267v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..41290549145e0db12001db3acf9ed1f7b9015ffa --- /dev/null +++ b/NNE3T4oBgHgl3EQfBglT/content/tmp_files/2301.04267v1.pdf.txt @@ -0,0 +1,2484 @@ +1 +Market Power Mitigation in Two-stage Electricity +Market with Supply Function and Quantity Bidding +Rajni Kant Bansal, Yue Chen, Pengcheng You, Enrique Mallada +Abstract—The main goal of a sequential two-stage electricity +market—e.g., day-ahead and real-time markets—is to operate +efficiently. However, the price difference across stages due to +inadequate competition and unforeseen circumstances leads to +undesirable price manipulation. To mitigate this, some Inde- +pendent System Operators (ISOs) proposed system-level market +power mitigation (MPM) policies in addition to existing local +policies. These policies aim to substitute noncompetitive bids with +a default bid based on estimated generator costs. However, these +policies may lead to unintended consequences when implemented +without accounting for the conflicting interest of participants. In +this paper, we model the competition between generators (bidding +supply functions) and loads (bidding quantity) in a two-stage +market with a stage-wise MPM policy. An equilibrium analysis +shows that a real-time MPM policy leads to equilibrium loss, +meaning no stable market outcome (Nash equilibrium) exists. +A day-ahead MPM policy, besides, leads to a Stackelberg-Nash +game with loads acting as leaders and generators as followers. +In this setting, loads become winners, i.e., their aggregate +payment is always less than competitive payments. Moreover, +comparison with standard market equilibrium highlights that +markets are better off without such policies. Finally, numerical +studies highlight the impact of heterogeneity and load size on +market equilibrium. +Index Terms—electricity market, two-stage settlement, supply +function bidding, Stackelberg game, equilibrium analysis +I. INTRODUCTION +M +OST wholesale energy markets in the US conduct +auctions to settle electricity transactions where suppli- +ers, e.g., generators, offer to supply energy as a function of +price, while consumers, e.g., utilities, offer to consume energy +to meet their demand. After all the bids are submitted, the +independent system operators (ISOs) or regional transmission +organizations (RTOs) clear the market to achieve a supply- +demand balance. Such a market operation may constitute +multiple stages at various time scales, e.g., week-ahead, day- +ahead, hour-ahead, 5-minute, etc. In practice, many ISOs +consider a two-stage settlement system, namely day-ahead +and real-time markets, as a norm in the market [1]. The first +stage, the day-ahead (forward) market, clears a day before the +delivery based on the hourly forecasts of resources for the next +day and accounts for the majority of energy trades. The second +stage, the real-time (spot) market, occurs at a faster timescale +R. K. Bansal and E. Mallada are with the Whiting School of Engi- +neering, Johns Hopkins University, Baltimore, MD 21218, USA (email: +{rbansal3,mallada}@jhu.edu) +Y. Chen is with the Department of Mechanical and Automation Engineering, +The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China +(email:yuechen@mae.cuhk.edu.hk) +P. You is with the College of Engineering, Peking University, Beijing, China +(email:pcyou@pku.edu.cn) +(typically every five minutes) and is considered a last resort +for participants to adjust their commitment following forecast +errors [2], [3]. +The main goal of such a sequential two-stage settlement +market is to operate efficiently and ensure proper incentives +to encourage participation in the market. However, the often +price difference between the two stages in practice, due to +intrinsic uncertainty in the forecast, unscheduled maintenance +or outage, etc., creates opportunities for price speculation +and arbitrage, which could be further exploited by strategic +participants to their benefit, signaling efficiency losses [4]– +[6]. To discourage suppliers from exploiting consumers, most +operators employ an inbuilt local market power mitigation +mechanism (LMPM) triggered at congestion during market +clearing [7], [8]. Despite this, some operators, like California +Independent System Operator (CAISO), have documented +periods of time with non-competitive bids (approximately 2% +hours in the case of CAISO [9]). It led to the development of +initiatives aimed at implementing system-level market power +mitigation (MPM), i.e., bid mitigation similar to LMPM, but +system-wide for each stage separately [10], [11]. +Even though it is reasonable but too aggressive to implement +such mitigation policies in both day-ahead and real-time, these +market modifications are being considered separately, starting +with the real-time stage in the first phase and followed by +the day-ahead. In particular, CAISO argues that real-time is +more susceptible to market power and day-ahead is relatively +competitive due to additional measures like virtual bidding +that add competitive pressure on market clearing [10]. Such +system-level policies, when implemented, substitute in, e.g., +real-time or day-ahead, any non-competitive bids with default +bids, which estimate generator costs based on the opera- +tor’s knowledge of technology, fuel prices, and operational +constraints [12, §30.11.2.3]. Although such market policies +are straightforward, their effect on market outcome remains +unknown if implemented without accounting for the con- +flicting interest of individual participants. This paper studies +the proposed system-level policies and discusses the possible +unintended effects. +Precisely, we study a sequential game formulation in a two- +stage market with an MPM policy to analyze the competition +between generators (bidding supply functions and seeking +to maximize individual profit) [13], [14] and loads (bidding +demand quantities and minimizing payment) [15] such that +the market operator substitutes generators’ bids with default +bids as per the policy. In this paper, we assume that an +operator accurately estimates the truthful cost of dispatching +the generator in a stage with an MPM policy such that the +arXiv:2301.04267v1 [math.OC] 11 Jan 2023 + +2 +market clears efficiently. We show that a real-time MPM +policy results in a loss of market equilibrium. However, the +complimentary case of a day-ahead MPM policy leads to a +form of Stackelberg-Nash game with loads leading generators +in their decision-making. A detailed Nash equilibrium analysis +for this case shows a stable market outcome that favors loads, +unlike the standard two-stage market. +Contributions of this work: The main contributions of this +paper are summarized below: +1) We show that a real-time MPM policy leads to a Nash +game in the day-ahead, while generators participate +truthfully in real-time. We characterize the competitive +equilibrium of such a game, which is optimal w.r.t the +social planner’s problem. However, competition between +price-anticipating participants does not result in a stable +market outcome, and a Nash equilibrium does not exist. +2) We then study the impact of a day-ahead MPM policy +that leads to a generalized Stackelberg-Nash game with +loads acting as leaders in the day-ahead market and +generators acting as followers in the real-time market. +The competitive equilibrium of such a game also aligns +with the social planner problem, while the Nash equi- +librium, assuming that generators are homogeneous and +bid symmetrically for closed-form analysis, results in +load-favoring market outcomes. +3) To understand the impact of such policies, we compare +the resulting day-ahead MPM policy market equilibrium +with equilibrium in a standard market, i.e., a two- +stage market without any mitigation policies, studied +extensively in the literature. The closed-form analysis +shows that prices across stages are the same for the two +cases. However, loads acting as leaders in a market with +a day-ahead MPM policy allocates higher demand in +day-ahead at the expense of generators’ profit. It leads to +an unintended consequence of additional market power +instead of mitigation compared to the standard two-stage +markets. +4) We further provide a detailed numerical study to il- +lustrate the impact of a day-ahead MPM policy. We +show that the Nash equilibrium converges to competitive +equilibrium as the number of participants increases in +the market. Furthermore, the case with heterogeneity +in generator cost shows that expensive generators are +least affected when benchmarked with the competitive +equilibrium. The case with significant diversity in load +participants reveals that a sufficiently smaller load could +earn money instead of making payments at the expense +of larger loads in the market. +Related work: To date, the problem of characterizing the +effect of system-level market power mitigation policies in two- +stage markets has not been considered by the literature. Sev- +eral works have investigated the competition between various +market players in different market settings and characterized +the effect of mixed bidding mechanisms. In particular, refer- +ences [13], [14], [16]–[18] focus on the strategic behaviors in +a single-stage market and characterize the resulting Nash equi- +librium, where generators bid linear supply function and max- +imize their profit and load is either inelastic [13], [14], elastic +and bids a supply function [14], [16], [17] or quantity [15], +[18] to minimize their payment or equivalently maximize +the utility. Further in this line of research, references [19]– +[22], looks at cross-group participation models in a two-stage +market setting, analyzing perfect competition [19], [20], where +participants accept existing prices in the market, and strategic +behavior [21], [22], where participants manipulate prices in +the market. Another line of work looks at MPM in electricity +markets. In particular, the role of forward contracting [23], +[24], demand shifting [25], [26], bidding capacity division +and constraint [27], promoting interconnection and capacity +regulation [28], [29] and LMP-based default energy bids [30], +as a tool for local market power mitigation. However, despite +the extensive studies on cross-group participation and local +market power mitigation, to the best of our knowledge, our +work is the first one to study the impact of a system-level +market power mitigation in a two-stage market and formally +analyze the effect of default bid substitution on the market +outcome. +Paper Organization: The rest of the paper is structured as +follows. In Section II we first introduce the social planner +problem, two-stage market model, and participants’ behavior, +and then define a two-stage market equilibrium. In Section III +we model the market power mitigation policy for each stage, +i.e., day-ahead and real-time, characterize the competitive and +Nash equilibrium, and compare it with the optimal solution +to the social planner problem. We first compare the impact +of MPM policies on market equilibrium and then compare it +with a standard market equilibrium in Section IV. Numerical +studies on market power for a day-ahead MPM policy and +conclusions are in Section V and VI, respectively. +Notation: We use standard notation f(a, b) to denote a +function of independent variables a and b. However, we use +f(a; b) to represent a function of independent variable a and +parameter b. +II. MARKET MODEL +In this section, we formulate the social planner problem +and describe the standard two-stage settlement market design, +where a generator submits a supply function and demand bids +quantities. We then formally define participants’ behavior, i.e., +price-taking or price-anticipating, and lay out a general market +equilibrium. +A. Social Planner Problem +Consider a single-interval two-stage settlement market +where a set G of generators participate with a set L of +inelastic loads to meet inelastic aggregate demand d ∈ R. +Each generator j ∈ G supplies gj ∈ R and each inelastic load +l ∈ L consumes dl ∈ R respectively, where � +l∈L dl = d. +We define G := |G| and L := |L| to denote the number +of generators and loads, respectively. Assuming a convex +cost function Cj(gj) for each generator j ∈ G, respectively, +the social planner problem — minimum cost of meeting +aggregate inelastic demand — is given by: + +3 +Fig. 1. Two-stage Market Mechanism +SOCIAL PLANNER +min +gj,j∈G +� +j∈G +Cj(gj) +(1a) +s.t. +� +l∈L +dl = +� +j∈G +gj +(1b) +where (1b) enforces the supply-demand balance in the two- +stage market. +B. Two-Stage Market Mechanism +In this subsection, we define the two-stage market clearing, +as shown in Figure 1. The net output gj of each generator j +is distributed between gd +j and gr +j such that +gj = gd +j + gr +j, +(2) +where gd +j , gr +j represent power output in day-ahead and real- +time markets, respectively. Similarly, each load l allocates its +inelastic demand dl over two stages such that +dd +l + dr +l = dl +(3) +where dd +l , dr +l represent load allocation in day-ahead and real- +time markets, respectively. +1) Day-Ahead Market: The power output of each generator +j ∈ G in the day-ahead market is denoted by gd +j . Each +generator j submits a supply function h : R × R → R, +parameterized by θd +j , that indicates willingness of generator +j to supply gd +j as a function of price +gd +j = h(λd; θd +j ) +(4) +where λd denotes the price in the day-ahead market. Each load +l ∈ L bids quantity dd +l in the day-ahead market. Based on the +bids (θd +j , dd +l ) from participants, the market operator clears the +day-ahead market to meet the supply-demand balance. +� +j∈G +h(λd; θd +j ) = dd +(5) +The optimal solution to the day-ahead dispatch problem (5) +gives the optimal dispatch (gd +j , dd +l ) and clearing prices λd to +all the participants. Each generator j ∈ G and load l ∈ L are +paid λdgd +j and λddd +l as part of the market settlement. +2) Real-Time Market: The power output of each generator +j in real-time market is denoted by gr +j and their bid is: +gr +j = h(λr; θr +j) +(6) +where λr denotes the price in the real-time market. The +supply function bid is parameterized by θr +j, indicating the +willingness of generator j to supply gr +j at the price λr. Each +load l ∈ L submits the quantity bids dr +l . Given the bids +(θr +j, dr +l ), the operator clears the real-time market to meet the +supply-demand balance. +� +j∈G +h(λr; θr +j) = dr +(7) +Similar to the day-ahead market clearing, the optimal solution +to the dispatch problem (7) gives the optimal dispatch and the +market clearing prices λr to all the participants, such that each +generator j ∈ G and load l ∈ L produces or consumes gr +j and +dr +l , and is paid or charged λrgr +j and λrdr +l , respectively, as part +of the market settlement. +3) Market Rules and Goal: In this section, we first define +a set of rules to account for degenerate cases in the market +mechanism and then discuss the goal of a two-stage market. +Rule 1: For v ∈ {d, r} and w ∈ {d, r}, if the net supply +and demand of the generators and loads in a stage follow +� +j∈G +h(λv; θv +j ) = 0, dv = 0 =⇒ λv = λw, v ̸= w +i.e., the clearing price in that stage is set to the clearing prices +of the other stage with a non-zero demand. +Rule 2: For v ∈ {d, r}, if the net supply and net demand +of the generators and loads in a stage follow +� +j∈G +h(λv; θv +j ) = 0, dv ̸= 0 =⇒ λv = 0 +i.e., the clearing price is set to zero, and demand is split evenly +across all the loads. +We are interested in two-stage market outcomes that satisfy +� +j∈G +(gd +j + gr +j) = � +j∈G +gj = � +l∈L +(dd +l + dr +l ) = � +l∈L +dl = d (8) +and solve the social planner problem (1). Though the market +outcome may deviate from the optimal social planner solution, +signaling efficiency losses due to price manipulation by partic- +ipants, we quantify such deviations to understand the behavior +of participants and the market outcome. +C. Participant Behaviour +In this section, for the purposes of our study, we introduce +two different types of rational participants’ behavior, price- +taking, and price-anticipating. Each generator j ∈ G seeks to +maximize their profit πj, given by: +Generator Profit +πj(gd +j , gr +j, λd, λr):=λrgr +j+ λdgd +j −Cj(gj) +(9) +Each load l ∈ L aims to minimize their payments ρl, as: +Load Payment +ρl(dd +l , dr +l , λd, λr) := λddd +l + λrdr +l +(10) +Substituting the coupling constraint for the load allocation +across two stages (3) in (10) we get, +ρl(dd +l , λd, λr) := λddd +l + λr(dl − dd +l ) +(11) +For each load l ∈ L, the allocation in the day-ahead market +dd +l determines its allocation in the real-time market dr +l due to +the demand inelasticity. + +Individual generator i E G +C;(gj) +Day-Ahead market clearing +Real-Time market clearing + h(X"; ,) = d" + h(αd; g) = dd +53! +jEg +- +- +dd t +Individual Load l E L +di = dd + dr4 +1) Price-Taking Participants: A price-taker participant is +defined below: +Definition 1: A market participant is price-taking if it ac- +cepts the existing prices in the market and does not anticipate +the impact of its bid on the market prices. +Given the prices in the day-ahead market λd and real-time +market λr, the generator individual problem is given by: +Price-taking Generator Bidding problem +max +gd +j ,gr +j +πj(gd +j , gr +j; λd, λr) +(12) +Similarly given the prices λd, λr, the individual bidding prob- +lem for load is given by: +Price-taking Load Bidding problem +min +dd +l +ρl(dd +l ; λd, λr) +(13) +We next define the price-anticipating (or strategic) participants. +2) Price-Anticipating Participants: +A price-anticipating +participant is defined below: +Definition 2: A market participant is price-anticipating +(strategic) if it anticipates the impact of its bid on the prices in +two stages and has complete knowledge of other participants’ +bids. +The individual problem of a price-anticipating generator is: +Strategic Generator Bidding problem +max +gd +j ,gr +j ,λd,λr πj +� +gd +j , gr +j, λd� +gd +j ; gd +−j, dd� +, λr� +gr +j; gr +−j, dr�� +(14a) +s.t. (5), (7) +(14b) +where gd +−j := � +k∈G,k̸=j gd +k, and gr +−j := � +k∈G,k̸=j gr +k . The +generator j maximizes its profit while anticipating the market +clearing prices in the day-ahead and real-time market (5),(7), +along with complete knowledge of load bids dd +l , dr +l , l ∈ L, and +other generators’ bids θd +k, θr +k, k ∈ G, k ̸= j. Similarly, the indi- +vidual problem for strategic load l with complete knowledge +of prices in two stages (5),(7) and other participants’ bids: +Strategic Load Bidding problem +min +dd +l ,λd,λr ρl +� +dd +l , λd� +dd +l ; gd +j , dd +−l +� +, λr� +dd +l ; gr +j, dr +−l +�� +(15a) +s.t. (5), (7) +(15b) +where the load l minimizes its payment in the market and +d +d +−l := � +l∈L,k̸=l dd +l , d +r +−l := � +l∈L,k̸=l dr +l . +D. Market Equilibrium +In this section, for the purpose of this study, we characterize +the market equilibrium in a two-stage settlement electricity +market. At the equilibrium, no participant has any incentive +to deviate from their bid, and the market clears, as defined +below. +Definition 3: We say the participant bids and market clearing +prices (θd +j , θr +j, j ∈ G, dd +l , dr +l , l ∈ L, λd, λr) in the day-ahead +and real-time respectively form a two-stage market equilibrium +if the following conditions are satisfied: +1) For each generator j ∈ G, the bid θd +j , θr +j maximizes their +individual profit. +2) For each load l ∈ L, the allocation dd +l , dr +l minimizes +their individual payment. +3) The inelastic demand d ∈ R is satisfied with the market- +clearing prices λd given by (5) and λr given by (7) over +the two-stages of the market. +We will study market equilibria as a tool to understand the +impact of MPM policies. +III. UNDERSTANDING IMPACT OF MPM POLICY +In this section, we first characterize the market equilibrium +in a standard two-stage market without any such mitigation +policy [22], then model mitigation policies and the resulting +market equilibrium. In particular, ISOs have significant prior +knowledge of market participants allowing them to evaluate +the competitiveness of energy bids. For example, operators +are aware of the generator’s technology, fuel prices, and oper- +ational constraints that can be used to estimate or bound the +generator’s cost [12, §30.11.2.3] within a reasonable threshold +under the mitigation policies. Although it is reasonable to +implement such mitigation policies in both day-ahead and +real-time, CAISO argues that real-time is more susceptible +to market power and day-ahead is relatively competitive [10]. +Therefore, these market modifications are being considered +separately, starting with the real-time stage in the first phase +and followed by the day-ahead. For ease of exposition, we +assume that the operator accurately estimates the truthful cost +of dispatching the generator in a stage with a mitigation policy +such that the market clears efficiently. Also, we assume a +quadratic cost function for each generator j, parameterized +by cost coefficient cj, +Cj(gj) = cj +2 gj +2 +(16) +We further assume that the supply function bid h(λv; θv +j ) of +each generator j is given by: +gv +j := h(λv; θv +j ) = θv +j λv, v ∈ {d, r} +(17) +where the parameter θv +j ∈ R explicitly indicate willingness of +generator j to produce gv +j per unit price λv. +A. Standard Two-stage Market +The role of participants in a standard market without any +mitigation policy is studied extensively in the literature [13], +[14], [22], [31]. Here, we cite the results from [22] that analyze +the role of strategic generators and inelastic demand in a +standard two-stage market and use them as a benchmark to +analyze the impact of a mitigation policy in the market. +1) Price-taking Participation and Competitive Equilibrium: +For the individual incentive problem in a two-stage market, +substituting the cost function (16) and supply function (17) +in (9), we get +πj(θd +j , θr +j;λd, λr)= θd +j λd2+θr +jλr2− cj +2 +� +θd +j λd+θr +jλr�2 +(18) +and the individual problem for price-taking generator j is: +max +θd +j ,θr +j +πj(θd +j , θr +j;λd, λr) +(19) + +5 +Similarly, the individual problem for load l is given by (11). +Given the prices, λd, λr, we next characterize the resulting +competitive equilibrium due to competition between price- +taking participants. +Theorem 1 (Proposition 1 [22]): A competitive equilibrium +in a two-stage market exists and is explicitly given by +θd +j + θr +j = 1 +cj +, θd +j ≥ 0, θr +j ≥ 0, ∀j ∈ G +(20) +dd +l + dr +l = dl, ∀l ∈ L +(21) +λd = λr = +d +� +j∈G c−1 +j +(22) +The resulting competitive equilibrium solves the social planner +problem (1). Moreover, it exists non-uniquely, and there is no +incentive for a load to allocate demand in the day-ahead market +due to equal prices in two stages. +2) Price-Anticipating Participation and Nash Equilibrium: +The individual problem of price-anticipating generator j and +price-anticipating load l is given by (14) and (15), respectively. +We next characterize the resulting Nash equilibrium in such a +market. +Theorem 2 (Proposition 4 [22]): Assuming strategic gener- +ators are homogeneous (cj := c, ∀j ∈ G) and make identical +bids (θv +j := θv, ∀j ∈ G, v ∈ {d, r}) at equilibrium. If there +are at least three firms, i.e., G ≥ 3, a Nash equilibrium in +a two-stage market exists. Further, this equilibrium is unique +and explicitly given by +θd +j =L(G − 1) + 1 +L(G − 1) +G − 2 +G − 1 +1 +c , θr +j= +1 +L + 1 +(G − 2)2 +(G − 1)2 +1 +c +(23) +dd +l = +L(G − 1) + 1 +L(L + 1)(G − 1)d, dr +l= dl − dd +l +(24) +λd = +L +L + 1 +G − 1 +G − 2 +c +Gd, λr = G − 1 +G − 2 +c +Gd +(25) +The resulting Nash equilibrium exists uniquely, where price- +anticipating loads anticipate the actions of generators and +allocate demand to exploit lower prices in the day-ahead +market. Thus prices are different in two stages. Moreover, the +net demand allocation in the day-ahead and real-time market +follows +dd ∈ (0.5d, d), dr ∈ (0, 0.5d). +B. Real-Time MPM Policy +In this section, we first discuss the modified market model, +the individual incentives of participants, and then characterize +market equilibrium for a real-time MPM policy. +1) Modeling Real-Time MPM Policy: In the case of a real- +time MPM policy, the market ignores generators’ bids in +real-time, as shown in Figure 2, and accurately estimates the +truthful market clearing, given the day-ahead dispatch gd +j +gr +j = h(λr; θr +j) = c−1 +j λr − gd +j +(26) +Using the two-stage generation and supply-demand balance (8) +and real-time dispatch (26) we get +λr = +d +� +j c−1 +j +(27) +Fig. 2. Two-stage Market Mechanism with Real-Time MPM +2) Price-taking Participation and Competitive Equilibrium: +For the individual incentive problem in a two-stage market +with real-time MPM policy, substituting the cost function (16), +day-ahead supply function (17), real-time true dispatch condi- +tion (26) and real-time clearing prices (27) in (9), we get +πj(θd +j ,λd)= θd +j λd2+ +d +� +j c−1 +j +� +c−1 +j d +� +j c−1 +j +− θd +j λd +� +− cj +2 +� +c−1 +j d +� +j c−1 +j +�2 += +� +λd − +d +� +j c−1 +j +� +θd +j λd + +c−1 +j +2 +� +d +� +j c−1 +j +�2 +(28) +Hence, an individual problem of a price-taking generator is: +max +θd +j +πj(θd +j ; λd) +(29) +Similarly, substituting the clearing price (27) in (11) we get, +ρl(dd +l , λd) := λddd +l + +d +� +j c−1 +j +(dl − dd +l ) +(30) +such that the individual problem for load l is given by: +min +dd +l +ρl(dd +l ; λd) +(31) +The competition between price-taking participants for indi- +vidual incentives leads to a set of competitive equilibria, as +characterized below. +Theorem 3: The competitive equilibrium in a two-stage +market with a real-time MPM policy exists, and given by: +gd +j + gr +j = +c−1 +j +� +j∈G c−1 +j +d, θd +j ∈ R≥0 ∀j ∈ G +(32a) +dd +l + dr +l = dl, ∀l ∈ L +(32b) +λd = λr = +1 +� +j∈G c−1 +j +d +(32c) +We provide proof of the theorem in Appendix A. At the +competitive equilibrium, the market clearing prices are equal +in the two stages, meaning there is no incentive for a load to +allocate demand in the day-ahead market, e.g., current market +practice. However, the equilibrium still attains the desirable +social planner objective. +Corollary 1: The competitive equilibrium in a two-stage +market with a real-time MPM policy (32) also solves the social +planner problem (1). + +Individual generator i E G +C(i) +Day-Ahead market clearing +Real-Time market clearing +h(d; g) = ogd = dd +jEg +jEg +- +jEg +Jdr +Individual Load l E L +di = dj + dr6 +3) Price-Anticipating Participation and Nash Equilibrium: +The individual problem of each price-anticipating generator j, +given by: +max +θd +j ,λd πj +� +θd +j , λd� +θd +j ; θ +d +−j, dd�� +(33a) +s.t. (5) +(33b) +where the generator j maximizes its profit in the two-stage +market. The individual problem of price-anticipating load is +given by: +min +dd +l ,λd ρl +� +dd +l , λd� +dd +l ; θd +j , d +d +−l +�� +(34a) +s.t. (5) +(34b) +where the load l minimizes its payment in the market. +We study the resulting sequential game where players +anticipate each other actions and prices in the market, and the +day-ahead clears before the real-time market. To this end, we +analyze the game backward, starting from the real-time market, +where prices are fixed due to MPM policy (27), followed +by the day-ahead market, where participants make decisions +for optimal individual incentives and compute the equilibrium +path. Generators do not bid in real-time, but loads are allowed +to bid in the market. However, load makes decisions simul- +taneously in the day-ahead market due to inelasticity, fixing +their bids in the real-time market, which affects the two-stage +market clearing. The following theorem characterizes the two- +stage Nash equilibrium that satisfies the Definition (3). +Theorem 4: The Nash equilibrium in a two-stage market +with a real-time MPM policy does not exist. +We provide proof of the theorem in Appendix B and a brief in- +sight below into the loss of equilibrium. The price-anticipating +participants compete with each other to manipulate prices in +the day-ahead given by substituting (17) in (5): +λd = +dd +� +j θd +j +while the prices in the real-time λr (27) is fixed. Loads bid +decreasing quantities dd +l to reduce clearing prices in the day- +ahead market and minimize the load payment. Simultaneously, +generators bid decreasing parameter θd +j to increase clearing +prices and maximize revenue. The competition between loads +and generators for individual incentives in the day-ahead mar- +ket drives all the demand to the real-time market, where gen- +erators operate truthfully. However, in our market mechanism, +loads then have the incentive to deviate and allocate demand in +the day ahead where prices are zero, meaning zero payment in +the market, see Rule 2. Such unilateral load deviations result in +deviations from generators to increase clearing prices in the +day-ahead market. Therefore the equilibrium does not exist. +Without such a market rule, the Nash equilibrium does exist +with undefined clearing prices in the day-ahead and all demand +allocated to the real-time market. Nevertheless, since day- +ahead accounts for a majority of energy trades, the resulting +equilibrium is undesirable. +Fig. 3. Two-stage Market Mechanism with Day-Ahead MPM +C. Day-Ahead MPM Policy +In this section, we define the individual incentive of par- +ticipants and characterize market equilibrium for a day-ahead +MPM policy. +1) Modeling Day-Ahead MPM Policy: In the case of a day- +ahead MPM policy, as shown in Figure 3, the market ignores +the generators’ bids and accurately estimates the truthful +market clearing in the day-ahead as given by: +gd +j = h(λd; θd +j ) = 1 +cj +λd +(35) +Moreover, using the day-ahead power balance constraint, we +have +λd = +dd +� +j c−1 +j +(36) +2) Price-taking Participation and Competitive Equilibrium: +For the individual incentive problem in a two-stage mar- +ket with a day-ahead MPM policy, substituting the clearing +price (36) in (9), we get +πj(θr +j;λr)= +c−1 +j dd2 +(� +j c−1 +j )2 +θr +jλr2− cj +2 +� +c−1 +j dd +� +j c−1 +j ++θr +jλr +�2 +(37) +and the individual problem for price-taking generator j is: +max +θr +j +πj(θr +j; λr) +(38) +Similarly, substituting the clearing price (36) in (11) we get +the individual problem for price-taking load l, +min +dd +l +ρl(dd +l ; λr) = min +dd +l +dd +� +j c−1 +j +dd +l + λr(dl − dd +l ) +(39) +Each price-taking generator j solves individual prob- +lem (38) and load l solve individual problem (39). The +resulting competitive equilibrium given the clearing prices λd +and λr is characterized below. +Theorem 5: The competitive equilibrium in a two-stage +market with a day-ahead MPM policy exists and is given by: +gd +j = +c−1 +j +� +j∈G c−1 +j +d, gr +j = 0, ∀j ∈ G +(40a) +dd +l + dr +l = dl, ∀l ∈ L, dd = d, dr = 0 +(40b) +θr +j = 0, λd = λr = +1 +� +j∈G c−1 +j +d +(40c) +We provide proof of the theorem in Appendix C. Unlike the +competitive equilibrium for a real-time MPM policy in (32) + +Individual generator i E G +X +Day-Ahead market clearing +Real-Time market clearing +h(αd; 0g)=d: +h(xr; o) =,xr = d +iEg +jEg +jEg +0.53 +- +d' +Individual Load l E L +lp +pp = lp7 +with equal prices across stages, the loads at equilibrium (40) +allocate all the demand in the day-ahead. The incentive for +day-ahead demand allocation is a desired market outcome +and is not generally satisfied by other market mechanisms. +Although dr = 0 and � +j θr +j = 0, unlike in Theorem 4, loads +do not deviate from the equilibrium by shifting demand to the +real-time market. The resulting equilibrium exists as price- +taking loads do not anticipate the effect of their bid on the +market prices, meaning the payment remains the same for +any allocation across the two stages. Moreover, the market +outcome (40) solves the social planner problem (1). +3) Price-Anticipating Participation and Nash Equilibrium: +The individual problem of each price-anticipating generator j, +given by: +max +θr +j ,λr πj +� +θr +j, λd� +dd� +, λr� +θr +j; θ +r +−j, dr�� +(41a) +s.t. (7) +(41b) +where generator j maximizes its profit in the market. The +individual problem of price-anticipating load l, is given by: +min +dd +l ,λr ρl +� +dd +l , λd� +dd +l ; d +d +−l +� +, λr� +dd +l ; θr +j, d +r +−l +�� +(42a) +s.t. (7) +(42b) +where load l minimizes its payment in the market. +In the market model with a day-ahead MPM policy, genera- +tors make decisions in real-time while load can make decisions +in the day-ahead. The resulting two-stage sequential game is +essentially a leader-follower Stackelberg-Nash game, where +generators are followers in the real-time market and loads +are leaders in the day-ahead market, and each participant +in their respective groups competes amongst themselves in +a Nash game. We follow the terminology used in [32] to +describe similar formulations in different markets. For the +closed form solution, we assume that generators are homo- +geneous in the sense that they share the same cost coefficient, +i.e. cj =: c, ∀j ∈ G and bid symmetrically in the market, +i.e. θr +j =: θr, ∀j ∈ G. Under these assumptions, the Nash +equilibrium is characterized below. +Theorem 6: Assume that generators are homogeneous and +bid symmetrically in the market. If more than two generators +are participating in the market i.e., G ≥ 3 and the number of +individual loads participating in the market satisfies L < G−2, +then the symmetric Nash equilibrium in a two-stage market +with a day-ahead MPM policy exists and is uniquely given +by: +gd +j = +L +L + 1 +G − 1 +G − 2 +d +G, gr +j= +� +1 − +L +L + 1 +G − 1 +G − 2 +� d +G +(43a) +dd +l = +1 +L + 1 +G − 1 +G − 2d, dr +l= +� +dl − +1 +L + 1 +G − 1 +G − 2d +� +(43b) +θr = 1 +c +�G − 2 +G − 1 − +L +L + 1 +� +(43c) +λd = +L +L + 1 +G − 1 +G − 2 +c +Gd, λr = G − 1 +G − 2 +c +Gd. +(43d) +Moreover, for L ≥ G − 2, a symmetric equilibrium does not +exist. +We provide proof of the Theorem in Appendix D. Unlike +the market with a real-time MPM policy, the Nash equilibrium +exists in the market with a day-ahead MPM policy. However, it +requires restrictive conditions on the number of participants in +the market and may not even exist in other cases. We discuss +these cases with no symmetric Nash equilibrium and provide +intuition into participants’ behavior in the market: +a) L > G − 2: In this case, the net demand is negative +in the real-time market. The first order condition implies that +each generator j acts as load, paying λrgr +j as part of the market +settlement since their optimal bid θr +j < 0 and the real-time +clearing price λr > 0. However, if the generators bid θr +j > 0, +then the linear supply function implies that each generator j +dispatch gr +j < 0 at the clearing prices λr < 0 earning revenue +in the market. However, this is not desirable from a load +perspective since they are making payments in the market and +they have the incentive to deviate to minimize their payment. +Hence, symmetric equilibrium with negative demand in the +real-time market does not exist as the symmetric bid θr +j > 0 +does not satisfy the first-order condition. The dependence of +the individual bid θr +j on the given bids from other participants +makes the closed-form analysis challenging, and any guarantee +of the existence of equilibrium is hard. +b) L = G − 2: In this case, no symmetric Nash equilib- +rium exists. Loads take advantage of the truthful participation +of generators in the day-ahead market and their ability to +anticipate the impact of bids on the clearing prices. Regardless +of generators’ bids, loads have the incentive to deviate by al- +locating demand in the real-time market with a lower clearing +price. +Corollary 2: For L < G − 2, the demand allocation at the +Nash equilibrium (43) in a two-stage market with a day-ahead +MPM policy is given by: +� +l∈L dd +l = +L +L+1 +G−1 +G−2d ∈ (0.5d, d) +(44a) +� +l∈L dr +l = +−L+G−2 +(L+1)(G−2)d ∈ (0, 0.5d) +(44b) +The proof uses the fact that L < G−2 implies that dr > 0. +IV. EQUILIBRIUM ANALYSIS +In this section, we study the properties of market equilib- +rium under the proposed policy framework and compare it +with the standard market equilibrium. +A. Comparison of a stage-wise MPM Policy +An MPM policy in real-time either provides no incentive +for loads to allocate demand in the day-ahead at competi- +tive equilibrium or leads to no Nash equilibrium. However, +an MPM policy in the day-ahead leads to a stable market +outcome, i.e., load allocates all or a fraction of demand in the +day-ahead market at competitive equilibrium (40b) and Nash +equilibrium (43b), respectively. This is summarized in Table I. +We further analyze the case of a day-ahead MPM policy to +study the strategic behavior of participants while regarding +the respective competitive equilibrium in Theorem 5 as a +benchmark. In the case of a day-ahead MPM policy, loads +act as leaders in the day-ahead and generators as followers in +real-time. The generator bids to manipulate prices leading to + +8 +TABLE I +COMPETITIVE EQUILIBRIUM (CE) AND NASH EQUILIBRIUM (NE) WITH A +STAGE-WISE MPM POLICY +Instance +Real-Time MPM +Day-Ahead MPM +CE +Non-unique equilibrium +Unique equilibrium +Solves social planner +Solves social planner +Arbitrary demand allocation +All demand in day-ahead +NE +Does not exist +Symmetric equilibrium +- +Homogeneous generators +- +Social Cost same as CE +- +Extra constraints on players +TABLE II +COMPARISON BETWEEN COMPETITIVE EQUILIBRIUM (CE) AND NASH +EQUILIBRIUM (NE) IN A MARKET WITH A DAY-AHEAD MPM POLICY +Case +Generators total profit +Loads total payment +CE +1 +2 +c +G d2 +c +G d2 +NE +1 +2 +c +G d2� +G +G−2 − (G−1)2 +(G−2)2 +2L +(L+1)2 +� +c +G d2� +G−1 +G−2 − (G−1)2 +(G−2)2 +L +(L+1)2 +� +inflated prices in real-time (43d) while the load allocates more +demand in the day-ahead (43b), increasing prices in the day- +ahead market. Though the market equilibrium deviates from +the competitive equilibrium (40), the social cost remains the +same due to the homogeneous participation of generators. Ta- +ble II summarizes the aggregate profit and aggregate payment +of generators and loads, respectively. +Corollary 3: The aggregate payment of loads and aggregate +profit of generators at symmetric Nash equilibrium (43) is +always less than that at the respective competitive equilib- +rium (40). +The corollary follows from comparing the aggregate profit +(payment) at Nash equilibrium to that at competitive equilib- +rium in Table II for L < G − 2. +B. Equilibrium comparison with a standard market +In this section, we compare the equilibrium in a day-ahead +MPM policy market to a standard market. The competitive +equilibrium remains the same for the two markets with equal +prices in the two stages. However, unlike in the case of a day- +ahead MPM policy, the competitive equilibrium in Theorem 1 +exists non-uniquely and there is no incentive for a load to +allocate demand in the day-ahead market. +Interestingly, at Nash equilibrium prices in the two stages +are the same for a day-ahead MPM policy market (43d) and a +standard market (25). However, demand allocation is different +in the two market settings, due to a leader-follower structure +between participants in the market with a day-ahead MPM +policy. Assuming L < G − 2, at Nash equilibrium we have +dd +DA−MP M − dd +Standard = +1 +L + 1 +� +L +G − 2 − +1 +G − 1 +� +d > 0 +meaning, a load is able to exploit the market further by +allocating more demand in the day-ahead market with a lower +clearing price. To understand the impact of price-anticipating +participants on market equilibrium, we compare the aggregate +profit (payment) in Table II and III, respectively. +We restrict our comparison for L < G − 2 only since the +Nash equilibrium in Theorem 6 does not exist otherwise. In +TABLE III +COMPARISON BETWEEN COMPETITIVE EQUILIBRIUM (CE) AND NASH +EQUILIBRIUM (NE) IN A STANDARD MARKET +Case +Generators total profit +Loads total payment +CE +1 +2 +c +G d2 +c +G d2 +NE +1 +2 +c +G d2� +G +G−2 − 2L(G−1)+2 +(L+1)2(G−2) +� +c +G d2� +G−1 +G−2 − +L(G−1)+1 +(L+1)2(G−2) +� +particular, for L = G − 3 the aggregate profit (payment) +as shown in row 2 of Table III at the Nash equilibrium +in Theorem 2 equals to that of the competitive equilibrium. +However, for L < G−3 the aggregate profit (payment) at Nash +equilibrium is always less than the competitive equilibrium, +meaning the loads are winners. The change in the normalized +aggregate profit (payment) at the Nash equilibrium between a +market with a day-ahead MPM policy and a standard market +is given by +µ(L, G) +� +1 − +L +G − 2 − L +� +< 0, for L < G − 2 +where µ(L, G) > 0 is some constant that depends on L and +G, and profit (payment) is normalized with the competitive +equilibrium. This implies that implementing a day-ahead MPM +policy has an adverse effect and at equilibrium, the market is +farther away from the competitive equilibrium in the presence +of price-anticipating participants. However, as the number of +participants increases, the difference tends to 0, since the +Nash equilibrium in both cases approaches the competitive +equilibrium, respectively. +Figure 4 compares the total profit (payment) normalized +with competitive equilibrium for a day-ahead MPM (DA- +MPM) policy market and a standard market, respectively, as +we change the number of loads (l ∈ L, L ∈ {1, . . . , G − 3}), +and generators (j ∈ G, G ∈ {4, . . . , 20}). The ratio decreases +monotonically as the number of generators increases, meaning +the increased competition between more generators to meet the +inelastic demand gives more power to loads, allowing them to +reduce their payment even further, as shown by the horizontal +rows in all panels in Figure 4. Moreover, the ratio increases +monotonically as the number of loads increases (for a large +enough number of generators), meaning the Nash equilibrium +tends towards the competitive equilibrium, as shown by the +vertical color columns in panels (a) and (b) in Figure 4. +Although loads are still winners in standard markets for +L < G−3, we observe a reversal in power for L > G−3 where +generators are now able to make a higher profit at the expense +of loads in the market, as shown in panels (c) and (d) in the +Figure 4. Unfortunately, with a day-ahead MPM policy, the +equilibrium no longer exists as shown by white colored cells in +panels (a) and (b). Finally, in the limit L → ∞ =⇒ G → ∞, +the Nash equilibrium converges to competitive equilibrium, +also shown in Table II. +V. NUMERICAL STUDY +We now analyze the impact of heterogeneity on individual +profit at Nash equilibrium in the market with a day-ahead +MPM. Since it is hard to analyze the heterogeneity cases + +9 +Fig. 4. +Total profit (a) and total payment (b) at Nash Equilibrium (NE) normalized with competitive equilibrium (CE) in a market with day-ahead MPM +(DA-MPM), and total profit (c) and total payment (d) at Nash Equilibrium (NE) normalized with competitive equilibrium (CE) in a standard market; white +cells denote no equilibrium +Fig. 5. Net (top) and normalized (bottom) individual profit at Nash Equilib- +rium (NE) normalized with competitive Equilibrium (CE) w.r.t cost coefficient +of generators +in closed-form, we run numerical best-response studies. To +this end, we consider the case of 2 price-anticipating loads +and 5 price-anticipating generators in a two-stage market. +The individual aggregate inelastic load is given by dl = +[99.4, 199.6]T MW. The cost coefficients of generators are +sampled 10, 000 times from a Gaussian distribution with mean +0.1 and variance 0.001, i.e. cj +∼ N(0.1, 0.001), ∀j +∈ +{1, ..., 5}. The top and bottom panel in Figure 5 plots the +absolute profit and the normalized profit (normalized with +the competitive equilibrium) at Nash equilibrium, respectively. +The cheaper generators earn a higher profit when compared +with the expensive generators with higher cost coefficients +at Nash equilibrium. However, the normalized profit ratio +in the bottom panel shows that expensive generators have a +higher value than cheaper ones, meaning that though expensive +generators have lower absolute profit, these are the least +exploited in the market. This result is counter-intuitive. +In Figure 6 we show the absolute (top panel) and normalized +(bottom panel) load payment w.r.t smaller load size. For this, +we keep the same number of loads and generators in the +market with varying load sizes for fixed net demand. We +again sample cost coefficients from Gaussian distribution with +mean 0.1 and variance 0.001, i.e. cj ∼ N(0.1, 0.001), ∀j ∈ +{1, ..., 5}. The top panel shows that though the net load +Fig. 6. Net (top) and normalized (bottom) load individual payment (bottom) +at Nash Equilibrium (NE) normalized with competitive Equilibrium (CE) w.r.t +size of smaller load d1, d1 < d2, d1 + d2 = d +payment remains the same as we change the size of the load, +the smaller load may even make a profit in the market at the ex- +pense of a higher load. More formally to develop intuition, in +the case of homogeneous generators, the normalized payment +ratio for individual load at Nash equilibrium in Theorem 6, is +given by +G − 1 +G − 2 +� +1 − +1 +(L + 1)2 +G − 1 +G − 2 +d +dl +� +which is negative for a sufficiently small load. In particular, the +smaller load has a negative normalized ratio at the expense of +a higher load (a ratio greater than 1), as shown in the bottom +panel of Figure 6. The larger load makes more payment at +Nash equilibrium than at the competitive equilibrium, while +the aggregate payment of the set of loads is still less than at +the competitive equilibrium. Though the heterogeneity in load +size does not affect the net payment or the group behavior +in the market, a smaller load makes negative payments at the +expense of larger loads and can exercise more market power. + +(normalized with CE) +Total T. +(normalized with CE) +Total p +(normalized with CE) +Total πT. +(normalized with CE) +‘j,NE:Standard +PI,NE:Standard +j,NE:DA-MPM +1234567890WB45 +1.8 + of Loads (L) +1.6 +1.4 +1.2 +1 +'ON +0.8 +0.6 +(a) +(b) +(c) +(d) +No. of Generators (G)Net (top) and normalized (with CE, bottom) individual profit at NE w.r.t cost coefficient c +Homogeneous generators +200 +Least Expensive +Most Expensive +(Net) +180 +j,NE +160 +140 +120 +0.07 +0.08 +0.09 +0.1 +0.11 +0.12 +0.13 +0.9 +(Normalized) +0.89 +0.88 +,NE +0.87 +0.07 +0.08 +0.09 +0.1 +0.11 +0.12 +0.13 +CNet (top) and normalized (with CE, bottom) individual load payment w.r.t load size d +Smaller load d, +Larger load d2 +Total load (d = d, +d,) +2000 +1500 +(Net) +1000 +500 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +(Normalized) +Smaller load d, +Larger load d2 +I,NE +-10 +p +-20 +-30 +-40 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +d,/d10 +VI. CONCLUSIONS +We study competition between generators (bid linear supply +function) and loads (bid quantity) in a two-stage settlement +electricity market with a stage-wise MPM policy. In the +proposed policy framework, CAISO substitutes generator bids +with default bids in the stage with an MPM policy, i.e., day- +ahead or real-time. To understand the participant behavior in +the market, we start with a real-time MPM policy and analyze +the sequential game, where generators only bid in the day- +ahead market. The resulting competitive equilibrium, price- +taker participants, align with the social planner problem, and +loads do not have any incentive to allocate demand in the +day-ahead due to equal prices across two stages. Moreover, +the Nash equilibrium, price-anticipating participants, fails to +exist, indicating an unstable market outcome. +In the case of a day-ahead MPM policy, on the other hand, +the competitive equilibrium aligns with the social planner +problem, and the load allocates all the demand in the day- +ahead market. Since the day-ahead market accounts for a +majority of energy trades, the incentive for day-ahead de- +mand allocation is desirable from the market perspective. +Finally, the Nash equilibrium shows that the generator fails +to manipulate prices to earn higher profit. Loads (leaders +in day-ahead) in the resulting generalized Stackelberg-Nash +game take advantage of the generators (followers in real- +time), lowering their aggregate payment. Counter-intuitively, +our comparison with a standard market equilibrium reveals +that a day-ahead MPM policy results in higher market power +due to such a leader-follower structure. 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Ho, “Multiplayer stackelberg- +nash game for nonlinear system via value iteration-based integral +reinforcement learning,” IEEE Transactions on Neural Networks and +Learning Systems, pp. 1–12, 2020. +APPENDIX A +PROOF OF THEOREM 3 +Under price-taking behavior, the individual problem for +loads (31) is a linear program with the closed-form solution +given by: +� +� +� +� +� +� +� +dd +l = ∞, dr +l = −∞, dd +l + dr +l = dl, if λd < +d +� +k c−1 +k +dd +l = −∞, dr +l = ∞, dd +l + dr +l = dl, if λd > +d +� +k c−1 +k +dd +l + dr +l = dl, +if λd = +d +� +k c−1 +k +(45) +where loads prefer the lower price in the market. The individ- +ual problem for generators (29) requires: +� +� +� +� +� +� +� +� +� +� +� +θd +j = −∞, if 0 ≤ λd < +d +� +k c−1 +k +θd +j = ∞, if λd < +d +� +k c−1 +k , andλd < 0 +θd +j = ∞, if λd > +d +� +k c−1 +k +θd +j ∈ R≥0, +if λd = +d +� +k c−1 +k +(46) +where generators prefer higher prices in the market and seek to +maximize profit. At the competitive equilibrium the day-ahead +supply function (17), real-time true dispatch condition (26), +real-time clearing prices (27), and the individual optimal so- +lution (45),(46) holds simultaneously and this is only possible +if the market price is equal in the two-stages, i.e., +λd = λr = +1 +� +k c−1 +k +d, s.t dl = dd +l + dr +l +From real-time true dispatch conditions we have +gr +j + gd +j = +c−1 +j +� +k c−1 +k +d +Thus a set of competitive equilibria exists. +APPENDIX B +PROOF OF THEOREM 4 +From the day-ahead market clearing we have +� +j∈G +θd +j λd = dd =⇒ λd = +1 +� +j∈G θd +j +dd +(47) +where we assume that � +j∈G θd +j ̸= 0. Substituting (47) in +generator individual profit optimization (33a), we get the +individual problem of strategic generator j as (we assume that +dd ̸= 0 and leave the discussion of dd = 0 for later): +max +θd +j +� dd +� +k θd +k +− +d +� +k c−1 +k +� θd +j dd +� +k θd +k ++ +c−1 +j +2 +� +d +� +k c−1 +k +�2 +(48a) +where we use shorthand notation � +j to denote the sum over +the set of generators for ease of analysis. Though the individual +problem is not necessarily concave in the domain, we can +analyze the optimal bidding behavior from the first-order and +second-order conditions. Writing the first-order condition, we +have +dπj +dθd +j += +� +θd +j +� +k θd +k +� +d +� +k c−1 +k +− +2dd +� +k θd +k +� ++ +� +dd +� +k θd +k +− +d +� +k c−1 +k +�� +dd +� +k θd +k +(49) +Now summing over j ∈ G to attain the turning point of (B), +we have +=⇒ (G − 2)(dd)2 − (G − 1) +� +j θd +j +� +j c−1 +j +ddd = 0 +(50a) +where we assume that |G| ≥ 2. For the assumption dd ̸= 0, +the potential turning point is given by +θd +j = 1 +G +�� +k +c−1 +k +� +G − 2 +G − 1 +dd +d +(51) +Similarly, substituting (47) in load individual payment opti- +mization (34), we get the individual problem of strategic load +l as - +min +dd +l +dd +� +j θd +j +dd +l + +d +� +j c−1 +j +(dl − dd +l ) +(52) +The unique optimal solution to the quadratic program (52) is +given by +dd +l = +1 +L + 1 +� +j θd +j +� +j c−1 +j +d, dr +l = dl − dd +l +(53) +At equilibrium (47),(51), and (53) must hold simultaneously. +This implies that +dd = 0, θd +j = 0 =⇒ λd = λr = +1 +� +j c−1 +j +d +where we use Rule 1 to define prices in the day-ahead market. +However, this is in contradiction to our assumption and can +be rejected. +In the case of dd = 0, +• If � +j θd ̸= 0, then solving (47) and (53) simultaneously +implies that � +j θd = 0, which contradicts our assump- +tion. +• If � +j θd = 0, then we define prices using the Rule 1 in +the day-ahead market. However, in this case, loads have +the incentive to deviate from the equilibrium by allocating +some demand in the day-ahead market since λd = 0, +meaning loads make zero payment in the market, using +Rule 2. +Therefore the equilibrium does not exist. Similarly, in the case +of only one generator, equilibrium does not exist. Though the +generator bids arbitrary small values in the day ahead to earn +increasing revenue, the load will also bid small quantities to +decrease its payment. Since the generator operates truthfully +in real-time, we attain the same equilibrium with all the +demand allocated to the real-time market. Again, loads have +the incentive to deviate and allocate demand in the day ahead +where prices are zero. This completes the proof of Theorem 4. + +12 +APPENDIX C +PROOF OF THEOREM 5 +Under price-taking behavior, the individual problem for +loads (39) is a linear program with the closed-form solution +given by: +� +� +� +� +� +� +� +dd +l = ∞, dr +l = −∞, dd +l + dr +l = dl, if +dd +� +k c−1 +k +< λr +dd +l = −∞, dr +l = ∞, dd +l + dr +l = dl, if +dd +� +k c−1 +k +> λr +dd +l + dr +l = dl, +if +dd +� +k c−1 +k += λr +(54) +where loads prefer the lower price in the market. Similarly, +solving concave individual problem of each generator (38) by +taking the derivative, we have +λr +� +(1−cjθr +j )λr− +dd +� +k c−1 +k +� +=0 =⇒ θr +j λr=c−1 +j λr− c−1 +j dd +� +k c−1 +k +(55) +where we assume λr ̸= 0. Summing (55) over j ∈ G and +using real-time market clearing +� +j∈G +θr +jλr = dr +(56) +we get +dr = +� +j +c−1 +j λr − +� +j c−1 +j dd +� +k c−1 +k +=⇒ λr = +d +� +j c−1 +j +(57) +At equilibrium (36), (54), (55), and (57) must hold simultane- +ously. This implies that +λd = λr = +d +� +j c−1 +j +, dd = d, dr = 0 +(58) +gd +j = +c−1 +j d +� +k c−1 +k +, gr +j = 0, θr +j = 0 +(59) +where we use Rule 1 to define prices in the real-time market. +Although dr = 0 and � +j θr +j = 0, loads do not have the +incentive to deviate by allocating demand in the real-time +market. Since loads do not anticipate the effect of their bid +on the market prices, the payment remains the same for any +allocation across the two stages. Therefore the equilibrium +exists, and this completes the proof of Theorem 5. +APPENDIX D +PROOF OF THEOREM 6 +Using the real-time clearing (56), we have +λr = +dr +� +j θr +j +(60) +where we assume that � +j θr +j ̸= 0. Substituting (60) in +generator individual problem (41a), the individual problem of +price-anticipating generator j is given by: +max +θr +j +c−1 +j dd2 +(� +k c−1 +k )2 + +θr +jdr2 +(� +k θr +k)2 − cj +2 +� +c−1 +j dd +� +k c−1 +k ++ θr +jdr +� +k θr +k +�2 +(61) +We again use first-order and second-order conditions to ana- +lyze the optimal bidding behavior since the individual problem +may not be concave in the domain. Writing the first-order +condition, we have +dπj +dθr +j += +dr +(� +k θr +k)3 +� +mr +j − nr +jθr +j +� +(62) +where +mr +j := dr � +k,k̸=j +θr +k − +dd +� +k c−1 +k +( +� +k,k̸=j +θr +k)2 +and +nr +j := dr + +dd +� +k c−1 +k +� +k,k̸=j +θr +k + cjdr � +k,k̸=j +θr +k +Assuming generators are homogeneous and bid symmetrically, +we can rewrite (62) as +dπj +dθr +j += +dr +G3θr2 [dr(G − 2) − cd(G − 1)θr] +(63) +then the turning point is given by +θr +p = G − 2 +G − 1 +dr +cd +(64) +Writing the second-order condition and evaluating for homo- +geneous generators that bid symmetrically, i.e., the turning +point (64), we have +d2πj +dθr +j +2 +���� +θr +j =θrp(dr) += +dr +(� +j θr +j)3 +� +˜mr +j + ˜nr +jθr +j +� ���� +θr +j =θrp(dr) +(65) += − c3(G − 1)4 +G4(G − 2)3 +� d +dr +�3 � +2 + (G − 2)dr +d +� +(66) +where +˜mr +j := − +4dr � +k,k̸=j θr +k +� +k θr +k ++ +2dd� +k,k̸=j θr +k +� +k c−1 +k +− +cjdr(� +k,k̸=j θr +k)2 +� +k θr +k +and +˜nr +j := +2dr +� +k θr +k ++ +2cjdr � +k,k̸=j θr +k +� +k θr +k +Now, loads acting as leaders anticipate the clearing prices +and optimal bids of generators in the real-time subgame +equilibrium, such that +λr = G − 1 +G − 2 +cd +G +(67) +where we substitute (64) in (60). Substituting (67) in load +individual problem (42), we have +min +dd +l +dd +� +j c−1 +j +dd +l + G − 1 +G − 2 +cd +G (dl − dd +l ) +(68) +The unique optimal solution to the quadratic program (68) is +given by +dd +l = +1 +L + 1 +G − 1 +G − 2d, dr +l = +� +dl − +1 +L + 1 +G − 1 +G − 2d +� +(69) +Assuming L < G − 2, +dr > 0 =⇒ d2πj +dθr +j +2 +���� +θr +j =θrp +< 0 + +13 +Thus the obtained equilibrium maximizes generators’ profit +and minimizes loads’ payment while the supply-demand bal- +ance is satisfied. However, if L > G − 2, then +dr < 0 =⇒ θr +p < 0 =⇒ d2πj +dθr +j +2 +���� +θr +j =θrp +> 0 +The obtained equilibrium minimizes generators’ profit, and +generators’ have the incentive to deviate from this equilibrium. +Therefore, symmetric equilibrium does not exist in this case. +Moreover, in the case of |G| < 3, generators have the incentive +to bid arbitrarily small values and earn arbitrarily large profits +in the market. +In the case of L = G − 2, at equilibrium dr = 0 which +contradicts our initial assumption. We analyze the case dr = 0 +separately, +1) If � +j θr +j ̸= 0 +=⇒ +λr = 0 and λd = +c +Gdd = +c +Gdd. +WLOG, we can assume that dr +l = 0, ∀l ∈ L, otherwise +load l with non-zero demand has the incentive to deviate +and participate in the real-time market to minimize its +payment. The payment of individual load l is then given +by +λddd +l + λrdr +l = c +Gd2 +However if load l unilaterally decides to deviate by +allocating demand in real-time, i.e., dr +l = ϵ then the +payment is given by +λddd +l + λrdr +l = c +G(d − ϵ)2 + +ϵ2 +� +j θj +which is smaller for small enough ϵ. Therefore the +equilibrium does not exist. +2) If � +j θr +j = 0, using Rule 1 we have λr = λd and +λd = +c +Gdd = +c +Gdd. However if load l unilaterally +decides to deviate by allocating demand in real-time i.e., +dr +l = ϵ then using Rule 2 λr = 0. Therefore load has +the incentive to deviate and allocate demand in the real- +time market with zero clearing price. Hence equilibrium +does not exist. +This completes the proof of the Theorem 6. + diff --git a/NNE3T4oBgHgl3EQfBglT/content/tmp_files/load_file.txt b/NNE3T4oBgHgl3EQfBglT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..522b989c1c85cdd752040d48eeb651bca3ed60b2 --- /dev/null +++ b/NNE3T4oBgHgl3EQfBglT/content/tmp_files/load_file.txt @@ -0,0 +1,825 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf,len=824 +page_content='1 Market Power Mitigation in Two-stage Electricity Market with Supply Function and Quantity Bidding Rajni Kant Bansal, Yue Chen, Pengcheng You, Enrique Mallada Abstract—The main goal of a sequential two-stage electricity market—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', day-ahead and real-time markets—is to operate efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, the price difference across stages due to inadequate competition and unforeseen circumstances leads to undesirable price manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' To mitigate this, some Inde- pendent System Operators (ISOs) proposed system-level market power mitigation (MPM) policies in addition to existing local policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' These policies aim to substitute noncompetitive bids with a default bid based on estimated generator costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, these policies may lead to unintended consequences when implemented without accounting for the conflicting interest of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In this paper, we model the competition between generators (bidding supply functions) and loads (bidding quantity) in a two-stage market with a stage-wise MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' An equilibrium analysis shows that a real-time MPM policy leads to equilibrium loss, meaning no stable market outcome (Nash equilibrium) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' A day-ahead MPM policy, besides, leads to a Stackelberg-Nash game with loads acting as leaders and generators as followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In this setting, loads become winners, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', their aggregate payment is always less than competitive payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, comparison with standard market equilibrium highlights that markets are better off without such policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Finally, numerical studies highlight the impact of heterogeneity and load size on market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Index Terms—electricity market, two-stage settlement, supply function bidding, Stackelberg game, equilibrium analysis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' INTRODUCTION M OST wholesale energy markets in the US conduct auctions to settle electricity transactions where suppli- ers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', generators, offer to supply energy as a function of price, while consumers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', utilities, offer to consume energy to meet their demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' After all the bids are submitted, the independent system operators (ISOs) or regional transmission organizations (RTOs) clear the market to achieve a supply- demand balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Such a market operation may constitute multiple stages at various time scales, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', week-ahead, day- ahead, hour-ahead, 5-minute, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In practice, many ISOs consider a two-stage settlement system, namely day-ahead and real-time markets, as a norm in the market [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The first stage, the day-ahead (forward) market, clears a day before the delivery based on the hourly forecasts of resources for the next day and accounts for the majority of energy trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The second stage, the real-time (spot) market, occurs at a faster timescale R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Bansal and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Mallada are with the Whiting School of Engi- neering, Johns Hopkins University, Baltimore, MD 21218, USA (email: {rbansal3,mallada}@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='edu) Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Chen is with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China (email:yuechen@mae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='hk) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' You is with the College of Engineering, Peking University, Beijing, China (email:pcyou@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='cn) (typically every five minutes) and is considered a last resort for participants to adjust their commitment following forecast errors [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The main goal of such a sequential two-stage settlement market is to operate efficiently and ensure proper incentives to encourage participation in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, the often price difference between the two stages in practice, due to intrinsic uncertainty in the forecast, unscheduled maintenance or outage, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', creates opportunities for price speculation and arbitrage, which could be further exploited by strategic participants to their benefit, signaling efficiency losses [4]– [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' To discourage suppliers from exploiting consumers, most operators employ an inbuilt local market power mitigation mechanism (LMPM) triggered at congestion during market clearing [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Despite this, some operators, like California Independent System Operator (CAISO), have documented periods of time with non-competitive bids (approximately 2% hours in the case of CAISO [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' It led to the development of initiatives aimed at implementing system-level market power mitigation (MPM), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', bid mitigation similar to LMPM, but system-wide for each stage separately [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Even though it is reasonable but too aggressive to implement such mitigation policies in both day-ahead and real-time, these market modifications are being considered separately, starting with the real-time stage in the first phase and followed by the day-ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In particular, CAISO argues that real-time is more susceptible to market power and day-ahead is relatively competitive due to additional measures like virtual bidding that add competitive pressure on market clearing [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Such system-level policies, when implemented, substitute in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', real-time or day-ahead, any non-competitive bids with default bids, which estimate generator costs based on the opera- tor’s knowledge of technology, fuel prices, and operational constraints [12, §30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Although such market policies are straightforward, their effect on market outcome remains unknown if implemented without accounting for the con- flicting interest of individual participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This paper studies the proposed system-level policies and discusses the possible unintended effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Precisely, we study a sequential game formulation in a two- stage market with an MPM policy to analyze the competition between generators (bidding supply functions and seeking to maximize individual profit) [13], [14] and loads (bidding demand quantities and minimizing payment) [15] such that the market operator substitutes generators’ bids with default bids as per the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In this paper, we assume that an operator accurately estimates the truthful cost of dispatching the generator in a stage with an MPM policy such that the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='04267v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='OC] 11 Jan 2023 2 market clears efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We show that a real-time MPM policy results in a loss of market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, the complimentary case of a day-ahead MPM policy leads to a form of Stackelberg-Nash game with loads leading generators in their decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' A detailed Nash equilibrium analysis for this case shows a stable market outcome that favors loads, unlike the standard two-stage market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Contributions of this work: The main contributions of this paper are summarized below: 1) We show that a real-time MPM policy leads to a Nash game in the day-ahead, while generators participate truthfully in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We characterize the competitive equilibrium of such a game, which is optimal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t the social planner’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, competition between price-anticipating participants does not result in a stable market outcome, and a Nash equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2) We then study the impact of a day-ahead MPM policy that leads to a generalized Stackelberg-Nash game with loads acting as leaders in the day-ahead market and generators acting as followers in the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The competitive equilibrium of such a game also aligns with the social planner problem, while the Nash equi- librium, assuming that generators are homogeneous and bid symmetrically for closed-form analysis, results in load-favoring market outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 3) To understand the impact of such policies, we compare the resulting day-ahead MPM policy market equilibrium with equilibrium in a standard market, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', a two- stage market without any mitigation policies, studied extensively in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The closed-form analysis shows that prices across stages are the same for the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, loads acting as leaders in a market with a day-ahead MPM policy allocates higher demand in day-ahead at the expense of generators’ profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' It leads to an unintended consequence of additional market power instead of mitigation compared to the standard two-stage markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 4) We further provide a detailed numerical study to il- lustrate the impact of a day-ahead MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We show that the Nash equilibrium converges to competitive equilibrium as the number of participants increases in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Furthermore, the case with heterogeneity in generator cost shows that expensive generators are least affected when benchmarked with the competitive equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The case with significant diversity in load participants reveals that a sufficiently smaller load could earn money instead of making payments at the expense of larger loads in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Related work: To date, the problem of characterizing the effect of system-level market power mitigation policies in two- stage markets has not been considered by the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Sev- eral works have investigated the competition between various market players in different market settings and characterized the effect of mixed bidding mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In particular, refer- ences [13], [14], [16]–[18] focus on the strategic behaviors in a single-stage market and characterize the resulting Nash equi- librium, where generators bid linear supply function and max- imize their profit and load is either inelastic [13], [14], elastic and bids a supply function [14], [16], [17] or quantity [15], [18] to minimize their payment or equivalently maximize the utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Further in this line of research, references [19]– [22], looks at cross-group participation models in a two-stage market setting, analyzing perfect competition [19], [20], where participants accept existing prices in the market, and strategic behavior [21], [22], where participants manipulate prices in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Another line of work looks at MPM in electricity markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In particular, the role of forward contracting [23], [24], demand shifting [25], [26], bidding capacity division and constraint [27], promoting interconnection and capacity regulation [28], [29] and LMP-based default energy bids [30], as a tool for local market power mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, despite the extensive studies on cross-group participation and local market power mitigation, to the best of our knowledge, our work is the first one to study the impact of a system-level market power mitigation in a two-stage market and formally analyze the effect of default bid substitution on the market outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Paper Organization: The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In Section II we first introduce the social planner problem, two-stage market model, and participants’ behavior, and then define a two-stage market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In Section III we model the market power mitigation policy for each stage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', day-ahead and real-time, characterize the competitive and Nash equilibrium, and compare it with the optimal solution to the social planner problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We first compare the impact of MPM policies on market equilibrium and then compare it with a standard market equilibrium in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Numerical studies on market power for a day-ahead MPM policy and conclusions are in Section V and VI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Notation: We use standard notation f(a, b) to denote a function of independent variables a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, we use f(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' b) to represent a function of independent variable a and parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' MARKET MODEL In this section, we formulate the social planner problem and describe the standard two-stage settlement market design, where a generator submits a supply function and demand bids quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We then formally define participants’ behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', price-taking or price-anticipating, and lay out a general market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Social Planner Problem Consider a single-interval two-stage settlement market where a set G of generators participate with a set L of inelastic loads to meet inelastic aggregate demand d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Each generator j ∈ G supplies gj ∈ R and each inelastic load l ∈ L consumes dl ∈ R respectively, where � l∈L dl = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We define G := |G| and L := |L| to denote the number of generators and loads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Assuming a convex cost function Cj(gj) for each generator j ∈ G, respectively, the social planner problem — minimum cost of meeting aggregate inelastic demand — is given by: 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Two-stage Market Mechanism SOCIAL PLANNER min gj,j∈G � j∈G Cj(gj) (1a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' � l∈L dl = � j∈G gj (1b) where (1b) enforces the supply-demand balance in the two- stage market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Two-Stage Market Mechanism In this subsection, we define the two-stage market clearing, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The net output gj of each generator j is distributed between gd j and gr j such that gj = gd j + gr j, (2) where gd j , gr j represent power output in day-ahead and real- time markets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Similarly, each load l allocates its inelastic demand dl over two stages such that dd l + dr l = dl (3) where dd l , dr l represent load allocation in day-ahead and real- time markets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 1) Day-Ahead Market: The power output of each generator j ∈ G in the day-ahead market is denoted by gd j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Each generator j submits a supply function h : R × R → R, parameterized by θd j , that indicates willingness of generator j to supply gd j as a function of price gd j = h(λd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θd j ) (4) where λd denotes the price in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Each load l ∈ L bids quantity dd l in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Based on the bids (θd j , dd l ) from participants, the market operator clears the day-ahead market to meet the supply-demand balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' � j∈G h(λd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θd j ) = dd (5) The optimal solution to the day-ahead dispatch problem (5) gives the optimal dispatch (gd j , dd l ) and clearing prices λd to all the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Each generator j ∈ G and load l ∈ L are paid λdgd j and λddd l as part of the market settlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2) Real-Time Market: The power output of each generator j in real-time market is denoted by gr j and their bid is: gr j = h(λr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θr j) (6) where λr denotes the price in the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The supply function bid is parameterized by θr j, indicating the willingness of generator j to supply gr j at the price λr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Each load l ∈ L submits the quantity bids dr l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Given the bids (θr j, dr l ), the operator clears the real-time market to meet the supply-demand balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' � j∈G h(λr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θr j) = dr (7) Similar to the day-ahead market clearing, the optimal solution to the dispatch problem (7) gives the optimal dispatch and the market clearing prices λr to all the participants, such that each generator j ∈ G and load l ∈ L produces or consumes gr j and dr l , and is paid or charged λrgr j and λrdr l , respectively, as part of the market settlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 3) Market Rules and Goal: In this section, we first define a set of rules to account for degenerate cases in the market mechanism and then discuss the goal of a two-stage market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Rule 1: For v ∈ {d, r} and w ∈ {d, r}, if the net supply and demand of the generators and loads in a stage follow � j∈G h(λv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θv j ) = 0, dv = 0 =⇒ λv = λw, v ̸= w i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', the clearing price in that stage is set to the clearing prices of the other stage with a non-zero demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Rule 2: For v ∈ {d, r}, if the net supply and net demand of the generators and loads in a stage follow � j∈G h(λv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θv j ) = 0, dv ̸= 0 =⇒ λv = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', the clearing price is set to zero, and demand is split evenly across all the loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We are interested in two-stage market outcomes that satisfy � j∈G (gd j + gr j) = � j∈G gj = � l∈L (dd l + dr l ) = � l∈L dl = d (8) and solve the social planner problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Though the market outcome may deviate from the optimal social planner solution, signaling efficiency losses due to price manipulation by partic- ipants, we quantify such deviations to understand the behavior of participants and the market outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Participant Behaviour In this section, for the purposes of our study, we introduce two different types of rational participants’ behavior, price- taking, and price-anticipating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Each generator j ∈ G seeks to maximize their profit πj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' given by: Generator Profit πj(gd j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' gr j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λr):=λrgr j+ λdgd j −Cj(gj) (9) Each load l ∈ L aims to minimize their payments ρl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' as: Load Payment ρl(dd l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' dr l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λr) := λddd l + λrdr l (10) Substituting the coupling constraint for the load allocation across two stages (3) in (10) we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' ρl(dd l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λr) := λddd l + λr(dl − dd l ) (11) For each load l ∈ L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' the allocation in the day-ahead market dd l determines its allocation in the real-time market dr l due to the demand inelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Individual generator i E G C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='(gj) Day-Ahead market clearing Real-Time market clearing h(X";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' ,) = d" h(αd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' g) = dd 53!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' jEg dd t Individual Load l E L di = dd + dr4 1) Price-Taking Participants: A price-taker participant is defined below: Definition 1: A market participant is price-taking if it ac- cepts the existing prices in the market and does not anticipate the impact of its bid on the market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Given the prices in the day-ahead market λd and real-time market λr, the generator individual problem is given by: Price-taking Generator Bidding problem max gd j ,gr j πj(gd j , gr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd, λr) (12) Similarly given the prices λd, λr, the individual bidding prob- lem for load is given by: Price-taking Load Bidding problem min dd l ρl(dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd, λr) (13) We next define the price-anticipating (or strategic) participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2) Price-Anticipating Participants: A price-anticipating participant is defined below: Definition 2: A market participant is price-anticipating (strategic) if it anticipates the impact of its bid on the prices in two stages and has complete knowledge of other participants’ bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The individual problem of a price-anticipating generator is: Strategic Generator Bidding problem max gd j ,gr j ,λd,λr πj � gd j , gr j, λd� gd j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' gd −j, dd� , λr� gr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' gr −j, dr�� (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (5), (7) (14b) where gd −j := � k∈G,k̸=j gd k, and gr −j := � k∈G,k̸=j gr k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The generator j maximizes its profit while anticipating the market clearing prices in the day-ahead and real-time market (5),(7), along with complete knowledge of load bids dd l , dr l , l ∈ L, and other generators’ bids θd k, θr k, k ∈ G, k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Similarly, the indi- vidual problem for strategic load l with complete knowledge of prices in two stages (5),(7) and other participants’ bids: Strategic Load Bidding problem min dd l ,λd,λr ρl � dd l , λd� dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' gd j , dd −l � , λr� dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' gr j, dr −l �� (15a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (5), (7) (15b) where the load l minimizes its payment in the market and d d −l := � l∈L,k̸=l dd l , d r −l := � l∈L,k̸=l dr l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Market Equilibrium In this section, for the purpose of this study, we characterize the market equilibrium in a two-stage settlement electricity market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' At the equilibrium, no participant has any incentive to deviate from their bid, and the market clears, as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Definition 3: We say the participant bids and market clearing prices (θd j , θr j, j ∈ G, dd l , dr l , l ∈ L, λd, λr) in the day-ahead and real-time respectively form a two-stage market equilibrium if the following conditions are satisfied: 1) For each generator j ∈ G, the bid θd j , θr j maximizes their individual profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2) For each load l ∈ L, the allocation dd l , dr l minimizes their individual payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 3) The inelastic demand d ∈ R is satisfied with the market- clearing prices λd given by (5) and λr given by (7) over the two-stages of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We will study market equilibria as a tool to understand the impact of MPM policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' UNDERSTANDING IMPACT OF MPM POLICY In this section, we first characterize the market equilibrium in a standard two-stage market without any such mitigation policy [22], then model mitigation policies and the resulting market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In particular, ISOs have significant prior knowledge of market participants allowing them to evaluate the competitiveness of energy bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' For example, operators are aware of the generator’s technology, fuel prices, and oper- ational constraints that can be used to estimate or bound the generator’s cost [12, §30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='3] within a reasonable threshold under the mitigation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Although it is reasonable to implement such mitigation policies in both day-ahead and real-time, CAISO argues that real-time is more susceptible to market power and day-ahead is relatively competitive [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore, these market modifications are being considered separately, starting with the real-time stage in the first phase and followed by the day-ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' For ease of exposition, we assume that the operator accurately estimates the truthful cost of dispatching the generator in a stage with a mitigation policy such that the market clears efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Also, we assume a quadratic cost function for each generator j, parameterized by cost coefficient cj, Cj(gj) = cj 2 gj 2 (16) We further assume that the supply function bid h(λv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θv j ) of each generator j is given by: gv j := h(λv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θv j ) = θv j λv, v ∈ {d, r} (17) where the parameter θv j ∈ R explicitly indicate willingness of generator j to produce gv j per unit price λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Standard Two-stage Market The role of participants in a standard market without any mitigation policy is studied extensively in the literature [13], [14], [22], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Here, we cite the results from [22] that analyze the role of strategic generators and inelastic demand in a standard two-stage market and use them as a benchmark to analyze the impact of a mitigation policy in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 1) Price-taking Participation and Competitive Equilibrium: For the individual incentive problem in a two-stage market, substituting the cost function (16) and supply function (17) in (9), we get πj(θd j , θr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='λd, λr)= θd j λd2+θr jλr2− cj 2 � θd j λd+θr jλr�2 (18) and the individual problem for price-taking generator j is: max θd j ,θr j πj(θd j , θr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='λd, λr) (19) 5 Similarly, the individual problem for load l is given by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Given the prices, λd, λr, we next characterize the resulting competitive equilibrium due to competition between price- taking participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Theorem 1 (Proposition 1 [22]): A competitive equilibrium in a two-stage market exists and is explicitly given by θd j + θr j = 1 cj , θd j ≥ 0, θr j ≥ 0, ∀j ∈ G (20) dd l + dr l = dl, ∀l ∈ L (21) λd = λr = d � j∈G c−1 j (22) The resulting competitive equilibrium solves the social planner problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, it exists non-uniquely, and there is no incentive for a load to allocate demand in the day-ahead market due to equal prices in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2) Price-Anticipating Participation and Nash Equilibrium: The individual problem of price-anticipating generator j and price-anticipating load l is given by (14) and (15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We next characterize the resulting Nash equilibrium in such a market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Theorem 2 (Proposition 4 [22]): Assuming strategic gener- ators are homogeneous (cj := c, ∀j ∈ G) and make identical bids (θv j := θv, ∀j ∈ G, v ∈ {d, r}) at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' If there are at least three firms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', G ≥ 3, a Nash equilibrium in a two-stage market exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Further, this equilibrium is unique and explicitly given by θd j =L(G − 1) + 1 L(G − 1) G − 2 G − 1 1 c , θr j= 1 L + 1 (G − 2)2 (G − 1)2 1 c (23) dd l = L(G − 1) + 1 L(L + 1)(G − 1)d, dr l= dl − dd l (24) λd = L L + 1 G − 1 G − 2 c Gd, λr = G − 1 G − 2 c Gd (25) The resulting Nash equilibrium exists uniquely, where price- anticipating loads anticipate the actions of generators and allocate demand to exploit lower prices in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Thus prices are different in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, the net demand allocation in the day-ahead and real-time market follows dd ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='5d, d), dr ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Real-Time MPM Policy In this section, we first discuss the modified market model, the individual incentives of participants, and then characterize market equilibrium for a real-time MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 1) Modeling Real-Time MPM Policy: In the case of a real- time MPM policy, the market ignores generators’ bids in real-time, as shown in Figure 2, and accurately estimates the truthful market clearing, given the day-ahead dispatch gd j gr j = h(λr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θr j) = c−1 j λr − gd j (26) Using the two-stage generation and supply-demand balance (8) and real-time dispatch (26) we get λr = d � j c−1 j (27) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Two-stage Market Mechanism with Real-Time MPM 2) Price-taking Participation and Competitive Equilibrium: For the individual incentive problem in a two-stage market with real-time MPM policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' substituting the cost function (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' day-ahead supply function (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' real-time true dispatch condi- tion (26) and real-time clearing prices (27) in (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' we get πj(θd j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='λd)= θd j λd2+ d � j c−1 j � c−1 j d � j c−1 j − θd j λd � − cj 2 � c−1 j d � j c−1 j �2 = � λd − d � j c−1 j � θd j λd + c−1 j 2 � d � j c−1 j �2 (28) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' an individual problem of a price-taking generator is: max θd j πj(θd j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd) (29) Similarly, substituting the clearing price (27) in (11) we get, ρl(dd l , λd) := λddd l + d � j c−1 j (dl − dd l ) (30) such that the individual problem for load l is given by: min dd l ρl(dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λd) (31) The competition between price-taking participants for indi- vidual incentives leads to a set of competitive equilibria, as characterized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Theorem 3: The competitive equilibrium in a two-stage market with a real-time MPM policy exists, and given by: gd j + gr j = c−1 j � j∈G c−1 j d, θd j ∈ R≥0 ∀j ∈ G (32a) dd l + dr l = dl, ∀l ∈ L (32b) λd = λr = 1 � j∈G c−1 j d (32c) We provide proof of the theorem in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' At the competitive equilibrium, the market clearing prices are equal in the two stages, meaning there is no incentive for a load to allocate demand in the day-ahead market, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', current market practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, the equilibrium still attains the desirable social planner objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Corollary 1: The competitive equilibrium in a two-stage market with a real-time MPM policy (32) also solves the social planner problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Individual generator i E G C(i) Day-Ahead market clearing Real-Time market clearing h(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' g) = ogd = dd jEg jEg jEg Jdr Individual Load l E L di = dj + dr6 3) Price-Anticipating Participation and Nash Equilibrium: The individual problem of each price-anticipating generator j, given by: max θd j ,λd πj � θd j , λd� θd j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θ d −j, dd�� (33a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (5) (33b) where the generator j maximizes its profit in the two-stage market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The individual problem of price-anticipating load is given by: min dd l ,λd ρl � dd l , λd� dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θd j , d d −l �� (34a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (5) (34b) where the load l minimizes its payment in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We study the resulting sequential game where players anticipate each other actions and prices in the market, and the day-ahead clears before the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' To this end, we analyze the game backward, starting from the real-time market, where prices are fixed due to MPM policy (27), followed by the day-ahead market, where participants make decisions for optimal individual incentives and compute the equilibrium path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Generators do not bid in real-time, but loads are allowed to bid in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, load makes decisions simul- taneously in the day-ahead market due to inelasticity, fixing their bids in the real-time market, which affects the two-stage market clearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The following theorem characterizes the two- stage Nash equilibrium that satisfies the Definition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Theorem 4: The Nash equilibrium in a two-stage market with a real-time MPM policy does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We provide proof of the theorem in Appendix B and a brief in- sight below into the loss of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The price-anticipating participants compete with each other to manipulate prices in the day-ahead given by substituting (17) in (5): λd = dd � j θd j while the prices in the real-time λr (27) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Loads bid decreasing quantities dd l to reduce clearing prices in the day- ahead market and minimize the load payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Simultaneously, generators bid decreasing parameter θd j to increase clearing prices and maximize revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The competition between loads and generators for individual incentives in the day-ahead mar- ket drives all the demand to the real-time market, where gen- erators operate truthfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, in our market mechanism, loads then have the incentive to deviate and allocate demand in the day ahead where prices are zero, meaning zero payment in the market, see Rule 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Such unilateral load deviations result in deviations from generators to increase clearing prices in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore the equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Without such a market rule, the Nash equilibrium does exist with undefined clearing prices in the day-ahead and all demand allocated to the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Nevertheless, since day- ahead accounts for a majority of energy trades, the resulting equilibrium is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Two-stage Market Mechanism with Day-Ahead MPM C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Day-Ahead MPM Policy In this section, we define the individual incentive of par- ticipants and characterize market equilibrium for a day-ahead MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 1) Modeling Day-Ahead MPM Policy: In the case of a day- ahead MPM policy, as shown in Figure 3, the market ignores the generators’ bids and accurately estimates the truthful market clearing in the day-ahead as given by: gd j = h(λd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θd j ) = 1 cj λd (35) Moreover, using the day-ahead power balance constraint, we have λd = dd � j c−1 j (36) 2) Price-taking Participation and Competitive Equilibrium: For the individual incentive problem in a two-stage mar- ket with a day-ahead MPM policy, substituting the clearing price (36) in (9), we get πj(θr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='λr)= c−1 j dd2 (� j c−1 j )2 +θr jλr2− cj 2 � c−1 j dd � j c−1 j +θr jλr �2 (37) and the individual problem for price-taking generator j is: max θr j πj(θr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λr) (38) Similarly, substituting the clearing price (36) in (11) we get the individual problem for price-taking load l, min dd l ρl(dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λr) = min dd l dd � j c−1 j dd l + λr(dl − dd l ) (39) Each price-taking generator j solves individual prob- lem (38) and load l solve individual problem (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The resulting competitive equilibrium given the clearing prices λd and λr is characterized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Theorem 5: The competitive equilibrium in a two-stage market with a day-ahead MPM policy exists and is given by: gd j = c−1 j � j∈G c−1 j d, gr j = 0, ∀j ∈ G (40a) dd l + dr l = dl, ∀l ∈ L, dd = d, dr = 0 (40b) θr j = 0, λd = λr = 1 � j∈G c−1 j d (40c) We provide proof of the theorem in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Unlike the competitive equilibrium for a real-time MPM policy in (32) Individual generator i E G X Day-Ahead market clearing Real-Time market clearing h(αd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 0g)=d: h(xr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' o) =,xr = d iEg jEg jEg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content="53 d' Individual Load l E L lp +pp = lp7 with equal prices across stages, the loads at equilibrium (40) allocate all the demand in the day-ahead." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The incentive for day-ahead demand allocation is a desired market outcome and is not generally satisfied by other market mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Although dr = 0 and � j θr j = 0, unlike in Theorem 4, loads do not deviate from the equilibrium by shifting demand to the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The resulting equilibrium exists as price- taking loads do not anticipate the effect of their bid on the market prices, meaning the payment remains the same for any allocation across the two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, the market outcome (40) solves the social planner problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 3) Price-Anticipating Participation and Nash Equilibrium: The individual problem of each price-anticipating generator j, given by: max θr j ,λr πj � θr j, λd� dd� , λr� θr j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θ r −j, dr�� (41a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (7) (41b) where generator j maximizes its profit in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The individual problem of price-anticipating load l, is given by: min dd l ,λr ρl � dd l , λd� dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' d d −l � , λr� dd l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θr j, d r −l �� (42a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (7) (42b) where load l minimizes its payment in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In the market model with a day-ahead MPM policy, genera- tors make decisions in real-time while load can make decisions in the day-ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The resulting two-stage sequential game is essentially a leader-follower Stackelberg-Nash game, where generators are followers in the real-time market and loads are leaders in the day-ahead market, and each participant in their respective groups competes amongst themselves in a Nash game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We follow the terminology used in [32] to describe similar formulations in different markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' For the closed form solution, we assume that generators are homo- geneous in the sense that they share the same cost coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' cj =: c, ∀j ∈ G and bid symmetrically in the market, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' θr j =: θr, ∀j ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Under these assumptions, the Nash equilibrium is characterized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Theorem 6: Assume that generators are homogeneous and bid symmetrically in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' If more than two generators are participating in the market i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' G ≥ 3 and the number of individual loads participating in the market satisfies L < G−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' then the symmetric Nash equilibrium in a two-stage market with a day-ahead MPM policy exists and is uniquely given by: gd j = L L + 1 G − 1 G − 2 d G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' gr j= � 1 − L L + 1 G − 1 G − 2 � d G (43a) dd l = 1 L + 1 G − 1 G − 2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' dr l= � dl − 1 L + 1 G − 1 G − 2d � (43b) θr = 1 c �G − 2 G − 1 − L L + 1 � (43c) λd = L L + 1 G − 1 G − 2 c Gd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' λr = G − 1 G − 2 c Gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (43d) Moreover, for L ≥ G − 2, a symmetric equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We provide proof of the Theorem in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Unlike the market with a real-time MPM policy, the Nash equilibrium exists in the market with a day-ahead MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, it requires restrictive conditions on the number of participants in the market and may not even exist in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We discuss these cases with no symmetric Nash equilibrium and provide intuition into participants’ behavior in the market: a) L > G − 2: In this case, the net demand is negative in the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The first order condition implies that each generator j acts as load, paying λrgr j as part of the market settlement since their optimal bid θr j < 0 and the real-time clearing price λr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, if the generators bid θr j > 0, then the linear supply function implies that each generator j dispatch gr j < 0 at the clearing prices λr < 0 earning revenue in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, this is not desirable from a load perspective since they are making payments in the market and they have the incentive to deviate to minimize their payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Hence, symmetric equilibrium with negative demand in the real-time market does not exist as the symmetric bid θr j > 0 does not satisfy the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The dependence of the individual bid θr j on the given bids from other participants makes the closed-form analysis challenging, and any guarantee of the existence of equilibrium is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' b) L = G − 2: In this case, no symmetric Nash equilib- rium exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Loads take advantage of the truthful participation of generators in the day-ahead market and their ability to anticipate the impact of bids on the clearing prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Regardless of generators’ bids, loads have the incentive to deviate by al- locating demand in the real-time market with a lower clearing price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Corollary 2: For L < G − 2, the demand allocation at the Nash equilibrium (43) in a two-stage market with a day-ahead MPM policy is given by: � l∈L dd l = L L+1 G−1 G−2d ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='5d, d) (44a) � l∈L dr l = −L+G−2 (L+1)(G−2)d ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='5d) (44b) The proof uses the fact that L < G−2 implies that dr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' EQUILIBRIUM ANALYSIS In this section, we study the properties of market equilib- rium under the proposed policy framework and compare it with the standard market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Comparison of a stage-wise MPM Policy An MPM policy in real-time either provides no incentive for loads to allocate demand in the day-ahead at competi- tive equilibrium or leads to no Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, an MPM policy in the day-ahead leads to a stable market outcome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', load allocates all or a fraction of demand in the day-ahead market at competitive equilibrium (40b) and Nash equilibrium (43b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This is summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We further analyze the case of a day-ahead MPM policy to study the strategic behavior of participants while regarding the respective competitive equilibrium in Theorem 5 as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In the case of a day-ahead MPM policy, loads act as leaders in the day-ahead and generators as followers in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The generator bids to manipulate prices leading to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='COMPETITIVE EQUILIBRIUM (CE) AND NASH EQUILIBRIUM (NE) WITH A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='STAGE-WISE MPM POLICY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Instance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Real-Time MPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Day-Ahead MPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='CE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Non-unique equilibrium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Unique equilibrium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Solves social planner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Solves social planner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Arbitrary demand allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='All demand in day-ahead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='NE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Does not exist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Symmetric equilibrium Homogeneous generators Social Cost same as CE Extra constraints on players ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='TABLE II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='COMPARISON BETWEEN COMPETITIVE EQUILIBRIUM (CE) AND NASH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='EQUILIBRIUM (NE) IN A MARKET WITH A DAY-AHEAD MPM POLICY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Case ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Generators total profit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='Loads total payment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='CE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='NE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G d2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G−2 − (G−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='(G−2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='(L+1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G d2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='G−2 − (G−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='(G−2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='(L+1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='inflated prices in real-time (43d) while the load allocates more ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='demand in the day-ahead (43b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' increasing prices in the day- ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Though the market equilibrium deviates from the competitive equilibrium (40), the social cost remains the same due to the homogeneous participation of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Ta- ble II summarizes the aggregate profit and aggregate payment of generators and loads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Corollary 3: The aggregate payment of loads and aggregate profit of generators at symmetric Nash equilibrium (43) is always less than that at the respective competitive equilib- rium (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The corollary follows from comparing the aggregate profit (payment) at Nash equilibrium to that at competitive equilib- rium in Table II for L < G − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Equilibrium comparison with a standard market In this section, we compare the equilibrium in a day-ahead MPM policy market to a standard market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The competitive equilibrium remains the same for the two markets with equal prices in the two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, unlike in the case of a day- ahead MPM policy, the competitive equilibrium in Theorem 1 exists non-uniquely and there is no incentive for a load to allocate demand in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Interestingly, at Nash equilibrium prices in the two stages are the same for a day-ahead MPM policy market (43d) and a standard market (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, demand allocation is different in the two market settings, due to a leader-follower structure between participants in the market with a day-ahead MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Assuming L < G − 2, at Nash equilibrium we have dd DA−MP M − dd Standard = 1 L + 1 � L G − 2 − 1 G − 1 � d > 0 meaning, a load is able to exploit the market further by allocating more demand in the day-ahead market with a lower clearing price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' To understand the impact of price-anticipating participants on market equilibrium, we compare the aggregate profit (payment) in Table II and III, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We restrict our comparison for L < G − 2 only since the Nash equilibrium in Theorem 6 does not exist otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In TABLE III COMPARISON BETWEEN COMPETITIVE EQUILIBRIUM (CE) AND NASH EQUILIBRIUM (NE) IN A STANDARD MARKET Case Generators total profit Loads total payment CE 1 2 c G d2 c G d2 NE 1 2 c G d2� G G−2 − 2L(G−1)+2 (L+1)2(G−2) � c G d2� G−1 G−2 − L(G−1)+1 (L+1)2(G−2) � particular, for L = G − 3 the aggregate profit (payment) as shown in row 2 of Table III at the Nash equilibrium in Theorem 2 equals to that of the competitive equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, for L < G−3 the aggregate profit (payment) at Nash equilibrium is always less than the competitive equilibrium, meaning the loads are winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The change in the normalized aggregate profit (payment) at the Nash equilibrium between a market with a day-ahead MPM policy and a standard market is given by µ(L, G) � 1 − L G − 2 − L � < 0, for L < G − 2 where µ(L, G) > 0 is some constant that depends on L and G, and profit (payment) is normalized with the competitive equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This implies that implementing a day-ahead MPM policy has an adverse effect and at equilibrium, the market is farther away from the competitive equilibrium in the presence of price-anticipating participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, as the number of participants increases, the difference tends to 0, since the Nash equilibrium in both cases approaches the competitive equilibrium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Figure 4 compares the total profit (payment) normalized with competitive equilibrium for a day-ahead MPM (DA- MPM) policy market and a standard market, respectively, as we change the number of loads (l ∈ L, L ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' , G − 3}), and generators (j ∈ G, G ∈ {4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' , 20}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The ratio decreases monotonically as the number of generators increases, meaning the increased competition between more generators to meet the inelastic demand gives more power to loads, allowing them to reduce their payment even further, as shown by the horizontal rows in all panels in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, the ratio increases monotonically as the number of loads increases (for a large enough number of generators), meaning the Nash equilibrium tends towards the competitive equilibrium, as shown by the vertical color columns in panels (a) and (b) in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Although loads are still winners in standard markets for L < G−3, we observe a reversal in power for L > G−3 where generators are now able to make a higher profit at the expense of loads in the market, as shown in panels (c) and (d) in the Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Unfortunately, with a day-ahead MPM policy, the equilibrium no longer exists as shown by white colored cells in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Finally, in the limit L → ∞ =⇒ G → ∞, the Nash equilibrium converges to competitive equilibrium, also shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' NUMERICAL STUDY We now analyze the impact of heterogeneity on individual profit at Nash equilibrium in the market with a day-ahead MPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Since it is hard to analyze the heterogeneity cases 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Total profit (a) and total payment (b) at Nash Equilibrium (NE) normalized with competitive equilibrium (CE) in a market with day-ahead MPM (DA-MPM), and total profit (c) and total payment (d) at Nash Equilibrium (NE) normalized with competitive equilibrium (CE) in a standard market;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' white cells denote no equilibrium Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Net (top) and normalized (bottom) individual profit at Nash Equilib- rium (NE) normalized with competitive Equilibrium (CE) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t cost coefficient of generators in closed-form, we run numerical best-response studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' To this end, we consider the case of 2 price-anticipating loads and 5 price-anticipating generators in a two-stage market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The individual aggregate inelastic load is given by dl = [99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='4, 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='6]T MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The cost coefficients of generators are sampled 10, 000 times from a Gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 and variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='001, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' cj ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='001), ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The top and bottom panel in Figure 5 plots the absolute profit and the normalized profit (normalized with the competitive equilibrium) at Nash equilibrium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The cheaper generators earn a higher profit when compared with the expensive generators with higher cost coefficients at Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, the normalized profit ratio in the bottom panel shows that expensive generators have a higher value than cheaper ones, meaning that though expensive generators have lower absolute profit, these are the least exploited in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This result is counter-intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In Figure 6 we show the absolute (top panel) and normalized (bottom panel) load payment w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t smaller load size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' For this, we keep the same number of loads and generators in the market with varying load sizes for fixed net demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We again sample cost coefficients from Gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 and variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='001, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' cj ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='001), ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The top panel shows that though the net load Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Net (top) and normalized (bottom) load individual payment (bottom) at Nash Equilibrium (NE) normalized with competitive Equilibrium (CE) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t size of smaller load d1, d1 < d2, d1 + d2 = d payment remains the same as we change the size of the load, the smaller load may even make a profit in the market at the ex- pense of a higher load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' More formally to develop intuition, in the case of homogeneous generators, the normalized payment ratio for individual load at Nash equilibrium in Theorem 6, is given by G − 1 G − 2 � 1 − 1 (L + 1)2 G − 1 G − 2 d dl � which is negative for a sufficiently small load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In particular, the smaller load has a negative normalized ratio at the expense of a higher load (a ratio greater than 1), as shown in the bottom panel of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The larger load makes more payment at Nash equilibrium than at the competitive equilibrium, while the aggregate payment of the set of loads is still less than at the competitive equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Though the heterogeneity in load size does not affect the net payment or the group behavior in the market, a smaller load makes negative payments at the expense of larger loads and can exercise more market power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (normalized with CE) Total T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (normalized with CE) Total p (normalized with CE) Total πT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' (normalized with CE) ‘j,NE:Standard PI,NE:Standard j,NE:DA-MPM 1234567890WB45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='8 of Loads (L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content="2 1 'ON 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='6 (a) (b) (c) (d) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' of Generators (G)Net (top) and normalized (with CE, bottom) individual profit at NE w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t cost coefficient c Homogeneous generators 200 Least Expensive Most Expensive (Net) 180 j,NE 160 140 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='9 (Normalized) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='88 ,NE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='13 CNet (top) and normalized (with CE, bottom) individual load payment w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t load size d Smaller load d, Larger load d2 Total load (d = d, +d,) 2000 1500 (Net) 1000 500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='5 (Normalized) Smaller load d, Larger load d2 I,NE 10 p 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='5 d,/d10 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' CONCLUSIONS We study competition between generators (bid linear supply function) and loads (bid quantity) in a two-stage settlement electricity market with a stage-wise MPM policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In the proposed policy framework, CAISO substitutes generator bids with default bids in the stage with an MPM policy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', day- ahead or real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' To understand the participant behavior in the market, we start with a real-time MPM policy and analyze the sequential game, where generators only bid in the day- ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The resulting competitive equilibrium, price- taker participants, align with the social planner problem, and loads do not have any incentive to allocate demand in the day-ahead due to equal prices across two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, the Nash equilibrium, price-anticipating participants, fails to exist, indicating an unstable market outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In the case of a day-ahead MPM policy, on the other hand, the competitive equilibrium aligns with the social planner problem, and the load allocates all the demand in the day- ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Since the day-ahead market accounts for a majority of energy trades, the incentive for day-ahead de- mand allocation is desirable from the market perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Finally, the Nash equilibrium shows that the generator fails to manipulate prices to earn higher profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Loads (leaders in day-ahead) in the resulting generalized Stackelberg-Nash game take advantage of the generators (followers in real- time), lowering their aggregate payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Counter-intuitively, our comparison with a standard market equilibrium reveals that a day-ahead MPM policy results in higher market power due to such a leader-follower structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, numerical studies for a day-ahead MPM policy show that in the case of heterogeneous generators, expensive generators are less affected in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Also, the load size diversity highlights the role of a sufficiently smaller load in exercising market power at the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 11 [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Qin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Freris, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Ho, “Multiplayer stackelberg- nash game for nonlinear system via value iteration-based integral reinforcement learning,” IEEE Transactions on Neural Networks and Learning Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 1–12, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' APPENDIX A PROOF OF THEOREM 3 Under price-taking behavior, the individual problem for loads (31) is a linear program with the closed-form solution given by: � � � � � � � dd l = ∞, dr l = −∞, dd l + dr l = dl, if λd < d � k c−1 k dd l = −∞, dr l = ∞, dd l + dr l = dl, if λd > d � k c−1 k dd l + dr l = dl, if λd = d � k c−1 k (45) where loads prefer the lower price in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The individ- ual problem for generators (29) requires: � � � � � � � � � � � θd j = −∞, if 0 ≤ λd < d � k c−1 k θd j = ∞, if λd < d � k c−1 k , andλd < 0 θd j = ∞, if λd > d � k c−1 k θd j ∈ R≥0, if λd = d � k c−1 k (46) where generators prefer higher prices in the market and seek to maximize profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' At the competitive equilibrium the day-ahead supply function (17), real-time true dispatch condition (26), real-time clearing prices (27), and the individual optimal so- lution (45),(46) holds simultaneously and this is only possible if the market price is equal in the two-stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', λd = λr = 1 � k c−1 k d, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='t dl = dd l + dr l From real-time true dispatch conditions we have gr j + gd j = c−1 j � k c−1 k d Thus a set of competitive equilibria exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 4 From the day-ahead market clearing we have � j∈G θd j λd = dd =⇒ λd = 1 � j∈G θd j dd (47) where we assume that � j∈G θd j ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Substituting (47) in generator individual profit optimization (33a), we get the individual problem of strategic generator j as (we assume that dd ̸= 0 and leave the discussion of dd = 0 for later): max θd j � dd � k θd k − d � k c−1 k � θd j dd � k θd k + c−1 j 2 � d � k c−1 k �2 (48a) where we use shorthand notation � j to denote the sum over the set of generators for ease of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Though the individual problem is not necessarily concave in the domain, we can analyze the optimal bidding behavior from the first-order and second-order conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Writing the first-order condition, we have dπj dθd j = � θd j � k θd k � d � k c−1 k − 2dd � k θd k � + � dd � k θd k − d � k c−1 k �� dd � k θd k (49) Now summing over j ∈ G to attain the turning point of (B), we have =⇒ (G − 2)(dd)2 − (G − 1) � j θd j � j c−1 j ddd = 0 (50a) where we assume that |G| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' For the assumption dd ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' the potential turning point is given by θd j = 1 G �� k c−1 k � G − 2 G − 1 dd d (51) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' substituting (47) in load individual payment opti- mization (34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' we get the individual problem of strategic load l as - min dd l dd � j θd j dd l + d � j c−1 j (dl − dd l ) (52) The unique optimal solution to the quadratic program (52) is given by dd l = 1 L + 1 � j θd j � j c−1 j d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' dr l = dl − dd l (53) At equilibrium (47),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='(51),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' and (53) must hold simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This implies that dd = 0, θd j = 0 =⇒ λd = λr = 1 � j c−1 j d where we use Rule 1 to define prices in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, this is in contradiction to our assumption and can be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In the case of dd = 0, If � j θd ̸= 0, then solving (47) and (53) simultaneously implies that � j θd = 0, which contradicts our assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' If � j θd = 0, then we define prices using the Rule 1 in the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, in this case, loads have the incentive to deviate from the equilibrium by allocating some demand in the day-ahead market since λd = 0, meaning loads make zero payment in the market, using Rule 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore the equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Similarly, in the case of only one generator, equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Though the generator bids arbitrary small values in the day ahead to earn increasing revenue, the load will also bid small quantities to decrease its payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Since the generator operates truthfully in real-time, we attain the same equilibrium with all the demand allocated to the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Again, loads have the incentive to deviate and allocate demand in the day ahead where prices are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 12 APPENDIX C PROOF OF THEOREM 5 Under price-taking behavior, the individual problem for loads (39) is a linear program with the closed-form solution given by: � � � � � � � dd l = ∞, dr l = −∞, dd l + dr l = dl, if dd � k c−1 k < λr dd l = −∞, dr l = ∞, dd l + dr l = dl, if dd � k c−1 k > λr dd l + dr l = dl, if dd � k c−1 k = λr (54) where loads prefer the lower price in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Similarly, solving concave individual problem of each generator (38) by taking the derivative, we have λr � (1−cjθr j )λr− dd � k c−1 k � =0 =⇒ θr j λr=c−1 j λr− c−1 j dd � k c−1 k (55) where we assume λr ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Summing (55) over j ∈ G and using real-time market clearing � j∈G θr jλr = dr (56) we get dr = � j c−1 j λr − � j c−1 j dd � k c−1 k =⇒ λr = d � j c−1 j (57) At equilibrium (36), (54), (55), and (57) must hold simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This implies that λd = λr = d � j c−1 j , dd = d, dr = 0 (58) gd j = c−1 j d � k c−1 k , gr j = 0, θr j = 0 (59) where we use Rule 1 to define prices in the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Although dr = 0 and � j θr j = 0, loads do not have the incentive to deviate by allocating demand in the real-time market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Since loads do not anticipate the effect of their bid on the market prices, the payment remains the same for any allocation across the two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore the equilibrium exists, and this completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' APPENDIX D PROOF OF THEOREM 6 Using the real-time clearing (56), we have λr = dr � j θr j (60) where we assume that � j θr j ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Substituting (60) in generator individual problem (41a), the individual problem of price-anticipating generator j is given by: max θr j c−1 j dd2 (� k c−1 k )2 + θr jdr2 (� k θr k)2 − cj 2 � c−1 j dd � k c−1 k + θr jdr � k θr k �2 (61) We again use first-order and second-order conditions to ana- lyze the optimal bidding behavior since the individual problem may not be concave in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Writing the first-order condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' we have dπj dθr j = dr (� k θr k)3 � mr j − nr jθr j � (62) where mr j := dr � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k − dd � k c−1 k ( � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k)2 and nr j := dr + dd � k c−1 k � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k + cjdr � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k Assuming generators are homogeneous and bid symmetrically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' we can rewrite (62) as dπj dθr j = dr G3θr2 [dr(G − 2) − cd(G − 1)θr] (63) then the turning point is given by θr p = G − 2 G − 1 dr cd (64) Writing the second-order condition and evaluating for homo- geneous generators that bid symmetrically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' the turning point (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' we have d2πj dθr j 2 ���� θr j =θrp(dr) = dr (� j θr j)3 � ˜mr j + ˜nr jθr j � ���� θr j =θrp(dr) (65) = − c3(G − 1)4 G4(G − 2)3 � d dr �3 � 2 + (G − 2)dr d � (66) where ˜mr j := − 4dr � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k � k θr k + 2dd� k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k � k c−1 k − cjdr(� k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k)2 � k θr k and ˜nr j := 2dr � k θr k + 2cjdr � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='k̸=j θr k � k θr k Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' loads acting as leaders anticipate the clearing prices and optimal bids of generators in the real-time subgame equilibrium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' such that λr = G − 1 G − 2 cd G (67) where we substitute (64) in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Substituting (67) in load individual problem (42),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' we have min dd l dd � j c−1 j dd l + G − 1 G − 2 cd G (dl − dd l ) (68) The unique optimal solution to the quadratic program (68) is given by dd l = 1 L + 1 G − 1 G − 2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' dr l = � dl − 1 L + 1 G − 1 G − 2d � (69) Assuming L < G − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' dr > 0 =⇒ d2πj dθr j 2 ���� θr j =θrp < 0 13 Thus the obtained equilibrium maximizes generators’ profit and minimizes loads’ payment while the supply-demand bal- ance is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However, if L > G − 2, then dr < 0 =⇒ θr p < 0 =⇒ d2πj dθr j 2 ���� θr j =θrp > 0 The obtained equilibrium minimizes generators’ profit, and generators’ have the incentive to deviate from this equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore, symmetric equilibrium does not exist in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Moreover, in the case of |G| < 3, generators have the incentive to bid arbitrarily small values and earn arbitrarily large profits in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' In the case of L = G − 2, at equilibrium dr = 0 which contradicts our initial assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' We analyze the case dr = 0 separately, 1) If � j θr j ̸= 0 =⇒ λr = 0 and λd = c Gdd = c Gdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' WLOG, we can assume that dr l = 0, ∀l ∈ L, otherwise load l with non-zero demand has the incentive to deviate and participate in the real-time market to minimize its payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' The payment of individual load l is then given by λddd l + λrdr l = c Gd2 However if load l unilaterally decides to deviate by allocating demand in real-time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', dr l = ϵ then the payment is given by λddd l + λrdr l = c G(d − ϵ)2 + ϵ2 � j θj which is smaller for small enough ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore the equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' 2) If � j θr j = 0, using Rule 1 we have λr = λd and λd = c Gdd = c Gdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' However if load l unilaterally decides to deviate by allocating demand in real-time i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=', dr l = ϵ then using Rule 2 λr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Therefore load has the incentive to deviate and allocate demand in the real- time market with zero clearing price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' Hence equilibrium does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} +page_content=' This completes the proof of the Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE3T4oBgHgl3EQfBglT/content/2301.04267v1.pdf'} diff --git a/P9AzT4oBgHgl3EQflf2o/content/tmp_files/2301.01549v1.pdf.txt b/P9AzT4oBgHgl3EQflf2o/content/tmp_files/2301.01549v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f06e378d424a4363065c89dd9cb68764e0ade34b --- /dev/null +++ b/P9AzT4oBgHgl3EQflf2o/content/tmp_files/2301.01549v1.pdf.txt @@ -0,0 +1,926 @@ +Matching Using Sufficient Dimension Reduction for +Heterogeneity Causal Effect Estimation +Haoran Zhao1#, Yinghao Zhang1#, Debo Cheng2,3∗, Chen Li4, Zaiwen Feng5,6,7,8,9,10∗ +1 College of Science, Huazhong Agricultural University, Wuhan, 430070, China +2 College of Computer Science and Engineering, Guangxi Normal University, Guilin, 541000, China +3 STEM, University of South Australia, Australia +4 Department of Computer Science and Systems Engineering, Kyushu Institute of Technology, Iizuka 820-8502, Japan +5 College of Informatics, Huazhong Agricultural University, Wuhan 430070, China +6 Hubei Key Laboratory of Agricultural Bioinformatics, Huazhong Agricultural University, Wuhan, 430070, China +7 Hubei Hongshan Laboratory, Huazhong Agricultural University, Wuhan 430070, China +8 National Key Laboratory of Crop Genetic Improvement, Huazhong Agricultural University, Wuhan, 430070, China +9 State Key Laboratory of Hybrid Rice, Wuhan University, 299 Bayi Rd, Wuhan, 430070, China +10 Macro Agricultural Research Institute, Huazhong Agricultural University, Wuhan 430070, China +*Correspondence: Debo Cheng (chengdb2016@gmail.com) and Zaiwen Feng (Zaiwen.Feng@mail.hzau.edu.cn) +Abstract—Causal inference plays an important role in under- +standing the underlying mechanisation of the data generation +process across various domains. It is challenging to estimate +the average causal effect and individual causal effects from +observational data with high-dimensional covariates due to the +curse of dimension and the problem of data sufficiency. The +existing matching methods can not effectively estimate individual +causal effect or solve the problem of dimension curse in causal +inference. To address this challenge, in this work, we prove that +the reduced set by sufficient dimension reduction (SDR) is a +balance score for confounding adjustment. Under the theorem, we +propose to use an SDR method to obtain a reduced representation +set of the original covariates and then the reduced set is used for +the matching method. In detail, a non-parametric model is used to +learn such a reduced set and to avoid model specification errors. +The experimental results on real-world datasets show that the +proposed method outperforms the compared matching methods. +Moreover, we conduct an experiment analysis and the results +demonstrate that the reduced representation is enough to balance +the imbalance between the treatment group and control group +individuals. +Index +Terms—Matching, +Sufficient +Dimension +Reduction, +Causal Inference, Individual Causal Effect. +I. INTRODUCTION +In recent decades, causal inference has gained increasing +attention across many areas, such as economic [1], health [2], +statistic [3] and computer science [4]. Causal effect estima- +tion plays an important role in revealing the causal strength +between two factors in causal inference. A randomised control +experiment is regarded as the standard gold for identifying +such causal strength. However, the randomised control exper- +iment is impractical due to the cost, time, or ethic [1], [4]. +Therefore, estimating causal effects from observational data is +an important alternative in causal inference [3], [5]–[8]. +Confounding bias is the main challenge in the causal effect +estimations from observational data [3], [9]. The commonly +#These authors contributed equally to this work +used approach is confounding adjustment to remove the +confounding bias when estimating the causal effects from +observational data [1]. One of the most popular confounding +adjustments is the matching method [10]. The core idea of the +matching method is to balance the distribution of covariates +between the treatment group and the control group. In detail, +the first step of matching is to seek units from the control +group with similar covariates to those in the treatment group +for constructing a matching pair, and then the causal effect +can be calculated based on the matched data. Matching can be +performed by selecting different functions of the covariates and +selecting different matching algorithms such as Mahalanobis +distance matching, full matching, nearest neighbour matching, +and genetic matching [10]. +The most widely used function is propensity scores which +are regarded as the probability of a unit receiving a treat- +ment [11], [12]. In general, the matching method involves +transforming multi-dimensional covariates into scalars by us- +ing propensity scores so as to overcome the difficulties in +matching based on original covariates [13], [14]. In fact, the +true propensity score is unknown in real applications, and thus +the estimation of the propensity score from data is required. +However, the errors of the model or model misspecified +inevitably occur in the calculation of the propensity score from +data [3], [15], [16]. Hence, the propensity score is not a good +solution for the matching method [17]. +Recently, sufficient dimension reduction has been success- +fully utilised for estimating causal effects from observa- +tional data [3], [5], [18]. For instance, Luo et al. proposed +a sufficient dimension reduction matching for causal effect +estimation [19]. The proposed matching method utilises a +sufficient dimension reduction method to reduce the original +covariates into reduced-dimensional covariates that retain the +advantages of both the original covariates and the propensity +arXiv:2301.01549v1 [cs.DS] 4 Jan 2023 + +score under mild assumptions. The advantage of the reduced- +dimensional covariates is the asymptotic stability and is su- +perior to that of the estimates obtained by using propensity +scores. However, this method is to obtain two sets of reduced +reduced-dimensional covariates by using sufficient dimension +reduction on sub-datasets, i.e. the treated samples and the +control samples, but not discovering an adjustment set over +the whole data. +In a data, if the number of samples is not big enough, +the performance of the proposed method will be decreased +significantly. Nabi et al. [5] proposed a semi-parametric causal +sufficient dimension reduction method to deal with multiple +treatments. Cheng et al. [18] proposed a CESD matching +method for the average causal effect estimation by using a +kernel dimension reduction method [20] to reduce the covari- +ates relative to the treatment variable, but not for heterogeneity +causal effect estimation. +In this work, we propose a novel Matching method based +on Inverse Regression Estimator (referred to as MIRE method) +for average and heterogeneity causal effect estimation from +observational data. In detail, our MIRE method utilises a suf- +ficient dimension reduction method, i.e. the inverse regression +estimator, to learn reduced-dimensional covariates relative to +the outcome variable over the whole data. Then, the MIRE +method utilises the reduced-dimensional covariates to conduct +a matching process for imputing the unobserved outcomes +(a.k.a counterfactual outcomes) [1], [3], [18]. Our experimental +analysis shows that the reduced-dimensional covariates are +well-balanced which is why MIRE addresses the confounding +bias very well. +To summarise, our work makes the following contributions. +• We tackle the problem of estimating heterogeneity causal +effect estimation from observational data with sufficient +dimension reduction. +• We propose a novel matching method based on the +inverse regression estimator, MIRE, for causal effect +estimation from observational data. +• Extensive experiments demonstrate that our proposed +matching method is more effective in terms of the causal +effect estimation from observational data. +II. BACKGROUND +A. Potential Outcome Model +Let T to be a binary treatment that includes Ti = 0 (a +controlled unit) and Ti = 1 (a treated unit), where i represents +an unit. The set of units which are not assigned to a certain +treatment is the control group, and the set of units which are +assigned to a certain treatment is the treatment group. The set +X is a set of pretreatment covariates, i.e. the covariates are +unchanged before and after that the treatment and outcome +variables are observed [1], [18]. The potential outcomes Yi +is defined as the outcome of unit i, i.e. Yi(0) and Yi(1) are +the potential outcomes of unit i unassigned and assigned to a +treatment. Note that both potential outcomes for a unit i can +not be observed at the same time. It belongs to a fundamental +challenging problem in causal inference [1], [21]. Under the +potential outcome model, the individual causal effect and +average causal effects can be defined as follows. +ITE = Yi(1) − Yi(0), +(1) +ATE = E[Y (1) − Y (0)]. +(2) +The following assumptions are usually required when we +utilise the potential outcome model to estimate the causal +effects from observational data. +Assumption 1 (Stable Unit Treatment Value Assumption +(SUTVA) [1]): The potential outcome of a unit is not affected +by whether or not other units are treated. That means, for each +unit, its potential outcomes only rely on the treatment T, and +there is no different form or version of each treatment level. +Assumption 2 (Unconfoundedness [1]): Conditioning on the +set of covariate X, the treatment T was independent of the +potential outcomes Y , formally Y (1), Y (0) ⫫ T∣X. +This assumption shows that all units with the set of covari- +ates X are randomly assigned to treatment. +Assumption 3 (Overlap [1]): For each unit, it has a non-zero +probability to being treated or control when given the set of +covariate X, i.e. 0 < P(T = t∣X) < 1, t = 0, 1. +This assumption shows that there is a probability that each +unit is assigned to a treatment t. Note that the assumptions +of Unconfoundedness and Overlap are usually called “the +ignorability assumption”. The ignorability assumption is not +testable directly from data since the counterfactual outcomes +are unmeasured [1]. Consequently, the set of covariates X +consists of all relevant and irrelevant covariates in terms of +estimating the causal effect of T on Y . Hence, it is necessary +to discover an adjustment set Z from X to accurately estimate +the causal effect of T on Y . The propensity score always plays +a role in causal effect estimation from observational data and +is defined below. +Definition 1 (Propensity score [1]): The propensity score is +defined as the conditional probability of a unit being assigned +to a treatment conditioning on the set of covariates X. +e(x) = P(T = 1∣X) +(3) +The balance score denoted as b(x) is proposed by Rosen- +baum and Rubin [12] that allows a class of functions to model +the covariate X. In practice, the balance score should be +satisfied the unconfoundedness assumption as well (Lemma +12.2 in [1]). +Y (1), Y (0) ⫫ T∣b(x) +(4) +The balance scores contain the propensity scores and the +original covariates, and others. Ignorability assumption also +suggests that units with the same or approximately equal +balance scores have the same distribution of covariates. + +B. Matching Method +The matching method is to identify the units in the control +group with a similar distribution of covariates to the units in +the treatment group, so that the potential outcomes of the units +in the control group are used to impute the missing potential +outcomes of units in the treatment group. The essential idea of +the matching method is to simulate the process of randomised +control experiments. Thus the matched units can be regarded +as the counterfactual outcome of units [13], [14], [21]. In a +matching method, the potential outcome of the i-th unit can +be obtained according to the formulas 5 and 6. +ˆYi(1) = { Yi, +if t = 1 +1 +∣ℓ(i)∣ ∑j∈ℓ(i) Yj, +if t = 0 +(5) +ˆYi(0) = { +1 +∣ℓ(i)∣ ∑j∈ℓ(i) Yj, +if t = 1 +Yi, +if t = 0 +(6) +where ℓ denotes that the sample set matches the i-th unit from +the control or treatment group. +In general, the distribution of covariates in the matched data +is more similar between two groups than before matching. +Therefore, the matched results can be used to calculate the +average causal effect so as to reduce the influence of con- +founding factors. +III. THE PROPOSED MIRE METHOD +In this section, we first prove that ψ(X) = XT β by +sufficient dimension reduction (SDR) is a balance score for +addressing the confounding bias when estimating the causal ef- +fects from observational data. Then we introduce our proposed +MIRE method (Matching based on the inverse regression +estimator) for causal effect estimation. +A. A Central DRS is a Sufficient Balance Score +Sufficient dimension reduction (SDR) is a dimension reduc- +tion method which is widely used for data processing [20], +[22], [23]. For a response variable Y and a set of covariates +X, SDR is to learn a function ψ(X) such that the original +covariates X can be reduced into a subspace XT β ∈ Rp×k +with k ≪ p. In this work, we assume that the function +ψ(X) = XT β is existed and can be expressed as follows. +Y ⫫ X∣XT β +(7) +where the column subspace of β is called the dimension +reduction space (DRS). Hence, it is important to learn the +subspace β in the SDR method. It is worth noting that XT β +obtained by using an SDR method to reduce the dimension +of the covariate X and can be viewed as a function of +ψ(X). When a subspace SY ∣Z is the intersection of all +other dimension reduction subspaces, the subspace SY ∣Z is +well-known as the central DRS [20], [22]. The central DRS +has the smallest dimension and unique dimension-reduction +subspace [24]. Thus, in our work, we would like to learn the +central DRS SY ∣Z. +Theorem 1: Given an observational data O that contains the +treatment T, the outcome Y , and the set of the pretreatment +variables X. Suppose that the central subspace SY ∣Z is ex- +isting and with r-dimensional, where r ≪ p. Then there is +an arbitrary basis matrix β ∈ Rp×k such that XT β satisfies +Y (1), Y (0) ⫫ T∣XT β and is a balancing score. +Proof 1: Under the pretreatment variable assumption, X +has not had a descendant node of either W or Y . Under +the ignorability assumption, the causal effect of T on Y +can be calculated unbiasedly based on confounding adjust- +ment or adjusting for a balance score. The existence of +a central subspace SY ∣Z ensures that there is an arbitrary +basis matrix β +∈ Rp×k such that Y +⫫ X∣XT β holds +according to the invariant property of central subspace [3], +[22]. Mathematically, XT β ≅ X holds. Moreover, the un- +confoundedness assumption, i.e. Y (1), Y (0) ⫫ T∣X holds. +So replacing X in Y (1), Y (0) ⫫ T∣X with XT β, we have +Y (1), Y (0) ⫫ T∣XT β. Therefore, XT β is a balance score +according to the invariant property of central subspace and +Y (1), Y (0) ⫫ T∣XT β. +Based on Theorem 1, XT β is a balance score for unbiased +causal effect estimation from observational data. That means, +it is sufficient to use XT β as balance scores in the matching +method for unbiased causal effect estimations. It is worth +noting that another advantage of SDR is able to reduce the +dimension of original covariates while retaining the important +information. +In this work, we adopt the inverse regression estimator (IRE) +method for learning the central DRS since IRE belonging to +the inverse regression (IR) method family is an optimal method +with the highest asymptotic efficiency [25]. +When Y is the continuous value, Y is discretised with its +range divided into h slices based on the previous notation [25]. +The central subspace β can be obtained by calculating the +following formula. +βξy = ∑h +y=1Span(ξy) +(8) +where ξy = Σ−1(E(X∣Y = y) − E(X)). +Furthermore, we assume that the linearity condition for +estimating central subspace., i.e. E(Z∣PSY ∣ZZ) = PSY ∣ZZ is +induced, based on which the central subspace is linked to the +inverse regression of Z on Y . +B. implementation of MIRE +In this study, we use the inverse regression estimator +(IRE) [25] to estimate the central DRS for our MIRE method. +The IRE method is to estimate the DRS by minimising the +objective function 9. First, the following equation is used to +calculate the quadratic discrepancy for the IRE method. +F IRE +k +(S, C) =(vec(ˆζ) − vec(SC))T ˆΓ−1 +ˆζ +(vec(ˆζ) − vec(SC)) +(9) +where ˆΓ−1 +ˆζ +is a nonsingular covariance matrix. The columns +of S ∈ Rp×k represent a basis for Span(ξ), and C ∈ Rk×(h−1) + +represents the coordinates of ξ relative to S. ˆζ satisfies +ˆζ ≡ βγDfA, where β ∈ Rp×k is a basis of DRS. A is a +nonstochastic matrix satisfies AT A = Ih−1 and AT 1h = 0. +γ is a vector such that ξ = βγ. Df is a diagonal matrix +with the elements of the vector f on the diagonal, where +ˆf = ( ˆf1, ..., ˆfh). vec(⋅) denotes the operator that constructs +a vector from a matrix by stacking its columns and can be +formalised as follows. +vec(C) =[(Ih−1 ⊗ ST )Vn(Ih−1 ⊗ S)]−1× +(Ih−1 ⊗ ST )Vnvec(ˆξ) +(10) +where ⊗ is the Kronecker product, which is an operation on +two matrices of arbitrary size resulting in a block matrix. Vn is +a positive-definite matrix and is equal to a consistent estimate +ˆΓ−1 +ˆξ +of Γ−1 +ˆζ . +ˆSd = QS(−d)[QS(−d)(cT +d ⊗ Ip)QS(−d)]−× +QS(−d)(cT +d ⊗ Ip)Vnαd +(11) +where αd = vec(ˆζ − S(−d)C(−d)), Cd is the d-th row of C, +C(−d) consists of all but the dth row of C, and QS(−d) projects +onto the orthogonal complement of Span(S(−d)) in the usual +inner product. +The matching process is performed according to the above- +mentioned matching steps, and the distance is measured by +using the following Mahalanobis distance. +Dij = (ψ(X)i − ψ(X)j)′Σ−1(ψ(X)i − ψ(X)j) +(12) +where Σ is the covariance matrix of all the units. In practice, +the observed data are used to calculate the covariance matrix. +The MIRE method is as shown in Algorithm 1. +After matching, the standardised difference in the mean of +the covariate balance. Based on the matched data, the causal +effect can be obtained. The average causal effects can be +estimated by using the formula 9. The individual causal effects +of the i-th unit can be estimated according to the equation +ˆYi(1) − ˆYi(0). +IV. EXPERIMENT +It is very difficult to evaluate the proposed causal effect +estimation methods by using the real-world datasets because +we cannot know the counterfactual outcomes in the real- +world datasets [1], [11]. To evaluate the performance of the +proposed MIRE method, we select two semi-synthetic datasets, +IHDP [26] and TWINS [27]. Both semi-synthetic datasets are +widely used for evaluating causal effect estimation methods. +Moreover, one real-world dataset, Jobs [28] is also used in our +experiments since the dataset have the empirical causal effects +in the literature. +To evaluate the performance of the proposed MIRE method, +seven commonly used causal effect estimation methods were +selected for comparison, including: NNM (Nearest neigh- +bor matching [13]), PSM (propensity score matching [12]), +BART (Bayesian Additive Regression Trees [26]), CF (Causal +Algorithm 1: MIRE +Input: Observational data O includes the set of +covariates X, treatment T and outcome Y . +Output: paired datasets +1 Initialise S ← (s1, ..., sd) randomly +2 Calculate the least squares coefficient for B using +equation (10) with fixed S +3 Assign e ← Fd(S, C) and iter ← 0 +4 for d ← 1 to k do +5 +S = (s1, ..., sk) +6 +Find a new sd by equation 11 +7 +ˆsd ← +ˆsd +∥ˆsd∥ +8 +Update S ← (s1, ..., ˆsd, ..., sk) +9 +C ← argc∗ min Fk(S, C∗) +10 +e ← Fk(S, C); iter ← iter + 1 +11 +if e no longer decreases then +12 +˜S ← S +13 +end +14 +˜S ← S +15 end +16 Calculate an ordered basis for Span(̃ +B) (i.e. ˆb1, ˆb2) +17 X∗ = X(ˆb1,ˆb2) +18 for i ← 1 to n1 do +19 +for j ← 1 to n2 do +20 +Dij = (X∗ +i − X∗ +j )′Σ−1(X∗ +i − X∗ +j ) +21 +end +22 +Find j with the smallest Dij +23 +Pair the units i, j. +24 end +Forest [29]), SDRM (Sufficient dimension reduction match- +ing [19]), BCF (Bayesian Causal Forest [30]), R-LASSO +(R-learner using LASSO Regression [31]). These methods +have been regarded as one of the most efficient causal effect +estimation methods as their ability of eliminating confounding +bias, i.e. the state-of-the-art method in causal effect estimation +method. +For the experiments on the IHDP, we use Precision in Es- +timating Heterogeneous Treatment Effects (PEHE) PEHE = +1 +N ∑N +i=1((yi1 − yi0) − (ˆyi1 − ˆyi0))2 as an evaluation criterion +for assessing the heterogeneous causal effects. For experiments +on the real dataset Jobs, we estimate the average causal effect +on the treated samples (ATT) since the empirical ATT is +known [32]. For experiments on TWINS, we estimated the +average causal effect (ATE). In addition, we use root-mean- +square error (RMSE) RMSE = +√ +1 +N ∑N +i=1 (yi− ˆyi)2 and +standard deviation (SD) as evaluation metrics for assessing +the performance of all methods. +A. Estimation of heterogeneous effects based on IHDP +The benchmark dataset IHDP in causal inference is from +a simulation study conducted in the work [26]. The data is +from the Infant Health and Development Program (IHDP), + +a program that began in 1985 to provide high-quality home +visiting services to low birth weight preterm infants. The +results of the program showed that after treatment (i.e., after +receiving the service), there was a significant increase in +cognitive test scores in the treatment group compared to the +control group at the age of 3. +A variety of covariates was collected in this study such as +child birth weight, head circumference, weeks of prematurity, +birth order, and neonatal health indicators, as well as mater- +nal behaviour during pregnancy and some indicators during +delivery. To simulate the imbalance between the treatment +and control groups, we discard the non-random portion of the +treatment group from the experimental data as suggested by +Hill [26], specifically all children of non-white mothers, while +leaving the control group intact. The potential outcomes for +each unit were then simulated by creating response surfaces +so that true individual causal effects could be concluded. And +because the response surface is known, the covariates that +generate the response surface can be adjusted to satisfy the +Ignorability assumption. We follow the response surface B +used by Hill [26]: +Y (0) ∼ N(exp((X + W)βB), 1) +Y (1) ∼ N(XβB − ωB, 1) +where W is an offset matrix with the same dimension as X +with every value equal to 0.5, βB is a vector of regression +coefficients (0, 0.1, 0.2, 0.3, 0.4) randomly sampled with +probabilities (0.5, 0.125, 0.125, 0.125, 0.125) for the 6 contin- +uous covariates and (0.6, 0.1, 0.1, 0.1, 0.1) for the 18 binary +covariates, ωB is an offset chosen to guarantee that ATT = 4. +Therefore, the true individual causal effect can be calculated +at this point. +Figure 1 shows the results of 1,000 simulations of this data +using different methods. For each method, we calculate the +average of the 1,000 results as the PEHE value. From this +figure, it can be seen clearly that MIRE has the best perfor- +mance among all the compared methods, which also shows the +effectiveness of the method. The second good performance of +methods is R-LASSO and SDRM. Meanwhile, the widely used +method, CF (Causal Forest) also has a good performance but +is worse than MIRE, R-LASSO and SDRM. The rest methods, +NNM (Nearest neighbour matching), PSM (propensity score +matching), BART (Bayesian Additive Regression Trees) and +BCF (Bayesian Causal Forest) have worse performance than +MIRE, R-LASSO, SDRM and CF. +B. Estimation of causal effects +1) Jobs: The Jobs dataset is a classic dataset established +by LaLonde in 1986 for causal inference [28]. The dataset +was derived from a temporary employment program aimed to +provide work skills to people who are faced with economic +hardship or lack job skills. The treatment is whether an indi- +vidual participates in the program, and the potential outcome +is the individual income in the year 1978. Covariates mainly +included education, age, race, marital status, income in 1974, +MIRE +R−LASSO +SDRM +CF +BCF +PSM +BART +NNM +2 +3 +4 +5 +PEHE +Methods +2 +3 +4 +5 +PEHE +Fig. 1. PEHE values estimated using different methods in the IHDP simulation +experiment. +and income in 1975. The average causal effect on the treated +samples in the Lalonde dataset was estimated as $886 with a +standard error of $448, and we used this estimate ($886) as a +criterion to evaluate our method [18], [28], [32], [33]. +Table I shows the results of estimating the average causal +effect on the treated samples on the Jobs dataset using different +methods. From table I, it can be seen that MIRE has a +good performance for ATT estimation, and the estimated ATT +($519.09) is close to the criterion ($886) with a small SD +($734.93) that is also close to the empirical SD ($448). The +two methods BART and BCF also show a good performance +on this dataset. While the CF estimated ATT has a larger +difference from the standard value. +Figure 2 shows the scatter plot after dimension reduction of +our MIRE on the jobs dataset, which shows the relationship +between the reduced variables and the response variable Y . +From Figure 2, it can be seen that both covariates after dimen- +sion reduction are significantly correlated with the response +variable Y . The result also shows the rationality of using +dimension reduction covariates for matching. +TABLE I +ESTIMATED AVERAGE CAUSAL EFFECT ON THE TREATED SAMPLES ON +JOBS DATASET. +Methods +Estimated ATT +RMSE +SD +NNM +198.16 +683.34 +1,280.70 +PSM +1,933.40 +1,090.60 +702.20 +BART +931.30 +2182.02 +1,032.00 +CF +182.24 +497.65 +890.61 +SDRM +1,740.98 +1,025.65 +710.55 +R-LASSO +1,271.63 +385.63 +825.23 +BCF +697.88 +544.90 +511.82 +MIRE +519.09 +734.92 +734.93 +2) TWINS: The TWINS benchmark dataset was created +based on the data of twins born in the United States between + +Y +60000 +60000 +40000 +40000 +20000 +20000 +0 +-25000 -20000 -15000 -10000 -5000 +0 +0 +0 +5000 +10000 +1st reduced covariate +2st reduced covariate +Y +Fig. 2. The relationships between reduced covariates and response variable +Y by our MIRE on Jobs dataset. +1983 and 2000, the treatment refers to the heavier weight of the +twins at birth [27]. The potential outcome is the mortality in +the first year of twin birth. In Louizos et al.’s study [34], one of +the two twins was selectively hidden, which was equivalent to +randomly assigning the treatment, thus making it similar to the +data of randomdised experiment. Then those twins with weight +less than 2kg were selected to establish a dataset. The dataset +included 40 covariates such as parents’ education, marital +status, race, and mother’s condition at the time of delivery. We +simulated the presence of confounding factors according to the +following formula [34]: Wi∣Xi ∼ Bern(Sigmiod(w′Xi) + +n), where W ∼ U(−0.1, 0.1)40×1, n ∼ N(0, 0.1). +The average causal effect of our established dataset is +−0.025. Table II shows the results of estimating the average +causal effect on the TWINS dataset using different methods. It +can be seen from the results shown in table II: MIRE, CF, BCF, +PSM both have really well performance. This also shows that +in the existence of confounders, MIRE is able to remove the +confounding bias as the stat-of-the-art causal effect estimators. +Figure 3 shows the scatter plot after dimension reduction +of the covariates on the TWINS dataset, which shows the +relationship between the reduced variables and the response +variable Y . From figure 3, we have that the covariates after +dimension reduction are significantly correlated with the re- +sponse variables. It also confirms that MIRE is effective for +estimating causal effects from observational data. +V. RELATED WORK +The potential outcome model is widely used in causal +inference [1], [13], [14], [21]. Our proposed MIRE method +builds on the potential outcome model with mild assumptions, +such as the assumptions of the pretreatment variables and +the ignorability assumption. The matching method is one of +the most commonly used methods in causal inference [13], +TABLE II +ESTIMATED AVERAGE CAUSAL EFFECT ON TWINS DATASET. +Methods +Estimated ATE +RMSE +SD +NNM +-0.0218 +0.0029 +0.0468 +PSM +-0.0252 +0.0043 +0.0502 +BART +-0.1794 +0.2793 +0.3188 +CF +-0.0252 +0.0034 +0.0408 +SDRM +-0.0230 +0.0032 +0.0480 +R-LASSO +-0.0623 +0.0221 +0.0901 +BCF +-0.0249 +0.0398 +0.0639 +MIRE +-0.0252 +0.0037 +0.0002 +Y +1.2 +1.0 +0.8 +0.4 +0.5 +0.0 +0.0 +−500 +0 +500 +1000 +1500 +1st reduced covariate +100 +200 +300 +400 +2st reduced covariate +Y +Fig. 3. The relationships between reduced covariates and response variable +Y after our MIRE on TWINS dataset. +[18], [19]. Stuart summarises the principles and steps of the +matching method as well as its application in many practical +problems [10]. In the following, we review some works that +are closely related to our proposed MIRE method. +It is difficult to find individuals with multiple dimensions +of the covariates between two groups (control and treatment) +during the matching process. To solve this problem, Rubin +and Rosenbaum [13] introduced a propensity score, defined +as the conditional probability that a unit is assigned to a +certain treatment under the covariate condition. In addition, +the propensity score has been proven to be a balance score [1]. +The most commonly used propensity score matching method +is to use the propensity score in place of the original covariate +in matching progress. In addition, propensity score matching +and Mahalanobis distance matching are combined to generate +GenMatch, and GenMatch uses a genetic search algorithm to +obtain weights so as to complete matching [10]. Luo and Zhu +used a sufficient dimension reduction (SDR) method to reduce +the dimension of the covariates in the treatment group and +control group, and then matched them based on Mahalanobis +distance [17]. +The most related work to MIRE is the SDR matching +proposed by the works [18], [19]. Luo and Zhu’s work [19] + +0000Xconsider reducing the sub-datasets over the treated units and +the control units to obtain two of the reduced-dimensional +covariates as the balance score for matching. The proposed +method maybe suffers from bias since dividing the whole +samples into two sub-datasets results in data insufficiency. +Cheng et al. [18] aim to estimate the average causal effect +from observational data, but not for heterogeneity causal effect +estimation. In contrast, our theoretical findings support a data- +driven method for heterogeneity causal effect estimation. +In recent, a large number of deep learning-based methods +have been proposed for estimating the causal effects from +observational data [35]–[39]. The main advantage of deep +learning-based methods is that the complex nonlinear relation- +ships between variables can be learned by neural networks +and the high-dimensional datasets can be addressed very +well. Nevertheless, a number of parameters turning are very +inefficient, and they do have not good interpretability. +Another line work on causal effect estimation from data +with latent confounders [7], [8], [39]. When an instrumental +variable (IV) is given, the causal effect of T on Y can be +calculated unbiasedly from data with latent variable too [29], +[40]–[43]. Because IV-based estimators do not rely on the +ignorability assumption, they are not directly related to our +MIRE method. +VI. CONCLUSION +In this work, we prove that the central DRS by a sufficient +dimensional reduction method is a balance score and is suffi- +cient to control for confounding bias in causal effect estima- +tion from observational data. Our findings provide theoretical +support for using the dimension-reduced covariates for match- +ing. Under the proposed theorem, we propose a data-driven +method, i.e. MIRE, to estimate the causal effects from obser- +vational data under mild assumptions. Firstly, MIRE utilises +the inverse regression estimator to reduce the dimensions of the +original covariates, and then uses the reduced-dimensional co- +variates for matching. The advantages of our proposed MIRE +have been verified through the experiments. First, the results +of average causal effect estimation based on the Jobs dataset +showed that the estimation results of our method were closer to +the criterion value ($886) recommended in the previous study +than those of other matching methods. Second, the results of +individual causal effect estimation showed that matching based +on dimension-reduced covariates made it easier for individuals +to be paired with another group in the matching process. Our +method displayed great advantages in estimating individual +causal effects over other matching methods. The estimation +results based on the IHDP dataset indicated that our method +could match more individuals during the matching process. +Compared with other causal inference methods, our method +also exhibited certain advantages in heterogeneous effect esti- +mation accuracy (expressed as PEHE) and confounding factor +control. +ACKNOWLEDGMENT +This research project was supported in part by the Ma- +jor Project of Hubei Hongshan Laboratory under Grant +2022HSZD031, and in part by the Innovation fund of Chi- +nese Marine Defense Technology Innovation Center under +Grant JJ-2021-722-04, and in part by the National Natural +Science Foundation of China under Grant Nos. 62076041 +and 61806027, and in part by the Fundamental Research +Funds for the Chinese Central Universities under Grant +2662020XXQD01, 2662022JC004, and in part by the open +funds of State Key Laboratory of Hybrid Rice, Wuhan Uni- +versity, and in part by the open funds of the National Key Lab- +oratory of Crop Genetic Improvement under Grant ZK202203, +Huzhong Agricultural University. +REFERENCES +[1] G. 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Pestman et al., “Instrumental variables: application +and limitations,” Epidemiology, vol. 17, no. 3, pp. 260–267, 2006. + diff --git a/P9AzT4oBgHgl3EQflf2o/content/tmp_files/load_file.txt b/P9AzT4oBgHgl3EQflf2o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92917a17143c8f892255c94d2cd128d63482245d --- /dev/null +++ b/P9AzT4oBgHgl3EQflf2o/content/tmp_files/load_file.txt @@ -0,0 +1,685 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf,len=684 +page_content='Matching Using Sufficient Dimension Reduction for Heterogeneity Causal Effect Estimation Haoran Zhao1#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Yinghao Zhang1#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Debo Cheng2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='3∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Chen Li4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Zaiwen Feng5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='10∗ 1 College of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Huazhong Agricultural University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 2 College of Computer Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Guangxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Guilin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 541000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 3 STEM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' University of South Australia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Australia 4 Department of Computer Science and Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Kyushu Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Iizuka 820-8502,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Japan 5 College of Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Huazhong Agricultural University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 6 Hubei Key Laboratory of Agricultural Bioinformatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Huazhong Agricultural University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 7 Hubei Hongshan Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Huazhong Agricultural University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 8 National Key Laboratory of Crop Genetic Improvement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Huazhong Agricultural University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 9 State Key Laboratory of Hybrid Rice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 299 Bayi Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China 10 Macro Agricultural Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Huazhong Agricultural University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Wuhan 430070,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' China Correspondence: Debo Cheng (chengdb2016@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='com) and Zaiwen Feng (Zaiwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='Feng@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='hzau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='cn) Abstract—Causal inference plays an important role in under- standing the underlying mechanisation of the data generation process across various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It is challenging to estimate the average causal effect and individual causal effects from observational data with high-dimensional covariates due to the curse of dimension and the problem of data sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The existing matching methods can not effectively estimate individual causal effect or solve the problem of dimension curse in causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' To address this challenge, in this work, we prove that the reduced set by sufficient dimension reduction (SDR) is a balance score for confounding adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Under the theorem, we propose to use an SDR method to obtain a reduced representation set of the original covariates and then the reduced set is used for the matching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In detail, a non-parametric model is used to learn such a reduced set and to avoid model specification errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The experimental results on real-world datasets show that the proposed method outperforms the compared matching methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Moreover, we conduct an experiment analysis and the results demonstrate that the reduced representation is enough to balance the imbalance between the treatment group and control group individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Index Terms—Matching, Sufficient Dimension Reduction, Causal Inference, Individual Causal Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' INTRODUCTION In recent decades, causal inference has gained increasing attention across many areas, such as economic [1], health [2], statistic [3] and computer science [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Causal effect estima- tion plays an important role in revealing the causal strength between two factors in causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' A randomised control experiment is regarded as the standard gold for identifying such causal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' However, the randomised control exper- iment is impractical due to the cost, time, or ethic [1], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Therefore, estimating causal effects from observational data is an important alternative in causal inference [3], [5]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Confounding bias is the main challenge in the causal effect estimations from observational data [3], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The commonly #These authors contributed equally to this work used approach is confounding adjustment to remove the confounding bias when estimating the causal effects from observational data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' One of the most popular confounding adjustments is the matching method [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The core idea of the matching method is to balance the distribution of covariates between the treatment group and the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In detail, the first step of matching is to seek units from the control group with similar covariates to those in the treatment group for constructing a matching pair, and then the causal effect can be calculated based on the matched data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Matching can be performed by selecting different functions of the covariates and selecting different matching algorithms such as Mahalanobis distance matching, full matching, nearest neighbour matching, and genetic matching [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The most widely used function is propensity scores which are regarded as the probability of a unit receiving a treat- ment [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In general, the matching method involves transforming multi-dimensional covariates into scalars by us- ing propensity scores so as to overcome the difficulties in matching based on original covariates [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In fact, the true propensity score is unknown in real applications, and thus the estimation of the propensity score from data is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' However, the errors of the model or model misspecified inevitably occur in the calculation of the propensity score from data [3], [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Hence, the propensity score is not a good solution for the matching method [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Recently, sufficient dimension reduction has been success- fully utilised for estimating causal effects from observa- tional data [3], [5], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' For instance, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' proposed a sufficient dimension reduction matching for causal effect estimation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The proposed matching method utilises a sufficient dimension reduction method to reduce the original covariates into reduced-dimensional covariates that retain the advantages of both the original covariates and the propensity arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='01549v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='DS] 4 Jan 2023 score under mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The advantage of the reduced- dimensional covariates is the asymptotic stability and is su- perior to that of the estimates obtained by using propensity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' However, this method is to obtain two sets of reduced reduced-dimensional covariates by using sufficient dimension reduction on sub-datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' the treated samples and the control samples, but not discovering an adjustment set over the whole data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In a data, if the number of samples is not big enough, the performance of the proposed method will be decreased significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Nabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' [5] proposed a semi-parametric causal sufficient dimension reduction method to deal with multiple treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' [18] proposed a CESD matching method for the average causal effect estimation by using a kernel dimension reduction method [20] to reduce the covari- ates relative to the treatment variable, but not for heterogeneity causal effect estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In this work, we propose a novel Matching method based on Inverse Regression Estimator (referred to as MIRE method) for average and heterogeneity causal effect estimation from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In detail, our MIRE method utilises a suf- ficient dimension reduction method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' the inverse regression estimator, to learn reduced-dimensional covariates relative to the outcome variable over the whole data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Then, the MIRE method utilises the reduced-dimensional covariates to conduct a matching process for imputing the unobserved outcomes (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='a counterfactual outcomes) [1], [3], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Our experimental analysis shows that the reduced-dimensional covariates are well-balanced which is why MIRE addresses the confounding bias very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' To summarise, our work makes the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' We tackle the problem of estimating heterogeneity causal effect estimation from observational data with sufficient dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' We propose a novel matching method based on the inverse regression estimator, MIRE, for causal effect estimation from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Extensive experiments demonstrate that our proposed matching method is more effective in terms of the causal effect estimation from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Potential Outcome Model Let T to be a binary treatment that includes Ti = 0 (a controlled unit) and Ti = 1 (a treated unit), where i represents an unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The set of units which are not assigned to a certain treatment is the control group, and the set of units which are assigned to a certain treatment is the treatment group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The set X is a set of pretreatment covariates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' the covariates are unchanged before and after that the treatment and outcome variables are observed [1], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The potential outcomes Yi is defined as the outcome of unit i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Yi(0) and Yi(1) are the potential outcomes of unit i unassigned and assigned to a treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Note that both potential outcomes for a unit i can not be observed at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It belongs to a fundamental challenging problem in causal inference [1], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Under the potential outcome model, the individual causal effect and average causal effects can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' ITE = Yi(1) − Yi(0), (1) ATE = E[Y (1) − Y (0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' (2) The following assumptions are usually required when we utilise the potential outcome model to estimate the causal effects from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Assumption 1 (Stable Unit Treatment Value Assumption (SUTVA) [1]): The potential outcome of a unit is not affected by whether or not other units are treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' That means, for each unit, its potential outcomes only rely on the treatment T, and there is no different form or version of each treatment level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Assumption 2 (Unconfoundedness [1]): Conditioning on the set of covariate X, the treatment T was independent of the potential outcomes Y , formally Y (1), Y (0) ⫫ T∣X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' This assumption shows that all units with the set of covari- ates X are randomly assigned to treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Assumption 3 (Overlap [1]): For each unit, it has a non-zero probability to being treated or control when given the set of covariate X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 0 < P(T = t∣X) < 1, t = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' This assumption shows that there is a probability that each unit is assigned to a treatment t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Note that the assumptions of Unconfoundedness and Overlap are usually called “the ignorability assumption”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The ignorability assumption is not testable directly from data since the counterfactual outcomes are unmeasured [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Consequently, the set of covariates X consists of all relevant and irrelevant covariates in terms of estimating the causal effect of T on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Hence, it is necessary to discover an adjustment set Z from X to accurately estimate the causal effect of T on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The propensity score always plays a role in causal effect estimation from observational data and is defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Definition 1 (Propensity score [1]): The propensity score is defined as the conditional probability of a unit being assigned to a treatment conditioning on the set of covariates X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' e(x) = P(T = 1∣X) (3) The balance score denoted as b(x) is proposed by Rosen- baum and Rubin [12] that allows a class of functions to model the covariate X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In practice, the balance score should be satisfied the unconfoundedness assumption as well (Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='2 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Y (1), Y (0) ⫫ T∣b(x) (4) The balance scores contain the propensity scores and the original covariates, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Ignorability assumption also suggests that units with the same or approximately equal balance scores have the same distribution of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Matching Method The matching method is to identify the units in the control group with a similar distribution of covariates to the units in the treatment group, so that the potential outcomes of the units in the control group are used to impute the missing potential outcomes of units in the treatment group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The essential idea of the matching method is to simulate the process of randomised control experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Thus the matched units can be regarded as the counterfactual outcome of units [13], [14], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In a matching method, the potential outcome of the i-th unit can be obtained according to the formulas 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' ˆYi(1) = { Yi, if t = 1 1 ∣ℓ(i)∣ ∑j∈ℓ(i) Yj, if t = 0 (5) ˆYi(0) = { 1 ∣ℓ(i)∣ ∑j∈ℓ(i) Yj, if t = 1 Yi, if t = 0 (6) where ℓ denotes that the sample set matches the i-th unit from the control or treatment group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In general, the distribution of covariates in the matched data is more similar between two groups than before matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Therefore, the matched results can be used to calculate the average causal effect so as to reduce the influence of con- founding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' THE PROPOSED MIRE METHOD In this section, we first prove that ψ(X) = XT β by sufficient dimension reduction (SDR) is a balance score for addressing the confounding bias when estimating the causal ef- fects from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Then we introduce our proposed MIRE method (Matching based on the inverse regression estimator) for causal effect estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' A Central DRS is a Sufficient Balance Score Sufficient dimension reduction (SDR) is a dimension reduc- tion method which is widely used for data processing [20], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' For a response variable Y and a set of covariates X, SDR is to learn a function ψ(X) such that the original covariates X can be reduced into a subspace XT β ∈ Rp×k with k ≪ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In this work, we assume that the function ψ(X) = XT β is existed and can be expressed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Y ⫫ X∣XT β (7) where the column subspace of β is called the dimension reduction space (DRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Hence, it is important to learn the subspace β in the SDR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It is worth noting that XT β obtained by using an SDR method to reduce the dimension of the covariate X and can be viewed as a function of ψ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' When a subspace SY ∣Z is the intersection of all other dimension reduction subspaces, the subspace SY ∣Z is well-known as the central DRS [20], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The central DRS has the smallest dimension and unique dimension-reduction subspace [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Thus, in our work, we would like to learn the central DRS SY ∣Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Theorem 1: Given an observational data O that contains the treatment T, the outcome Y , and the set of the pretreatment variables X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Suppose that the central subspace SY ∣Z is ex- isting and with r-dimensional, where r ≪ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Then there is an arbitrary basis matrix β ∈ Rp×k such that XT β satisfies Y (1), Y (0) ⫫ T∣XT β and is a balancing score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Proof 1: Under the pretreatment variable assumption, X has not had a descendant node of either W or Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Under the ignorability assumption, the causal effect of T on Y can be calculated unbiasedly based on confounding adjust- ment or adjusting for a balance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The existence of a central subspace SY ∣Z ensures that there is an arbitrary basis matrix β ∈ Rp×k such that Y ⫫ X∣XT β holds according to the invariant property of central subspace [3], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Mathematically, XT β ≅ X holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Moreover, the un- confoundedness assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Y (1), Y (0) ⫫ T∣X holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' So replacing X in Y (1), Y (0) ⫫ T∣X with XT β, we have Y (1), Y (0) ⫫ T∣XT β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Therefore, XT β is a balance score according to the invariant property of central subspace and Y (1), Y (0) ⫫ T∣XT β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Based on Theorem 1, XT β is a balance score for unbiased causal effect estimation from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' That means, it is sufficient to use XT β as balance scores in the matching method for unbiased causal effect estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It is worth noting that another advantage of SDR is able to reduce the dimension of original covariates while retaining the important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In this work, we adopt the inverse regression estimator (IRE) method for learning the central DRS since IRE belonging to the inverse regression (IR) method family is an optimal method with the highest asymptotic efficiency [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' When Y is the continuous value, Y is discretised with its range divided into h slices based on the previous notation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The central subspace β can be obtained by calculating the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' βξy = ∑h y=1Span(ξy) (8) where ξy = Σ−1(E(X∣Y = y) − E(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Furthermore, we assume that the linearity condition for estimating central subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' E(Z∣PSY ∣ZZ) = PSY ∣ZZ is induced, based on which the central subspace is linked to the inverse regression of Z on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' implementation of MIRE In this study, we use the inverse regression estimator (IRE) [25] to estimate the central DRS for our MIRE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The IRE method is to estimate the DRS by minimising the objective function 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' First, the following equation is used to calculate the quadratic discrepancy for the IRE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' F IRE k (S, C) =(vec(ˆζ) − vec(SC))T ˆΓ−1 ˆζ (vec(ˆζ) − vec(SC)) (9) where ˆΓ−1 ˆζ is a nonsingular covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The columns of S ∈ Rp×k represent a basis for Span(ξ), and C ∈ Rk×(h−1) represents the coordinates of ξ relative to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' ˆζ satisfies ˆζ ≡ βγDfA, where β ∈ Rp×k is a basis of DRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' A is a nonstochastic matrix satisfies AT A = Ih−1 and AT 1h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' γ is a vector such that ξ = βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Df is a diagonal matrix with the elements of the vector f on the diagonal, where ˆf = ( ˆf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', ˆfh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' vec(⋅) denotes the operator that constructs a vector from a matrix by stacking its columns and can be formalised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' vec(C) =[(Ih−1 ⊗ ST )Vn(Ih−1 ⊗ S)]−1× (Ih−1 ⊗ ST )Vnvec(ˆξ) (10) where ⊗ is the Kronecker product, which is an operation on two matrices of arbitrary size resulting in a block matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Vn is a positive-definite matrix and is equal to a consistent estimate ˆΓ−1 ˆξ of Γ−1 ˆζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' ˆSd = QS(−d)[QS(−d)(cT d ⊗ Ip)QS(−d)]−× QS(−d)(cT d ⊗ Ip)Vnαd (11) where αd = vec(ˆζ − S(−d)C(−d)), Cd is the d-th row of C, C(−d) consists of all but the dth row of C, and QS(−d) projects onto the orthogonal complement of Span(S(−d)) in the usual inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The matching process is performed according to the above- mentioned matching steps, and the distance is measured by using the following Mahalanobis distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Dij = (ψ(X)i − ψ(X)j)′Σ−1(ψ(X)i − ψ(X)j) (12) where Σ is the covariance matrix of all the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In practice, the observed data are used to calculate the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The MIRE method is as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' After matching, the standardised difference in the mean of the covariate balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Based on the matched data, the causal effect can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The average causal effects can be estimated by using the formula 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The individual causal effects of the i-th unit can be estimated according to the equation ˆYi(1) − ˆYi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' EXPERIMENT It is very difficult to evaluate the proposed causal effect estimation methods by using the real-world datasets because we cannot know the counterfactual outcomes in the real- world datasets [1], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' To evaluate the performance of the proposed MIRE method, we select two semi-synthetic datasets, IHDP [26] and TWINS [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Both semi-synthetic datasets are widely used for evaluating causal effect estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Moreover, one real-world dataset, Jobs [28] is also used in our experiments since the dataset have the empirical causal effects in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' To evaluate the performance of the proposed MIRE method, seven commonly used causal effect estimation methods were selected for comparison, including: NNM (Nearest neigh- bor matching [13]), PSM (propensity score matching [12]), BART (Bayesian Additive Regression Trees [26]), CF (Causal Algorithm 1: MIRE Input: Observational data O includes the set of covariates X, treatment T and outcome Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Output: paired datasets 1 Initialise S ← (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', sd) randomly 2 Calculate the least squares coefficient for B using equation (10) with fixed S 3 Assign e ← Fd(S, C) and iter ← 0 4 for d ← 1 to k do 5 S = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', sk) 6 Find a new sd by equation 11 7 ˆsd ← ˆsd ∥ˆsd∥ 8 Update S ← (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', ˆsd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', sk) 9 C ← argc∗ min Fk(S, C∗) 10 e ← Fk(S, C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' iter ← iter + 1 11 if e no longer decreases then 12 ˜S ← S 13 end 14 ˜S ← S 15 end 16 Calculate an ordered basis for Span(̃ B) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' ˆb1, ˆb2) 17 X∗ = X(ˆb1,ˆb2) 18 for i ← 1 to n1 do 19 for j ← 1 to n2 do 20 Dij = (X∗ i − X∗ j )′Σ−1(X∗ i − X∗ j ) 21 end 22 Find j with the smallest Dij 23 Pair the units i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 24 end Forest [29]), SDRM (Sufficient dimension reduction match- ing [19]), BCF (Bayesian Causal Forest [30]), R-LASSO (R-learner using LASSO Regression [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' These methods have been regarded as one of the most efficient causal effect estimation methods as their ability of eliminating confounding bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' the state-of-the-art method in causal effect estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' For the experiments on the IHDP, we use Precision in Es- timating Heterogeneous Treatment Effects (PEHE) PEHE = 1 N ∑N i=1((yi1 − yi0) − (ˆyi1 − ˆyi0))2 as an evaluation criterion for assessing the heterogeneous causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' For experiments on the real dataset Jobs, we estimate the average causal effect on the treated samples (ATT) since the empirical ATT is known [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' For experiments on TWINS, we estimated the average causal effect (ATE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In addition, we use root-mean- square error (RMSE) RMSE = √ 1 N ∑N i=1 (yi− ˆyi)2 and standard deviation (SD) as evaluation metrics for assessing the performance of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Estimation of heterogeneous effects based on IHDP The benchmark dataset IHDP in causal inference is from a simulation study conducted in the work [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The data is from the Infant Health and Development Program (IHDP), a program that began in 1985 to provide high-quality home visiting services to low birth weight preterm infants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The results of the program showed that after treatment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=', after receiving the service), there was a significant increase in cognitive test scores in the treatment group compared to the control group at the age of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' A variety of covariates was collected in this study such as child birth weight, head circumference, weeks of prematurity, birth order, and neonatal health indicators, as well as mater- nal behaviour during pregnancy and some indicators during delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' To simulate the imbalance between the treatment and control groups, we discard the non-random portion of the treatment group from the experimental data as suggested by Hill [26], specifically all children of non-white mothers, while leaving the control group intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The potential outcomes for each unit were then simulated by creating response surfaces so that true individual causal effects could be concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' And because the response surface is known, the covariates that generate the response surface can be adjusted to satisfy the Ignorability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' We follow the response surface B used by Hill [26]: Y (0) ∼ N(exp((X + W)βB), 1) Y (1) ∼ N(XβB − ωB, 1) where W is an offset matrix with the same dimension as X with every value equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='5, βB is a vector of regression coefficients (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='4) randomly sampled with probabilities (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='125) for the 6 contin- uous covariates and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1) for the 18 binary covariates, ωB is an offset chosen to guarantee that ATT = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Therefore, the true individual causal effect can be calculated at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Figure 1 shows the results of 1,000 simulations of this data using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' For each method, we calculate the average of the 1,000 results as the PEHE value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' From this figure, it can be seen clearly that MIRE has the best perfor- mance among all the compared methods, which also shows the effectiveness of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The second good performance of methods is R-LASSO and SDRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Meanwhile, the widely used method, CF (Causal Forest) also has a good performance but is worse than MIRE, R-LASSO and SDRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The rest methods, NNM (Nearest neighbour matching), PSM (propensity score matching), BART (Bayesian Additive Regression Trees) and BCF (Bayesian Causal Forest) have worse performance than MIRE, R-LASSO, SDRM and CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Estimation of causal effects 1) Jobs: The Jobs dataset is a classic dataset established by LaLonde in 1986 for causal inference [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The dataset was derived from a temporary employment program aimed to provide work skills to people who are faced with economic hardship or lack job skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The treatment is whether an indi- vidual participates in the program, and the potential outcome is the individual income in the year 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Covariates mainly included education, age, race, marital status, income in 1974, MIRE R−LASSO SDRM CF BCF PSM BART NNM 2 3 4 5 PEHE Methods 2 3 4 5 PEHE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' PEHE values estimated using different methods in the IHDP simulation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' and income in 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The average causal effect on the treated samples in the Lalonde dataset was estimated as $886 with a standard error of $448, and we used this estimate ($886) as a criterion to evaluate our method [18], [28], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Table I shows the results of estimating the average causal effect on the treated samples on the Jobs dataset using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' From table I, it can be seen that MIRE has a good performance for ATT estimation, and the estimated ATT ($519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='09) is close to the criterion ($886) with a small SD ($734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='93) that is also close to the empirical SD ($448).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The two methods BART and BCF also show a good performance on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' While the CF estimated ATT has a larger difference from the standard value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Figure 2 shows the scatter plot after dimension reduction of our MIRE on the jobs dataset, which shows the relationship between the reduced variables and the response variable Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' From Figure 2, it can be seen that both covariates after dimen- sion reduction are significantly correlated with the response variable Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The result also shows the rationality of using dimension reduction covariates for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' TABLE I ESTIMATED AVERAGE CAUSAL EFFECT ON THE TREATED SAMPLES ON JOBS DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Methods Estimated ATT RMSE SD NNM 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='16 683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='34 1,280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='70 PSM 1,933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='40 1,090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='60 702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='20 BART 931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='30 2182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='02 1,032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='00 CF 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='24 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='65 890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='61 SDRM 1,740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='98 1,025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='65 710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='55 R-LASSO 1,271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='63 385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='63 825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='23 BCF 697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='88 544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='90 511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='82 MIRE 519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='09 734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='92 734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='93 2) TWINS: The TWINS benchmark dataset was created based on the data of twins born in the United States between Y 60000 60000 40000 40000 20000 20000 0 25000 -20000 -15000 -10000 -5000 0 0 0 5000 10000 1st reduced covariate 2st reduced covariate Y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The relationships between reduced covariates and response variable Y by our MIRE on Jobs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 1983 and 2000, the treatment refers to the heavier weight of the twins at birth [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The potential outcome is the mortality in the first year of twin birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In Louizos et al.’s study [34], one of the two twins was selectively hidden, which was equivalent to randomly assigning the treatment, thus making it similar to the data of randomdised experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Then those twins with weight less than 2kg were selected to establish a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The dataset included 40 covariates such as parents’ education, marital status, race, and mother’s condition at the time of delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' We simulated the presence of confounding factors according to the following formula [34]: Wi∣Xi ∼ Bern(Sigmiod(w′Xi) + n), where W ∼ U(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1)40×1, n ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The average causal effect of our established dataset is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Table II shows the results of estimating the average causal effect on the TWINS dataset using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It can be seen from the results shown in table II: MIRE, CF, BCF, PSM both have really well performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' This also shows that in the existence of confounders, MIRE is able to remove the confounding bias as the stat-of-the-art causal effect estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Figure 3 shows the scatter plot after dimension reduction of the covariates on the TWINS dataset, which shows the relationship between the reduced variables and the response variable Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' From figure 3, we have that the covariates after dimension reduction are significantly correlated with the re- sponse variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It also confirms that MIRE is effective for estimating causal effects from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' RELATED WORK The potential outcome model is widely used in causal inference [1], [13], [14], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Our proposed MIRE method builds on the potential outcome model with mild assumptions, such as the assumptions of the pretreatment variables and the ignorability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The matching method is one of the most commonly used methods in causal inference [13], TABLE II ESTIMATED AVERAGE CAUSAL EFFECT ON TWINS DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Methods Estimated ATE RMSE SD NNM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0468 PSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0502 BART 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='1794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='2793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='3188 CF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0408 SDRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0480 R-LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0623 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0901 BCF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0639 MIRE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0002 Y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='0 −500 0 500 1000 1500 1st reduced covariate 100 200 300 400 2st reduced covariate Y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The relationships between reduced covariates and response variable Y after our MIRE on TWINS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Stuart summarises the principles and steps of the matching method as well as its application in many practical problems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In the following, we review some works that are closely related to our proposed MIRE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' It is difficult to find individuals with multiple dimensions of the covariates between two groups (control and treatment) during the matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' To solve this problem, Rubin and Rosenbaum [13] introduced a propensity score, defined as the conditional probability that a unit is assigned to a certain treatment under the covariate condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In addition, the propensity score has been proven to be a balance score [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The most commonly used propensity score matching method is to use the propensity score in place of the original covariate in matching progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In addition, propensity score matching and Mahalanobis distance matching are combined to generate GenMatch, and GenMatch uses a genetic search algorithm to obtain weights so as to complete matching [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Luo and Zhu used a sufficient dimension reduction (SDR) method to reduce the dimension of the covariates in the treatment group and control group, and then matched them based on Mahalanobis distance [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The most related work to MIRE is the SDR matching proposed by the works [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Luo and Zhu’s work [19] 0000Xconsider reducing the sub-datasets over the treated units and the control units to obtain two of the reduced-dimensional covariates as the balance score for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The proposed method maybe suffers from bias since dividing the whole samples into two sub-datasets results in data insufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' [18] aim to estimate the average causal effect from observational data, but not for heterogeneity causal effect estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In contrast, our theoretical findings support a data- driven method for heterogeneity causal effect estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' In recent, a large number of deep learning-based methods have been proposed for estimating the causal effects from observational data [35]–[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The main advantage of deep learning-based methods is that the complex nonlinear relation- ships between variables can be learned by neural networks and the high-dimensional datasets can be addressed very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Nevertheless, a number of parameters turning are very inefficient, and they do have not good interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Another line work on causal effect estimation from data with latent confounders [7], [8], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' When an instrumental variable (IV) is given, the causal effect of T on Y can be calculated unbiasedly from data with latent variable too [29], [40]–[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Because IV-based estimators do not rely on the ignorability assumption, they are not directly related to our MIRE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' CONCLUSION In this work, we prove that the central DRS by a sufficient dimensional reduction method is a balance score and is suffi- cient to control for confounding bias in causal effect estima- tion from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Our findings provide theoretical support for using the dimension-reduced covariates for match- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Under the proposed theorem, we propose a data-driven method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' MIRE, to estimate the causal effects from obser- vational data under mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Firstly, MIRE utilises the inverse regression estimator to reduce the dimensions of the original covariates, and then uses the reduced-dimensional co- variates for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The advantages of our proposed MIRE have been verified through the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' First, the results of average causal effect estimation based on the Jobs dataset showed that the estimation results of our method were closer to the criterion value ($886) recommended in the previous study than those of other matching methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Second, the results of individual causal effect estimation showed that matching based on dimension-reduced covariates made it easier for individuals to be paired with another group in the matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Our method displayed great advantages in estimating individual causal effects over other matching methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' The estimation results based on the IHDP dataset indicated that our method could match more individuals during the matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' Compared with other causal inference methods, our method also exhibited certain advantages in heterogeneous effect esti- mation accuracy (expressed as PEHE) and confounding factor control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} +page_content=' ACKNOWLEDGMENT This research project was supported in part by the Ma- jor Project of Hubei Hongshan Laboratory under Grant 2022HSZD031, and in part by the Innovation fund of Chi- nese Marine Defense Technology Innovation Center under Grant JJ-2021-722-04, and in part by the National Natural Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQflf2o/content/2301.01549v1.pdf'} 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Applied Sciences and Engineering +ECCOMAS Congress 2022 +5-–9 June 2022, Oslo, Norway +LARGE-EDDY SIMULATIONS OF TURBULENT +COMPRESSIBLE SUPERSONIC JET FLOWS USING +DISCONTINUOUS GALERKIN METHODS +Diego F. Abreu1,†,∗, Carlos Junqueira-Junior2, Eron T. V. Dauricio1,‡ and +Jo˜ao Luiz F. Azevedo3 +1 Instituto Tecnol´ogico de Aeron´autica, 12228–900, S˜ao Jos´e dos Campos, SP, Brazil, +†mecabreu@yahoo.com.br , ‡eron.tiago90@gmail.com +2 Arts et M´etiers Institute of Technology, DynFluid, CNAM, HESAM University, 151 +Boulevard de l’Hˆopital, 75013, Paris, France, junior.junqueira@ensam.eu +3 Instituto de Aeron´autica e Espa¸co, 12228–904, S˜ao Jos´e dos Campos, SP, Brazil, +joaoluiz.azevedo@gmail.com +Key words: Large-Eddy Simulation, Turbulent Flow, Jet Flow, Discontinuous Galerkin Meth- +ods +Abstract. In this work, a discontinuous Galerkin scheme is employed to perform LES simu- +lations of supersonic jet flows. A total of four simulations are performed with different meshes +and order of accuracy. +The number of degrees of freedom from the simulations varies from +50 × 106 to 400 × 106. The results indicate that by increasing the resolution of simulation, in +general, the results got closer to experimental data. The jet lipline is the only region in which +this behavior is not observed. It investigated a procedure of using lower-order simulations to +initialize high-order simulations and consequently reduce the total time of the simulation using +high-order schemes. This strategy is successful and allows to perform high-order simulations +with only 5% more computational effort than a complete second-order simulation. +1 +INTRODUCTION +The Reynolds-Averaged Navier-Stokes (RANS) formulation has difficulty representing some +types of fluid motions predominantly governed by free shear flows or wall-bounded flows with +separated boundary layers. This difficulty is related to constructive assumptions of the for- +mulation, characterized by the modeling of all turbulent quantities. +The recent progress of +computational power is enabling the employment of large-eddy simulations (LES) to simulate +the problems that RANS formulation fails to model important aspects of the flow. Recent studies +show the capability of LES simulations for reproducing free shear layer [4, 21] and detached flows +[13, 22]. Another advantage of using LES is its capability to produce high-frequency unsteady +information, which is necessary for aerodynamics, acoustics, loads, and heat transfer analyses. +The authors are interested in the simulation of supersonic jet flows for performing aerody- +namic analyses of the shear layer regarding velocity and pressure fluctuations to improve the +design of nozzles and adjacent structures. Different numerical options are employed to obtain +1 +arXiv:2301.01773v1 [physics.flu-dyn] 4 Jan 2023 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +the solution of LES formulation for jet flows. For example, low-order accuracy [18] and high- +order accuracy [3, 9] finite difference schemes on structured meshes were employed to perform +LES simulations of subsonic and supersonic jet flows. +Low-order finite volume approach on +unstructured meshes [23, 6] is another option employed. +Due to the employment of structured meshes, the finite difference schemes have difficulty +handling complex geometries. The finite volume schemes are applied to unstructured meshes, +which make it easier to represent complex geometries, however, it is difficult to implement +high-order discretizations with these schemes [15]. In this context, the discontinuous Galerkin +schemes are gaining relevance, because they are easily implemented with high-order accuracy +discretizations and can be employed with unstructured meshes. Some work is already simulating +jet flows with discontinuous Galerkin schemes [1, 8] or using similar strategies, for example, the +Flux Reconstruction schemes [25]. +The discontinuous Galerkin schemes have multiple options for implementation. For example, +one may choose to represent the solution by nodal or modal polynomials. +It is possible to +choose between different options of test functions that could better suits different types of +elements, which are utilized to discretize the computational domain. One set of choices for the +discontinuous Galerkin formulation is named discontinuous Galerkin spectral element method +(DGSEM) [19, 16]. The DGSEM, implemented in a numerical framework called FLEXI [20], +was investigated for performing LES of a supersonic round jet flows with Mach number equal +to 1.4 and Reynolds number based on jet inlet diameter of 1.58 × 106 [1]. +The simulations were performed with two numerical meshes with 6.2 × 106 and 1.8 × 106 +elements with second-order and third-order accurate discretizations, respectively. The two sim- +ulations were performed with nearly 50×106 degrees of freedom (DOF). They presented similar +results, with the simulation performed with third-order accuracy requiring twice the time to +perform the same simulation time as the second-order accurate simulation. When comparing +the results to experimental data, excessive dissipation is observed, which led to shorter potential +cores. The potential core of the jet is the length in the centerline of the jet where the velocity +reaches 0.95 of jet velocity. Other aspects of the flow, for example, the root mean square (RMS) +of velocity fluctuations in the centerline and lipline of the jet, also presented some differences +with experimental data. +In this work, the results obtained using a new mesh are presented. The new mesh has a larger +refinement and improved topology than the meshes utilized in previous work. The new mesh +is simulated with second-order and third-order accurate discretizations. Discussions regarding +the quality and improvement of the simulations are presented. A discussion of computational +efficiency utilizing discontinuous Galerkin methods is also performed to develop guidelines for +future works. +2 +NUMERICAL FORMULATION +2.1 +GOVERNING EQUATIONS +The work has an interest in the solution of the filtered Navier-Stokes equations. The filtering +strategy is based on a spatial filtering process that separates the flow into a resolved part ¯(·) +and a non-resolved part (·)′. Implicit filter size is obtained from the mesh size. The filtered +2 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +Navier-Stokes equations in conservative form can be written by +∂ ¯Q +∂t + ∇ · F(¯Q, ∇¯Q) = 0, +(1) +where ¯Q = [¯ρ, ¯ρ˜u, ¯ρ˜v, ¯ρ ˜w, ¯ρ ˇE]T is the vector of filtered conserved variables and F is the flux +vector. The flux vector can be divided into the Euler fluxes and the viscous flux, F = Fe − Fv. +The fluxes with the filtered variables may be written as +Fe +i = +� +����� +¯ρ˜ui +¯ρ˜u˜ui + δ1i¯p +¯ρ˜v˜ui + δ2i¯p +¯ρ ˜w˜ui + δ3i¯p +(¯ρ ˇE + ¯p)˜ui +� +����� +Fv +i = +� +����� +0 +τ mod +1i +τ mod +2i +τ mod +3i +˜ujτ mod +ij +− qmod +i +� +����� +, for i = 1, 2, 3, +(2) +where ˜ui or (˜u, ˜v, ˜w) are the Favre averaged velocity components, ¯ρ is the filtered density, ¯p +is the filtered pressure and ¯ρ ˇE is the filtered total energy per unit volume. The terms τ mod +ij +and qmod +i +are the modified viscous stress tensor and heat flux vector, respectively, and δij is the +Kronecker delta. The filtered total energy per unit volume, according to the definition proposed +by Vreman [28] in its ”system I” approach, is given by +¯ρ ˇE = +¯p +γ − 1 + 1 +2 ¯ρ˜ui˜ui. +(3) +The filtered pressure, Favre averaged temperature and filtered density are correlated using +the ideal gas equation of state ¯p = ¯ρR ˜T, and R is the gas constant, written as R = cp − cv. The +properties cp and cv are the specific heat at constant pressure and volume, respectively. The +modified viscous stress tensor may be written as +τ mod +ij += (µ + µSGS) +� ∂˜ui +∂xj ++ ∂˜uj +∂xi +� +− 2 +3(µ + µSGS) +�∂˜uk +∂xk +� +δij +(4) +where µ is the dynamic viscosity coefficient, calculated by Sutherland’s Law, and µSGS is the +SGS dynamic viscosity coefficient, which is provided by the subgrid-scale model. The strategy +of modeling the subgrid-scale contribution as an additional dynamic viscosity coefficient is based +on the Boussinesq hyphotesis. The modified heat flux vector, using the same modeling strategy, +is given by +qmod +i += −(k + kSGS) ∂ ˜T +∂xi +(5) +where k is the thermal conductivity coefficient of the fluid and kSGS is the SGS thermal con- +ductivity coefficient given by +kSGS = µSGScp +PrSGS +(6) +and PrSGS is the SGS Prandtl number. The present work employs the static Smagorinsky model +[26] in order to calculate the subgrid-scale contribution. +3 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +2.2 +NODAL DISCONTINUOUS GALERKIN METHOD +The nodal discontinuous Galerkin method used in this work is based on the modeling called +discontinuous Galerkin spectral element method [19, 16]. In this modeling strategy, the domain +is divided into multiple hexahedral elements. This choice of elements permits the interpolating +polynomial to be defined as a tensor product basis with degree N in each space direction. This +set of options leads to an algorithm that can be easily implemented and also produce a high +level of computational efficiency due to reduced calculations. +In this method, the elements from the physical domain are mapped onto a reference unit +cube elements E = [−1, 1]3. The equations, presented in (1) need also to be mapped to this new +reference domain, leading to +J ∂ ¯Q +∂t + ∇ξ · ¯F = 0, +(7) +where ∇ξ is the divergence operator with respect to the reference element coordinates, ξ = +(ξ1, ξ2, ξ3)T , J = |∂x/∂ξ| is the Jacobian of the coordinate transformation and ¯F is the con- +travariant flux vector. +The discontinuous Galerkin formulation is obtained multiplying (7) by the test function +ψ = ψ(ξ) and integrating over the reference element E +� +E +J ∂ ¯Q +∂t ψdξ + +� +E +∇ξ · ¯Fψdξ = 0. +(8) +It is possible to obtain the weak form of the scheme by integrating by parts the second term in +(8) +∂ +∂t +� +E +J ¯Qψdξ + +� +∂E +( ¯F · ⃗N)∗ψdS − +� +E +¯F · (∇ξψ)dξ = 0, +(9) +where ⃗N is the unit normal vector of the reference element faces. Because the discontinuous +Galerkin scheme allows discontinuities in the interfaces, the surface integral above is ill-defined. +In this case, a numerical flux, ¯F∗, is defined, and a Riemann solver is used to compute the value +of this flux based on the discontinuous solutions given by the elements sharing the interface. +For the nodal form of the discontinuous Galerkin formulation, the solution in each element +is approximated by a polynomial interpolation of the form +¯Q(ξ) ≈ +N +� +p,q,r=0 +¯Qh(ξ1 +p, ξ2 +q, ξ3 +r, t)φpqr(ξ), +(10) +where ¯Qh(ξ1 +p, ξ2 +q, ξ3 +r, t) is the value of the vector of conserved variables at each interpolation node +in the reference element and φpqr(ξ) is the interpolating polynomial. For hexahedral elements, +the interpolating polynomial is a tensor product basis with degree N in each space direction +φpqr(ξ) = lp(ξ1)lq(ξ2)lr(ξ3), +lp(ξ1) = +Np +� +i=0 +i̸=p +ξ1 − ξ1 +i +ξ1p − ξ1 +i +. +(11) +The definitions presented are applicable to other two directions. +4 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +The numerical scheme used in the simulation additionally presents the split formulation [24], +with the discrete form [11], to enhance the stability of the simulation. The split formulation +is employed for Euler fluxes only. The solution and the fluxes are interpolated and integrated +at the nodes of a Gauss-Lobatto Legende quadrature, which presents the summation-by-parts +property, that is necessary to employ the split formulation. +The Riemann solver used in the simulations is a Roe scheme with entropy fix [14] to ensure +that the second law of thermodynamics is respected, even with the split formulation. For the +viscous flux, since the discontinuous Galerkin scheme is not suitable for discretizing the high +order derivative operator, the lifting scheme of Bassi and Rebay [2] is used, which is also known +for BR2. The time marching method chosen is a five-stage, fourth-order explicit Runge-Kutta +scheme [7]. The shock waves that appear in the simulation are stabilized using the finite-volume +sub-cell shock-capturing method [27]. The shock indicator of Jameson, Schmidt, and Turkel [17] +is utilized. +3 +EXPERIMENTAL CONFIGURATION +The experimental work [5] provides a good characterization of the flow properties for jet +flows. Many configurations were analyzed. In this work, the interest is to simulate the fully +expanded free jet flow configuration with a Mach number of 1.4. In this configuration the jet +flow has a static pressure in the nozzle exit section that equals the ambient static pressure with +a supersonic velocity, for this reason, it is possible to avoid the use of nozzle wall geometries and +also the shock waves are weaker when compared to other operating conditions. +The experimental apparatus for analyzed configuration is composed of a convergent-divergent +nozzle designed with the method of characteristics [5]. The nozzle exit diameter is 50.8 mm. +The Reynolds number based on nozzle exit diameter is approximately 1.58 × 106, which is large +when compared to other jet experiments available in the literature. +The data acquisition in the tests applies Time-Resolved Particle Image Velocimetry (TRPIV) +operated primarily with a 10 kHz sample rate. The experiment uses two sets of cameras, one +positioned to capture the flow along the nozzle centerline and the other positioned to capture +the flow of the mixing layer along the nozzle lipline. +4 +NUMERICAL SETUP +4.1 +GEOMETRY AND MESH CONFIGURATION +The geometry used for the calculations in the work presents a divergent shape and axis length +of 40D, where D is the jet inlet diameter and has external diameters of 16D and 25D. Figure 1 +illustrates a 2-D representation of the computational domain indicating the inlet surface in red, +the far-field region in blue, the lipline in gray, and the centerline in black. +The computational grids used in the work are named M-1, M-2, and M-3. The M-1 and M-2 +meshes are adaptations of the mesh utilized in previous work [18] due to the different restrictions +of each computational code. The M-3 mesh is generated with topological differences from M-1 +and M-2 meshes. The M-3 mesh topology presents a high refinement level around the jet inlet +boundary external diameter that transitions to a uniform distribution when moving forward in +the longitudinal direction. In addition to the new topology, the M-3 mesh also presents a larger +number of elements. The mesh generation uses a multiblock strategy since the FLEXI solver +5 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +only handles hexahedral elements. +Fig. 2 exhibits a cut plane of the M-2 and M-3 meshes. M-2 mesh is presented to illustrate +the topological differences between the two strategies. M-1 mesh is not presented because it +only differs in the number of elements from M-2 mesh. The M-1 and M-2 meshes have a total +of 6.2 × 106 and 1.8 × 106 elements that are simulated with second and third-order accuracy, +respectively, resulting in simulations with 50× 106 DOF. The M-2 mesh has 15.4 × 106 elements +and is simulated with second and third-order accuracy, resulting in approximately 120×106 and +410×106 DOF. All the meshes utilized in the work are generated with the GMSH [12] generator. +Figure 1: 2-D schematic representation of the computational domain used on the jet flow simu- +lations. +(a) M-2 mesh. +(b) M-3 mesh. +Figure 2: Visualization of the half-plane longitudinal cut planes for the meshes used in the +present work. +6 + +Sponge region +neY +7Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +4.2 +BOUNDARY CONDITIONS +different reference states to characterize the jet inflow, (·)jet, and the far-field, (·)ff. The +inflow and the far-field surfaces are indicated in Fig. 1 in red and blue, respectively. A weakly +enforced solution of a Riemann problem with a Dirichlet condition is enforced at the boundaries. +The flow is characterized as perfectly expanded and unheated, i.e. pjet/pff = Tjet/Tff = 1, +where p stands for pressure and T for temperature. The Mach number of the jet at the inlet is +Mjet = 1.4 and the Reynolds number based on the diameter of the nozzle is Rejet = 1.58 × 106. +A small velocity component with Mff = 0.01 in the streamwise direction is imposed at the far- +field to avoid numerical issues. A sponge zone [10] is employed close to all far-field boundaries +to dump any oscillation that could reach the boundaries. +4.3 +SIMULATION SETTINGS +A total of 4 simulations are compared in this work. The development of the simulations +utilized 3 different meshes with two orders of accuracy obtained by changing the degree of the +polynomial representing the solution. The S-1 simulation utilizes the M-1 mesh with second- +order accuracy. The S-2 simulation utilizes the M-2 mesh with third-order accuracy. The S-3 +and S-4 simulations utilize the M-3 mesh with second and third-order accuracy, respectively. +Table 1 summarizes the simulations performed and the total number of degrees of freedom in +each of them. +Table 1: Summary of simulations settings. +Simulation +Meshes +Order of +DOF/cell +Cells +Total # of DOF +Accuracy +(106) +(106) +S-1 +M-1 +2nd order +8 +6.2 +≈ 50 +S-2 +M-2 +3rd order +27 +1.8 +≈ 50 +S-3 +M-3 +2nd order +8 +15.4 +≈ 120 +S-4 +M-3 +3rd order +27 +15.4 +≈ 410 +4.4 +CALCULATION OF STATISTICAL PROPERTIES +Two different approaches are taken to perform the 4 simulations. +In the first approach, +utilized for S-1, S-2, and S-3 simulations, the procedure involves three steps. The first one is to +clean off the domain since the computation starts with a quiescent flow initial condition. The +simulations run three flow-through times (FTT) to develop the jet flow. One FTT is the time +required for one particle with the jet velocity to cross the computational domain. In the sequence, +the simulations run an additional three FTT to produce a statistically steady condition. Then, +in the last step, data are collected with a sample of approximately 250 kHz for another FTT +to obtain the statistical properties of the flow. In the second approach, utilized for the S-4 +simulation, the solution obtained from the S-3 simulation is utilized as the initial condition. +The simulation is performed for 0.5 FTT to clean the second-order accuracy solution and allow +it to provide a third-order accuracy solution. Then 2 additional FTT are simulated to extract +data for the analysis. The cost of S4 simulation is higher than other simulations and the authors +7 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +had some difficulties to stabilize the simulation, which consumed some available computational +resources, for this reason it was not possible to run 3 FTT to obtain the statistics. +The mean and the root mean square (RMS) fluctuations of properties of the flow are cal- +culated along the centerline, lipline, and different domain surfaces in the streamwise direction. +The centerline is defined as the line in the center of the geometry y/D = 0, whereas the lipline +is a surface parallel to the centerline and located at the nozzle diameter, y/D = 0.5. The results +from the lipline are an azimuthal mean from six equally spaced positions. The four surfaces +in the streamwise positions are x/D = 2.5, x/D = 5.0, x/D = 10.0, and x/D = 15.0. Fig. 3 +illustrates a snapshot of the jet flow with the lines and surfaces of data extraction. Mach number +contours are presented in the figure. +Figure 3: Snapshot of the jet simulation with the two longitudinal lines and three crossflow lines +along which data is extracted. Mach number contours are shown. +5 +RESULTS +5.1 +ANALYSIS OF NUMERICAL RESULTS +The results from S1, S2, S3, and S4 simulations are presented in this section, which is di- +vided into two parts to group different types of comparisons. In the first part, contours of mean +longitudinal velocity, RMS longitudinal velocity fluctuation, and mean density are presented for +each simulation. In the second part, the distribution of mean longitudinal velocity and RMS +of longitudinal velocity fluctuation are presented along the jet centerline and lipline for the +four simulations and compared to experimental data. In the final results, the mean longitudi- +nal velocity, RMS of longitudinal velocity fluctuation, RMS of radial velocity fluctuation, and +shear-stress tensor are presented in four spanwise lines for all the simulations and compared to +experimental data. +In the first part, three main aspects can be analyzed from the different contours investigated +and each contour is better to discuss one of the three aspects. The length of the potential core +cannot be directly assessed from visual inspection, so the authors prefer to refer to the region +of high velocity that can be easily inspected from the results of mean longitudinal velocity. The +development of the shear layer can be visualized in all results, however, the one in which its +intensity can be better visualized is in the results of RMS of longitudinal velocity fluctuation. +The last aspect that can be assessed from the contours is the development of the series of shock +and expansion waves in the early stages of the jet. +8 + +Mach number +X/D=2.5 +X/D=5 +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +Lipline +CenterlineDiego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +Figure 4 presents the contours for the mean longitudinal velocity for all simulations. In Figs. +4a and 4b the contours of velocity are very similar, with the high velocity region in Fig. 4b being +slightly longer. Analyzing the results in Fig. 4c, one can observe that the high-velocity region +has increased significantly when compared to previous results. The improvement in the results +obtained shows the importance of distributing elements where they are necessary. Finally, in Fig. +4d, the results from S4 simulation are presented. It is possible to observe that the high-velocity +region is the longest among all the simulations, which is indicative that it was lacking resolution +in previous simulations to adequately capture the development of the jet flow by adding too +much dissipation. +(a) S1 simulation. +(b) S2 simulation. +(c) S3 simulation. +(d) S4 simulation. +Figure 4: Contours of mean longitudinal velocity component on cutplane in z/D = 0. +In Fig. 5 the contours of RMS of longitudinal velocity fluctuation is presented. Once more, +the results presented in Figs. 5a and 5b from S1 and S2 simulations are similar, with the shear- +layer development starting approximately 1D far from the jet inlet section. Just after the initial +of the shear layer development, one can observe that the peak of RMS fluctuation occurs, which +can be related to the large difference between the velocities and possibly the transition of the +shear layer from laminar to turbulent. The results presented in Fig. 5c from the S3 simulation +have significant differences from the other two previously discussed. The development of the +shear layer is starting closer to the jet inlet section with smaller peaks of RMS of velocity +fluctuation and with a smaller spreading. +One can visualize that the two mixing layers are +crossing in the center of the jet farther in Fig. 5c than in Figs. 5a and 5b even presenting +a sooner development. Analyzing the results in Fig. 5d, it can be observed that tendencies +from the previous investigation continued to increase, which means that the beginning of the +development of the shear layer got even closer to the jet inlet section and the crossing of the two +mixing layers is happening farther from jet inlet section than the results from S3 simulation, +Fig. 5c. S4 simulation is the one with smaller spreading and early development of the shear +layer among all the simulations. +Finally, in Fig. 6, the contours of mean density are presented for all simulations. In these +9 + +0.15 +0. 1- +0.05 +1.0 +Y (m)0 +/Ujet +-0.05 +0.2 +-0.1- +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15- +0. 1- +0.05 +1.0 +0.8 +Y (m)0 +/Ujet +加 +-0.05 +-0.1- +0.2 +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15- +0. 1- +0.05 +1.0 +Y (m)0 +/Ujet +-0.05 +0.2 +-0.1- +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15 +0. 1- +0.05 +1.0 +Y (m)0 +/Ujet +-0.05 +-0.1- +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.9Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +(a) S1 simulation. +(b) S2 simulation. +(c) S3 simulation. +(d) S4 simulation. +Figure 5: Contours of RMS of longitudinal velocity fluctuation on cutplane in z/D = 0. +results, it is possible to better visualize the development of the series of shocks and expansion +waves. Different from what has been observed in Figs. 4 and 5, where the results from S1 and +S2 simulation are very similar, in Figs. 6a and 6b the results of mean pressure for S1 and S2 +simulations, one can observe clear differences regarding the series of shock and expansion waves. +In Fig. 6a only three sets of shocks and expansion waves are clearly visible, while in Fig. 6b it +is possible to observe more than 6 sets. It is possible to observe also that the sets of shocks +and expansion waves from the S2 simulation are stronger than those from the S1 simulation. +Analyzing Fig. 6c, it is possible to observe that the S3 simulation produced even more sets +of shocks and expansion waves than the S2 simulation, Fig. 6b, with larger intensity, that is +evaluated by the level of variation of density produced by the shocks and expansion waves. +Another aspect that can be observed is that the first set of shock and expansion waves in the +S3 simulation is occurring closer to the jet inlet section than in S2 and S1 simulations, and +appears to be a relation to the first set of shock and expansion waves with the beginning of the +development of the shear layer. In Fig. 6d the results from S4 simulation is presented. One +can observe the largest set of shock and expansion waves among all the simulations and also a +thinner representation of the shocks and expansion waves, which can be closely related to the +increased resolution of the simulation. It is also possible to observe a reduction in the intensity +of the sets of shock and expansion waves when compared to the S3 simulation, Fig. 6c. The +first part of this section is concluded with the comparison of the contours of mean longitudinal +velocity, RMS of longitudinal velocity fluctuation, and mean density among the simulations. In +the second part, the numerical results are compared with experimental data. +In the second part, in Fig. 7, the numerical results of all simulations are compared to the +experimental data. +In Fig. 7a the distribution of mean longitudinal velocity < U > /Uj is +presented in the centerline of the jet. +One can observe in the figure that results from S1 +and S2 simulations are almost equal. +Results from the S3 simulation present a significant +improvement when compared to previously performed simulations and the S4 simulation could +10 + +0.15 +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujet +-0.05 +0.10 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15- +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujet +0.10 +-0.05 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15- +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujet +0.10 +-0.05 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x9%) +0.6 +0.7 +0.8 +0.90.15 +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujei +-0.05 +0.10 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.9Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +(a) S1 simulation. +(b) S2 simulation. +(c) S3 simulation. +(d) S4 simulation. +Figure 6: Contours of mean density on cutplane in z/D = 0. +almost capture the behavior observed in the experiments. In Fig. 7b the RMS of longitudinal +velocity fluctuation urms/Uj is presented in the centerline of the jet. It is possible to observe +once more in these results how well the resolution influenced the numerical results regarding the +proximity to experimental data. The differences between S1 and S2 simulations are small, the +S3 simulation got closer to experimental data and the S4 simulation once more presented the +best match with experimental data. It can be observed in the results from the S4 simulation +a double peak that does not appear in any other simulation or the experimental data. The +authors believe that this result is a consequence of the fewer FFTs of the simulation in which +data was gathered. Once more data could be used, authors believe that this characteristic would +disappear and only one peak would be formed. +While the results for the jet centerline present always improvements in the simulations with +increased resolution, in the lipline that behavior is not always observed. In Fig. 7c, where the +mean longitudinal velocity < U > /Uj is presented in the lipline of the jet one may observe +that far from the jet inlet section, the increased resolution produced a monotonic improvement +in the numerical results, close to the jet inlet section, S3 simulation was the one that could +better capture experimental data. However, it is not from the mean results where the greatest +differences are observed. When analyzing the distribution of the RMS of longitudinal velocity +distribution along the lipline of the jet, Fig. 7d, one can observe that monotonically the increased +resolution pushed the results away from experimental data. While in the experimental data it +is possible to observe a smooth growth of RMS of longitudinal velocity fluctuation and almost +a plateau from x/D = 5 to x/D = 15, in all the simulations there is a sudden increase in the +RMS of longitudinal velocity fluctuation and after the peak, it is possible only to observe a +reduction on the values. The authors believe that the differences observed in these results are +related to the choice of the boundary condition imposed for the jet inlet that represents neither +the boundary layer profile from the nozzle nor the turbulent intensity in the nozzle exit section. +The results in Fig. 8 present different statistical properties of the flow in different longitudinal +11 + +0.15 +0. 1 +0.05 +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0.1 +0.2 +0.3 +0.4 +x9%) +0.6 +0.7 +0.8 +0.90.15 +0. 1 +0.05 +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x9%) +0.6 +0.7 +0.8 +0.90.15 +0. 1 +0.05- +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0.1 +0.2 +0.3 +0.4 +x9§) +0.6 +0.7 +0.8 +0.90.15- +0.1 +0.05 +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0.1 +0.2 +0.3 +0.4 +x9§) +0.6 +0.7 +0.8 +0.9Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +(a) Centerline +(b) Centerline +(c) Lipline +(d) Lipline +Figure 7: Results of mean streamwise velocity component distribution (left) and RMS of stream- +wise velocity fluctuation (right) in the jet centerline y/D = 0 (top) and lipline y/D = 0.5 +(bottom). +positions. The first set of results, in Figs. 8a to 8d, concerns the mean of longitudinal velocity. +The S1 simulation is in agreement with the experimental data at x/D = 2.5, Fig. 8a. In the +position x/D = 5, Fig. 8b all the simulations produce very similar results. Moving forward and +analyzing the results in position x/D = 10, Fig. 8c it is possible to observe a similar behavior +between S1 and S2 simulations, S3 simulation presenting improvements to the other two and S4 +simulation presenting the best match with experimental data. In the last position x/D = 15, +Fig. 8a, a monotonically improvement is observed with increased resolution the the simulations. +12 + +1.2 +S1 +S2 +S3 +S4 +/U, +Exp +0.8 +0.6 +0.4 +0 +5 +10 +15 +20 +x/D0.15 +0.1 +rms +S1 +S2 +0.05 +S3 +S4 +Exp +0 +5 +10 +15 +20 +x/D0.8 +S1 +0.7 +S2 +S3 +0.6 +S4 +/U, +Exp +0.5 +0.4 +0.3 +0.2 +0 +5 +10 +15 +20 +x/D0.2 +S1 +S2 +S3 +0.15 +S4 +Exp +rms +0.1 +u +0.05 +0 +0 +5 +10 +15 +20 +x/DDiego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +(a) x/D = 2.5 +(b) x/D = 5 +(c) x/D = 10 +(d) x/D = 15 +(e) x/D = 2.5 +(f) x/D = 5 +(g) x/D = 10 +(h) x/D = 15 +(i) x/D = 2.5 +(j) x/D = 5 +(k) x/D = 10 +(l) x/D = 15 +(m) x/D = 2.5 +(n) x/D = 5 +(o) x/D = 10 +(p) x/D = 15 +Figure 8: Profiles of mean streamwise velocity component, RMS of streamwise velocity fluctua- +tion, RMS of radial velocity fluctuation, and mean shear-stress tensor component (from top to +bottom) at four streamwise positions x/D = 2.5, x/D = 5, x/D = 10 and x/D = 15 (from left +to right). +13 + +1.5 +1 +0.5 +0 +S1 +-0.5 +S2 +S3 +-1 +S4 +Exp +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +0 +-0.5 +1 +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +y/D +0 +S1 +-0.5 +S2 +S3 +.1 +S4 +Exp +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +y/D +0 +-0.5 +-1 +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +y/D +0 +S1 +-0.5 +S2 +S3 +.1 +S4 +Exp +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +S1 +S2 +-0.5 +S3 +-1 +S4 +Exp +-1.5 +-0.01 +0 +0.01 +/U?1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +-0.01 +0 +0.01 +/U?1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +-0.01 +0 +0.01 +/U?1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +-0.01 +0 +0.01 +/U?Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +The profiles of RMS values of longitudinal velocity fluctuation are presented in Figs. 8e to 8h. +The simulation results at x/D = 2.5, Fig. 8e, present a similar profile with differences only in +the peak of RMS of longitudinal velocity fluctuation with a monotonically decrease in the peak +with increased resolution and proximity to experimental data. Similar behavior is observed in +the next position at x/D = 5, Fig. 8f. In the position x/D = 10, Fig. 8g the main aspects of the +flow are captured except for the smaller values of RMS of longitudinal velocity fluctuation in +the center of the jet that is only badly captured by S4 simulation. The results for all simulations +and experimental data are very similar in the last position x/D = 15, Fig. 8h +Profiles of RMS values of radial velocity component fluctuation are presented in Figs. 8i +to 8l. +They exhibit similar behavior as the longitudinal velocity fluctuation. +One can also +observe the positive effects of the increased resolution on the profiles of the mean shear-stress +tensor component, Figs. 8m to 8p. The profiles from the S4 simulation are in good agreement +with the experimental data and they indicate considerable improvement when compared to the +simulations with smaller resolution. +This result concludes our analysis of the numerical results from the simulations. +It was +possible to observe, in general, that the improved resolution of the simulations produced better +results compared to experimental data. +The simulation with the highest resolution, the S4 +simulation, was the one that better matched the experimental data. Only in the jet lipline, this +behavior is not observed. The authors have strong confidence that this has nothing to do with +the effect of the resolution, instead, it may be strongly related to the choice of the boundary +condition for the jet inlet condition, that does not reproduce the boundary layer developed inside +the nozzle nor the turbulent intensity in the region. To improve the quality of the simulations in +the jet lipline, a new boundary condition or a different simulation strategy should be adopted. +5.2 +ANALYSIS OF COMPUTATIONAL EFFORT +At this point of the work, it is important to discuss some other aspects of computational +effort to be able to improve the computational efficiency of the simulations. The main parameter +utilized to measure the efficiency of a simulation is the Performance Index PID, which can be +calculated by +PID = +wall clock time ncores +nDOF ntimesteps nRK−stages +, +(12) +where wall clock time is the time the simulation needed to perform ntimesteps time steps, ncores +is the quantity of cores used in the simulation, nDOF is the number of DOF of the simulation +and nRK−stages is the number of stages from the Runge-Kutta scheme. The PID was calculated +for all four simulations and the results are presented in Tab. 2. +Table 2: Summary of Performance Index PID from all simulations. +Simulation +PID (µs) +S1 +8 +S2 +15 +S3 +5 +S4 +2 +14 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +It is important to clarify to the author that the numerical solver presented some improvements +during the execution of the simulations and they can be related to the improvements in the PID +reduction from S3 and S4 simulations compared to S1 and S2 simulations. It is also possible to +argue that with the increased number of degrees of freedom it is expected that more computation +is performed with a similar number of cores, which also should increase the efficiency of the +simulations. If we compare only S1 and S2 simulations, for the same number of DOF, the third- +order accurate simulations cost almost twice the effort of a second-order accurate simulation. If +these values are employed in the two simulations procedure performed in this analysis, the total +cost of the simulation can be compared. +The first simulation procedure involves the whole calculation of the 9 FTT with a mesh of +50 × 106 elements that produces ≈ 400 × 106 DOF when simulated with second-order accurate +discretization, which could produce a very similar result to those of S4 simulation. +If the +procedure for S4 simulation could be completely performed, it could initially start its 5 FTT +with second-order accurate discretization with a total of 120 × 106 DOF. At this point, we +do not consider the effect of the number of degrees of freedom in the PID, only the order of +discretization. For this 5 FTT, the total time would be 3.34× smaller than those to perform the +first simulation procedure due to the reduced number of degrees of freedom. Then, it is possible +to consider the next 4 FTT calculated with third-order accurate discretization and a total of +≈ 400 × 106 DOF. In this 4FTT the cost of the third-order accurate discretization is twice of +the second-order accurate simulation. If the total time of the second procedure is calculated, it +costs ≈ 5% more than the procedure with second-order accuracy. +What is discussed here is that, once it is possible to start the high-order simulation with a +previous result from another order of accuracy with the same mesh, it is possible to reduce the +time required to obtain the desired data with high-order simulation and consequently reduce +the cost of the total simulation. In the proposed procedure, the cost of the total third-order +accurate simulation was only 5% larger than those of a second-order accurate simulation. This +result is expressive and very interesting for high-order simulations. +Another important point to present is that the computational code utilized presented very +good scalability with the number of cores. The tests for the S4 simulation varied the number of +cores from a few hundred cores to a few thousand cores and the PID was always close to 2µs. +6 +CONCLUDING REMARKS +In this work, the employment of a discontinuous Galerkin framework called FLEXI was +investigated for the LES simulation of supersonic free round jets. A total of four simulations are +performed with 3 different meshes and second and third-order accuracy. The range of degrees +of freedom from the simulations varies from 50 × 106 to 400 × 106. All the simulations are +performed for the same geometric model and with the same boundary conditions. +The results of the simulations are firstly compared visually between themselves to compare +how they are capturing the main features of the flow: extension of the high-velocity region, +development of the shear layer, and development of the sets of shocks and expansion waves. +The results showed that with increased resolution the high-velocity regions got longer. The +development of the shear layer starts closer to the jet inlet section and presents a smaller +spreading. The number of sets of shocks and expansion waves increased and the visual of the +15 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. Azevedo +shocks and expansion waves are thinner. +The numerical results showed that, in general, the increase in the resolution of the simulation, +especially the number of degrees of freedom, produced better results when compared to experi- +mental data. This behavior is observed in the results of mean longitudinal velocity distribution +and RMS of longitudinal velocity fluctuation distribution in the centerline. It is also observed +in the results from the four spanwise planes. The only results that do not follow this behavior +are the mean longitudinal and RMS of longitudinal velocity fluctuation distributions along the +jet lipline. In these regions, with increased resolutions, the results are pushed away from exper- +imental data. The authors believe this behavior is related to the lack of resolution from the jet +inlet boundary condition that could not reproduce the boundary layer and turbulent intensity +from the experiments. +The analysis of the computational effort of the simulation showed that even utilizing a high- +order method that costs more than a second-order method for the same number of degrees of +freedom it was possible to reproduce a third-order simulation with only 5% more computational +cost of total simulation by initializing the simulation with a smaller order of accuracy. +The work reached is objective of identifying the guidelines for performing LES simulations +of supersonic jet flows using a discontinuous Galerkin scheme with adequate results with a +reasonable computational cost. The open point on the jet inlet condition is the next step in the +development of the work. +ACKNOWLEDGMENTS +The authors acknowledge the support for the present research provided by Conselho Nacional +de Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, under the Research Grant No. 309985/2013- +7. The work is also supported by the computational resources from the Center for Mathematical +Sciences Applied to Industry, CeMEAI, funded by Funda¸c˜ao de Amparo `a Pesquisa do Estado +de S˜ao Paulo, FAPESP, under the Research Grant No. 2013/07375-0. +The authors further +acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for pro- +viding HPC resources of the SDumont supercomputer. This work was also granted access to the +HPC resources of IDRIS under the allocation 2020-A0092A12067 / 2021-A0112A12067 made +by GENCI. The first author acknowledges authorization by his employer, Embraer S.A., which +has allowed his participation in the present research effort. The doctoral scholarship provide by +FAPESP to the third author, under the Grant No. 2018/05524-1, is thankfully acknowledged. +Additional support to the fourth author under the FAPESP Research Grant No. 2013/07375- +0 is also gratefully acknowledged. +This study was financed in part by the Coordena¸c˜ao de +Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001. +REFERENCES +[1] Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio, and Jo˜ao Luiz F. Azevedo. +A comparison of low and high-order methods for the simulation of supersonic jet flows. In +Proceedings of the 26th ABCM International Congress of Mechanical Engineering, COBEM +Paper No. 2021-0388, Florian´opolis, Santa Catarina, Brazil, November 2021. +[2] F Bassi and S Rebay. A high-order accurate discontinuous finite element method for the +16 + +Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and Jo˜ao Luiz F. 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PhD thesis, University of Twente, Twente, Netherlands, 1995. +18 + diff --git a/PdAzT4oBgHgl3EQfzv5V/content/tmp_files/load_file.txt b/PdAzT4oBgHgl3EQfzv5V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..439981c45f33f6585de4aa0d1cf5090093c2e083 --- /dev/null +++ b/PdAzT4oBgHgl3EQfzv5V/content/tmp_files/load_file.txt @@ -0,0 +1,919 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf,len=918 +page_content='The 8th European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS Congress 2022 5-–9 June 2022, Oslo, Norway LARGE-EDDY SIMULATIONS OF TURBULENT COMPRESSIBLE SUPERSONIC JET FLOWS USING DISCONTINUOUS GALERKIN METHODS Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu1,†,∗, Carlos Junqueira-Junior2, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio1,‡ and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo3 1 Instituto Tecnol´ogico de Aeron´autica, 12228–900, S˜ao Jos´e dos Campos, SP, Brazil, †mecabreu@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='br , ‡eron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='tiago90@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='com 2 Arts et M´etiers Institute of Technology, DynFluid, CNAM, HESAM University, 151 Boulevard de l’Hˆopital, 75013, Paris, France, junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='junqueira@ensam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='eu 3 Instituto de Aeron´autica e Espa¸co, 12228–904, S˜ao Jos´e dos Campos, SP, Brazil, joaoluiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='azevedo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='com Key words: Large-Eddy Simulation, Turbulent Flow, Jet Flow, Discontinuous Galerkin Meth- ods Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this work, a discontinuous Galerkin scheme is employed to perform LES simu- lations of supersonic jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' A total of four simulations are performed with different meshes and order of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The number of degrees of freedom from the simulations varies from 50 × 106 to 400 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results indicate that by increasing the resolution of simulation, in general, the results got closer to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The jet lipline is the only region in which this behavior is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It investigated a procedure of using lower-order simulations to initialize high-order simulations and consequently reduce the total time of the simulation using high-order schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This strategy is successful and allows to perform high-order simulations with only 5% more computational effort than a complete second-order simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 1 INTRODUCTION The Reynolds-Averaged Navier-Stokes (RANS) formulation has difficulty representing some types of fluid motions predominantly governed by free shear flows or wall-bounded flows with separated boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This difficulty is related to constructive assumptions of the for- mulation, characterized by the modeling of all turbulent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The recent progress of computational power is enabling the employment of large-eddy simulations (LES) to simulate the problems that RANS formulation fails to model important aspects of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Recent studies show the capability of LES simulations for reproducing free shear layer [4, 21] and detached flows [13, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Another advantage of using LES is its capability to produce high-frequency unsteady information, which is necessary for aerodynamics, acoustics, loads, and heat transfer analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The authors are interested in the simulation of supersonic jet flows for performing aerody- namic analyses of the shear layer regarding velocity and pressure fluctuations to improve the design of nozzles and adjacent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Different numerical options are employed to obtain 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='01773v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='flu-dyn] 4 Jan 2023 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo the solution of LES formulation for jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' For example, low-order accuracy [18] and high- order accuracy [3, 9] finite difference schemes on structured meshes were employed to perform LES simulations of subsonic and supersonic jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Low-order finite volume approach on unstructured meshes [23, 6] is another option employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Due to the employment of structured meshes, the finite difference schemes have difficulty handling complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The finite volume schemes are applied to unstructured meshes, which make it easier to represent complex geometries, however, it is difficult to implement high-order discretizations with these schemes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this context, the discontinuous Galerkin schemes are gaining relevance, because they are easily implemented with high-order accuracy discretizations and can be employed with unstructured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Some work is already simulating jet flows with discontinuous Galerkin schemes [1, 8] or using similar strategies, for example, the Flux Reconstruction schemes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The discontinuous Galerkin schemes have multiple options for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' For example, one may choose to represent the solution by nodal or modal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is possible to choose between different options of test functions that could better suits different types of elements, which are utilized to discretize the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' One set of choices for the discontinuous Galerkin formulation is named discontinuous Galerkin spectral element method (DGSEM) [19, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The DGSEM, implemented in a numerical framework called FLEXI [20], was investigated for performing LES of a supersonic round jet flows with Mach number equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 and Reynolds number based on jet inlet diameter of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='58 × 106 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The simulations were performed with two numerical meshes with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 × 106 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 × 106 elements with second-order and third-order accurate discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The two sim- ulations were performed with nearly 50×106 degrees of freedom (DOF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' They presented similar results, with the simulation performed with third-order accuracy requiring twice the time to perform the same simulation time as the second-order accurate simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' When comparing the results to experimental data, excessive dissipation is observed, which led to shorter potential cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The potential core of the jet is the length in the centerline of the jet where the velocity reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='95 of jet velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Other aspects of the flow, for example, the root mean square (RMS) of velocity fluctuations in the centerline and lipline of the jet, also presented some differences with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this work, the results obtained using a new mesh are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The new mesh has a larger refinement and improved topology than the meshes utilized in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The new mesh is simulated with second-order and third-order accurate discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Discussions regarding the quality and improvement of the simulations are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' A discussion of computational efficiency utilizing discontinuous Galerkin methods is also performed to develop guidelines for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 2 NUMERICAL FORMULATION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='1 GOVERNING EQUATIONS The work has an interest in the solution of the filtered Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The filtering strategy is based on a spatial filtering process that separates the flow into a resolved part ¯(·) and a non-resolved part (·)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Implicit filter size is obtained from the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The filtered 2 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo Navier-Stokes equations in conservative form can be written by ∂ ¯Q ∂t + ∇ · F(¯Q, ∇¯Q) = 0, (1) where ¯Q = [¯ρ, ¯ρ˜u, ¯ρ˜v, ¯ρ ˜w, ¯ρ ˇE]T is the vector of filtered conserved variables and F is the flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The flux vector can be divided into the Euler fluxes and the viscous flux, F = Fe − Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The fluxes with the filtered variables may be written as Fe i = � ����� ¯ρ˜ui ¯ρ˜u˜ui + δ1i¯p ¯ρ˜v˜ui + δ2i¯p ¯ρ ˜w˜ui + δ3i¯p (¯ρ ˇE + ¯p)˜ui � ����� Fv i = � ����� 0 τ mod 1i τ mod 2i τ mod 3i ˜ujτ mod ij − qmod i � ����� , for i = 1, 2, 3, (2) where ˜ui or (˜u, ˜v, ˜w) are the Favre averaged velocity components, ¯ρ is the filtered density, ¯p is the filtered pressure and ¯ρ ˇE is the filtered total energy per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The terms τ mod ij and qmod i are the modified viscous stress tensor and heat flux vector, respectively, and δij is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The filtered total energy per unit volume, according to the definition proposed by Vreman [28] in its ”system I” approach, is given by ¯ρ ˇE = ¯p γ − 1 + 1 2 ¯ρ˜ui˜ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (3) The filtered pressure, Favre averaged temperature and filtered density are correlated using the ideal gas equation of state ¯p = ¯ρR ˜T, and R is the gas constant, written as R = cp − cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The properties cp and cv are the specific heat at constant pressure and volume, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The modified viscous stress tensor may be written as τ mod ij = (µ + µSGS) � ∂˜ui ∂xj + ∂˜uj ∂xi � − 2 3(µ + µSGS) �∂˜uk ∂xk � δij (4) where µ is the dynamic viscosity coefficient, calculated by Sutherland’s Law, and µSGS is the SGS dynamic viscosity coefficient, which is provided by the subgrid-scale model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The strategy of modeling the subgrid-scale contribution as an additional dynamic viscosity coefficient is based on the Boussinesq hyphotesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The modified heat flux vector, using the same modeling strategy, is given by qmod i = −(k + kSGS) ∂ ˜T ∂xi (5) where k is the thermal conductivity coefficient of the fluid and kSGS is the SGS thermal con- ductivity coefficient given by kSGS = µSGScp PrSGS (6) and PrSGS is the SGS Prandtl number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The present work employs the static Smagorinsky model [26] in order to calculate the subgrid-scale contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 3 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 NODAL DISCONTINUOUS GALERKIN METHOD The nodal discontinuous Galerkin method used in this work is based on the modeling called discontinuous Galerkin spectral element method [19, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this modeling strategy, the domain is divided into multiple hexahedral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This choice of elements permits the interpolating polynomial to be defined as a tensor product basis with degree N in each space direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This set of options leads to an algorithm that can be easily implemented and also produce a high level of computational efficiency due to reduced calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this method, the elements from the physical domain are mapped onto a reference unit cube elements E = [−1, 1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The equations, presented in (1) need also to be mapped to this new reference domain, leading to J ∂ ¯Q ∂t + ∇ξ · ¯F = 0, (7) where ∇ξ is the divergence operator with respect to the reference element coordinates, ξ = (ξ1, ξ2, ξ3)T , J = |∂x/∂ξ| is the Jacobian of the coordinate transformation and ¯F is the con- travariant flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The discontinuous Galerkin formulation is obtained multiplying (7) by the test function ψ = ψ(ξ) and integrating over the reference element E � E J ∂ ¯Q ∂t ψdξ + � E ∇ξ · ¯Fψdξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (8) It is possible to obtain the weak form of the scheme by integrating by parts the second term in (8) ∂ ∂t � E J ¯Qψdξ + � ∂E ( ¯F · ⃗N)∗ψdS − � E ¯F · (∇ξψ)dξ = 0, (9) where ⃗N is the unit normal vector of the reference element faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Because the discontinuous Galerkin scheme allows discontinuities in the interfaces, the surface integral above is ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this case, a numerical flux, ¯F∗, is defined, and a Riemann solver is used to compute the value of this flux based on the discontinuous solutions given by the elements sharing the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' For the nodal form of the discontinuous Galerkin formulation, the solution in each element is approximated by a polynomial interpolation of the form ¯Q(ξ) ≈ N � p,q,r=0 ¯Qh(ξ1 p, ξ2 q, ξ3 r, t)φpqr(ξ), (10) where ¯Qh(ξ1 p, ξ2 q, ξ3 r, t) is the value of the vector of conserved variables at each interpolation node in the reference element and φpqr(ξ) is the interpolating polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' For hexahedral elements, the interpolating polynomial is a tensor product basis with degree N in each space direction φpqr(ξ) = lp(ξ1)lq(ξ2)lr(ξ3), lp(ξ1) = Np � i=0 i̸=p ξ1 − ξ1 i ξ1p − ξ1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (11) The definitions presented are applicable to other two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo The numerical scheme used in the simulation additionally presents the split formulation [24], with the discrete form [11], to enhance the stability of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The split formulation is employed for Euler fluxes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The solution and the fluxes are interpolated and integrated at the nodes of a Gauss-Lobatto Legende quadrature, which presents the summation-by-parts property, that is necessary to employ the split formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The Riemann solver used in the simulations is a Roe scheme with entropy fix [14] to ensure that the second law of thermodynamics is respected, even with the split formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' For the viscous flux, since the discontinuous Galerkin scheme is not suitable for discretizing the high order derivative operator, the lifting scheme of Bassi and Rebay [2] is used, which is also known for BR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The time marching method chosen is a five-stage, fourth-order explicit Runge-Kutta scheme [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The shock waves that appear in the simulation are stabilized using the finite-volume sub-cell shock-capturing method [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The shock indicator of Jameson, Schmidt, and Turkel [17] is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 3 EXPERIMENTAL CONFIGURATION The experimental work [5] provides a good characterization of the flow properties for jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Many configurations were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this work, the interest is to simulate the fully expanded free jet flow configuration with a Mach number of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this configuration the jet flow has a static pressure in the nozzle exit section that equals the ambient static pressure with a supersonic velocity, for this reason, it is possible to avoid the use of nozzle wall geometries and also the shock waves are weaker when compared to other operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The experimental apparatus for analyzed configuration is composed of a convergent-divergent nozzle designed with the method of characteristics [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The nozzle exit diameter is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The Reynolds number based on nozzle exit diameter is approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='58 × 106, which is large when compared to other jet experiments available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The data acquisition in the tests applies Time-Resolved Particle Image Velocimetry (TRPIV) operated primarily with a 10 kHz sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The experiment uses two sets of cameras, one positioned to capture the flow along the nozzle centerline and the other positioned to capture the flow of the mixing layer along the nozzle lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4 NUMERICAL SETUP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='1 GEOMETRY AND MESH CONFIGURATION The geometry used for the calculations in the work presents a divergent shape and axis length of 40D, where D is the jet inlet diameter and has external diameters of 16D and 25D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 1 illustrates a 2-D representation of the computational domain indicating the inlet surface in red, the far-field region in blue, the lipline in gray, and the centerline in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The computational grids used in the work are named M-1, M-2, and M-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The M-1 and M-2 meshes are adaptations of the mesh utilized in previous work [18] due to the different restrictions of each computational code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The M-3 mesh is generated with topological differences from M-1 and M-2 meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The M-3 mesh topology presents a high refinement level around the jet inlet boundary external diameter that transitions to a uniform distribution when moving forward in the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In addition to the new topology, the M-3 mesh also presents a larger number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The mesh generation uses a multiblock strategy since the FLEXI solver 5 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo only handles hexahedral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 2 exhibits a cut plane of the M-2 and M-3 meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' M-2 mesh is presented to illustrate the topological differences between the two strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' M-1 mesh is not presented because it only differs in the number of elements from M-2 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The M-1 and M-2 meshes have a total of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 × 106 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 × 106 elements that are simulated with second and third-order accuracy, respectively, resulting in simulations with 50× 106 DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The M-2 mesh has 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 × 106 elements and is simulated with second and third-order accuracy, resulting in approximately 120×106 and 410×106 DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' All the meshes utilized in the work are generated with the GMSH [12] generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 1: 2-D schematic representation of the computational domain used on the jet flow simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (a) M-2 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (b) M-3 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 2: Visualization of the half-plane longitudinal cut planes for the meshes used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6 Sponge region neY 7Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 BOUNDARY CONDITIONS different reference states to characterize the jet inflow, (·)jet, and the far-field, (·)ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The inflow and the far-field surfaces are indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 1 in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' A weakly enforced solution of a Riemann problem with a Dirichlet condition is enforced at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The flow is characterized as perfectly expanded and unheated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' pjet/pff = Tjet/Tff = 1, where p stands for pressure and T for temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The Mach number of the jet at the inlet is Mjet = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 and the Reynolds number based on the diameter of the nozzle is Rejet = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='58 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' A small velocity component with Mff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='01 in the streamwise direction is imposed at the far- field to avoid numerical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' A sponge zone [10] is employed close to all far-field boundaries to dump any oscillation that could reach the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='3 SIMULATION SETTINGS A total of 4 simulations are compared in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The development of the simulations utilized 3 different meshes with two orders of accuracy obtained by changing the degree of the polynomial representing the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The S-1 simulation utilizes the M-1 mesh with second- order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The S-2 simulation utilizes the M-2 mesh with third-order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The S-3 and S-4 simulations utilize the M-3 mesh with second and third-order accuracy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Table 1 summarizes the simulations performed and the total number of degrees of freedom in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Table 1: Summary of simulations settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Simulation Meshes Order of DOF/cell Cells Total # of DOF Accuracy (106) (106) S-1 M-1 2nd order 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 ≈ 50 S-2 M-2 3rd order 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 ≈ 50 S-3 M-3 2nd order 8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 ≈ 120 S-4 M-3 3rd order 27 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 ≈ 410 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 CALCULATION OF STATISTICAL PROPERTIES Two different approaches are taken to perform the 4 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the first approach, utilized for S-1, S-2, and S-3 simulations, the procedure involves three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The first one is to clean off the domain since the computation starts with a quiescent flow initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The simulations run three flow-through times (FTT) to develop the jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' One FTT is the time required for one particle with the jet velocity to cross the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the sequence, the simulations run an additional three FTT to produce a statistically steady condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Then, in the last step, data are collected with a sample of approximately 250 kHz for another FTT to obtain the statistical properties of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the second approach, utilized for the S-4 simulation, the solution obtained from the S-3 simulation is utilized as the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The simulation is performed for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 FTT to clean the second-order accuracy solution and allow it to provide a third-order accuracy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Then 2 additional FTT are simulated to extract data for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The cost of S4 simulation is higher than other simulations and the authors 7 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo had some difficulties to stabilize the simulation, which consumed some available computational resources, for this reason it was not possible to run 3 FTT to obtain the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The mean and the root mean square (RMS) fluctuations of properties of the flow are cal- culated along the centerline, lipline, and different domain surfaces in the streamwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The centerline is defined as the line in the center of the geometry y/D = 0, whereas the lipline is a surface parallel to the centerline and located at the nozzle diameter, y/D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results from the lipline are an azimuthal mean from six equally spaced positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The four surfaces in the streamwise positions are x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5, x/D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='0, x/D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='0, and x/D = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 3 illustrates a snapshot of the jet flow with the lines and surfaces of data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Mach number contours are presented in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 3: Snapshot of the jet simulation with the two longitudinal lines and three crossflow lines along which data is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Mach number contours are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5 RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='1 ANALYSIS OF NUMERICAL RESULTS The results from S1, S2, S3, and S4 simulations are presented in this section, which is di- vided into two parts to group different types of comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the first part, contours of mean longitudinal velocity, RMS longitudinal velocity fluctuation, and mean density are presented for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the second part, the distribution of mean longitudinal velocity and RMS of longitudinal velocity fluctuation are presented along the jet centerline and lipline for the four simulations and compared to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the final results, the mean longitudi- nal velocity, RMS of longitudinal velocity fluctuation, RMS of radial velocity fluctuation, and shear-stress tensor are presented in four spanwise lines for all the simulations and compared to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the first part, three main aspects can be analyzed from the different contours investigated and each contour is better to discuss one of the three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The length of the potential core cannot be directly assessed from visual inspection, so the authors prefer to refer to the region of high velocity that can be easily inspected from the results of mean longitudinal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The development of the shear layer can be visualized in all results, however, the one in which its intensity can be better visualized is in the results of RMS of longitudinal velocity fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The last aspect that can be assessed from the contours is the development of the series of shock and expansion waves in the early stages of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8 Mach number X/D=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 X/D=5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 Lipline CenterlineDiego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo Figure 4 presents the contours for the mean longitudinal velocity for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4a and 4b the contours of velocity are very similar, with the high velocity region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4b being slightly longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Analyzing the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4c, one can observe that the high-velocity region has increased significantly when compared to previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The improvement in the results obtained shows the importance of distributing elements where they are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4d, the results from S4 simulation are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is possible to observe that the high-velocity region is the longest among all the simulations, which is indicative that it was lacking resolution in previous simulations to adequately capture the development of the jet flow by adding too much dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (a) S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (b) S2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (c) S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (d) S4 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 4: Contours of mean longitudinal velocity component on cutplane in z/D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5 the contours of RMS of longitudinal velocity fluctuation is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Once more, the results presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5a and 5b from S1 and S2 simulations are similar, with the shear- layer development starting approximately 1D far from the jet inlet section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Just after the initial of the shear layer development, one can observe that the peak of RMS fluctuation occurs, which can be related to the large difference between the velocities and possibly the transition of the shear layer from laminar to turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5c from the S3 simulation have significant differences from the other two previously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The development of the shear layer is starting closer to the jet inlet section with smaller peaks of RMS of velocity fluctuation and with a smaller spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' One can visualize that the two mixing layers are crossing in the center of the jet farther in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5c than in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5a and 5b even presenting a sooner development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Analyzing the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5d, it can be observed that tendencies from the previous investigation continued to increase, which means that the beginning of the development of the shear layer got even closer to the jet inlet section and the crossing of the two mixing layers is happening farther from jet inlet section than the results from S3 simulation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' S4 simulation is the one with smaller spreading and early development of the shear layer among all the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6, the contours of mean density are presented for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In these 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='9Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo (a) S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (b) S2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (c) S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (d) S4 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 5: Contours of RMS of longitudinal velocity fluctuation on cutplane in z/D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' results, it is possible to better visualize the development of the series of shocks and expansion waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Different from what has been observed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 4 and 5, where the results from S1 and S2 simulation are very similar, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6a and 6b the results of mean pressure for S1 and S2 simulations, one can observe clear differences regarding the series of shock and expansion waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6a only three sets of shocks and expansion waves are clearly visible, while in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6b it is possible to observe more than 6 sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is possible to observe also that the sets of shocks and expansion waves from the S2 simulation are stronger than those from the S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Analyzing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6c, it is possible to observe that the S3 simulation produced even more sets of shocks and expansion waves than the S2 simulation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6b, with larger intensity, that is evaluated by the level of variation of density produced by the shocks and expansion waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Another aspect that can be observed is that the first set of shock and expansion waves in the S3 simulation is occurring closer to the jet inlet section than in S2 and S1 simulations, and appears to be a relation to the first set of shock and expansion waves with the beginning of the development of the shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6d the results from S4 simulation is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' One can observe the largest set of shock and expansion waves among all the simulations and also a thinner representation of the shocks and expansion waves, which can be closely related to the increased resolution of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is also possible to observe a reduction in the intensity of the sets of shock and expansion waves when compared to the S3 simulation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The first part of this section is concluded with the comparison of the contours of mean longitudinal velocity, RMS of longitudinal velocity fluctuation, and mean density among the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the second part, the numerical results are compared with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the second part, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 7, the numerical results of all simulations are compared to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 7a the distribution of mean longitudinal velocity < U > /Uj is presented in the centerline of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' One can observe in the figure that results from S1 and S2 simulations are almost equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Results from the S3 simulation present a significant improvement when compared to previously performed simulations and the S4 simulation could 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='9Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo (a) S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (b) S2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (c) S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' (d) S4 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Figure 6: Contours of mean density on cutplane in z/D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' almost capture the behavior observed in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 7b the RMS of longitudinal velocity fluctuation urms/Uj is presented in the centerline of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is possible to observe once more in these results how well the resolution influenced the numerical results regarding the proximity to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The differences between S1 and S2 simulations are small, the S3 simulation got closer to experimental data and the S4 simulation once more presented the best match with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It can be observed in the results from the S4 simulation a double peak that does not appear in any other simulation or the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The authors believe that this result is a consequence of the fewer FFTs of the simulation in which data was gathered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Once more data could be used, authors believe that this characteristic would disappear and only one peak would be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' While the results for the jet centerline present always improvements in the simulations with increased resolution, in the lipline that behavior is not always observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 7c, where the mean longitudinal velocity < U > /Uj is presented in the lipline of the jet one may observe that far from the jet inlet section, the increased resolution produced a monotonic improvement in the numerical results, close to the jet inlet section, S3 simulation was the one that could better capture experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' However, it is not from the mean results where the greatest differences are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' When analyzing the distribution of the RMS of longitudinal velocity distribution along the lipline of the jet, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 7d, one can observe that monotonically the increased resolution pushed the results away from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' While in the experimental data it is possible to observe a smooth growth of RMS of longitudinal velocity fluctuation and almost a plateau from x/D = 5 to x/D = 15, in all the simulations there is a sudden increase in the RMS of longitudinal velocity fluctuation and after the peak, it is possible only to observe a reduction on the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The authors believe that the differences observed in these results are related to the choice of the boundary condition imposed for the jet inlet that represents neither the boundary layer profile from the nozzle nor the turbulent intensity in the nozzle exit section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8 present different statistical properties of the flow in different longitudinal 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='15 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='9Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo (a) Centerline (b) Centerline (c) Lipline (d) Lipline Figure 7: Results of mean streamwise velocity component distribution (left) and RMS of stream- wise velocity fluctuation (right) in the jet centerline y/D = 0 (top) and lipline y/D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The first set of results, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8a to 8d, concerns the mean of longitudinal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The S1 simulation is in agreement with the experimental data at x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the position x/D = 5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8b all the simulations produce very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Moving forward and analyzing the results in position x/D = 10, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8c it is possible to observe a similar behavior between S1 and S2 simulations, S3 simulation presenting improvements to the other two and S4 simulation presenting the best match with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the last position x/D = 15, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8a, a monotonically improvement is observed with increased resolution the the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 S1 S2 S3 S4 /U, Exp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 0 5 10 15 20 x/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='1 rms S1 S2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='05 S3 S4 Exp 0 5 10 15 20 x/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='8 S1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='7 S2 S3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='6 S4 /U, Exp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 0 5 10 15 20 x/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 S1 S2 S3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='15 S4 Exp rms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='1 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='05 0 0 5 10 15 20 x/DDiego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo (a) x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 (b) x/D = 5 (c) x/D = 10 (d) x/D = 15 (e) x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 (f) x/D = 5 (g) x/D = 10 (h) x/D = 15 (i) x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 (j) x/D = 5 (k) x/D = 10 (l) x/D = 15 (m) x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 (n) x/D = 5 (o) x/D = 10 (p) x/D = 15 Figure 8: Profiles of mean streamwise velocity component, RMS of streamwise velocity fluctua- tion, RMS of radial velocity fluctuation, and mean shear-stress tensor component (from top to bottom) at four streamwise positions x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5, x/D = 5, x/D = 10 and x/D = 15 (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 0 S1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 S2 S3 1 S4 Exp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='01 /U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo The profiles of RMS values of longitudinal velocity fluctuation are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8e to 8h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The simulation results at x/D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8e, present a similar profile with differences only in the peak of RMS of longitudinal velocity fluctuation with a monotonically decrease in the peak with increased resolution and proximity to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Similar behavior is observed in the next position at x/D = 5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the position x/D = 10, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8g the main aspects of the flow are captured except for the smaller values of RMS of longitudinal velocity fluctuation in the center of the jet that is only badly captured by S4 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results for all simulations and experimental data are very similar in the last position x/D = 15, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8h Profiles of RMS values of radial velocity component fluctuation are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8i to 8l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' They exhibit similar behavior as the longitudinal velocity fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' One can also observe the positive effects of the increased resolution on the profiles of the mean shear-stress tensor component, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 8m to 8p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The profiles from the S4 simulation are in good agreement with the experimental data and they indicate considerable improvement when compared to the simulations with smaller resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This result concludes our analysis of the numerical results from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It was possible to observe, in general, that the improved resolution of the simulations produced better results compared to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The simulation with the highest resolution, the S4 simulation, was the one that better matched the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Only in the jet lipline, this behavior is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The authors have strong confidence that this has nothing to do with the effect of the resolution, instead, it may be strongly related to the choice of the boundary condition for the jet inlet condition, that does not reproduce the boundary layer developed inside the nozzle nor the turbulent intensity in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' To improve the quality of the simulations in the jet lipline, a new boundary condition or a different simulation strategy should be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='2 ANALYSIS OF COMPUTATIONAL EFFORT At this point of the work, it is important to discuss some other aspects of computational effort to be able to improve the computational efficiency of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The main parameter utilized to measure the efficiency of a simulation is the Performance Index PID, which can be calculated by PID = wall clock time ncores nDOF ntimesteps nRK−stages , (12) where wall clock time is the time the simulation needed to perform ntimesteps time steps, ncores is the quantity of cores used in the simulation, nDOF is the number of DOF of the simulation and nRK−stages is the number of stages from the Runge-Kutta scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The PID was calculated for all four simulations and the results are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Table 2: Summary of Performance Index PID from all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Simulation PID (µs) S1 8 S2 15 S3 5 S4 2 14 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo It is important to clarify to the author that the numerical solver presented some improvements during the execution of the simulations and they can be related to the improvements in the PID reduction from S3 and S4 simulations compared to S1 and S2 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is also possible to argue that with the increased number of degrees of freedom it is expected that more computation is performed with a similar number of cores, which also should increase the efficiency of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' If we compare only S1 and S2 simulations, for the same number of DOF, the third- order accurate simulations cost almost twice the effort of a second-order accurate simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' If these values are employed in the two simulations procedure performed in this analysis, the total cost of the simulation can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The first simulation procedure involves the whole calculation of the 9 FTT with a mesh of 50 × 106 elements that produces ≈ 400 × 106 DOF when simulated with second-order accurate discretization, which could produce a very similar result to those of S4 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' If the procedure for S4 simulation could be completely performed, it could initially start its 5 FTT with second-order accurate discretization with a total of 120 × 106 DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' At this point, we do not consider the effect of the number of degrees of freedom in the PID, only the order of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' For this 5 FTT, the total time would be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='34× smaller than those to perform the first simulation procedure due to the reduced number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Then, it is possible to consider the next 4 FTT calculated with third-order accurate discretization and a total of ≈ 400 × 106 DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In this 4FTT the cost of the third-order accurate discretization is twice of the second-order accurate simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' If the total time of the second procedure is calculated, it costs ≈ 5% more than the procedure with second-order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' What is discussed here is that, once it is possible to start the high-order simulation with a previous result from another order of accuracy with the same mesh, it is possible to reduce the time required to obtain the desired data with high-order simulation and consequently reduce the cost of the total simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In the proposed procedure, the cost of the total third-order accurate simulation was only 5% larger than those of a second-order accurate simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This result is expressive and very interesting for high-order simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Another important point to present is that the computational code utilized presented very good scalability with the number of cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The tests for the S4 simulation varied the number of cores from a few hundred cores to a few thousand cores and the PID was always close to 2µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 6 CONCLUDING REMARKS In this work, the employment of a discontinuous Galerkin framework called FLEXI was investigated for the LES simulation of supersonic free round jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' A total of four simulations are performed with 3 different meshes and second and third-order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The range of degrees of freedom from the simulations varies from 50 × 106 to 400 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' All the simulations are performed for the same geometric model and with the same boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results of the simulations are firstly compared visually between themselves to compare how they are capturing the main features of the flow: extension of the high-velocity region, development of the shear layer, and development of the sets of shocks and expansion waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The results showed that with increased resolution the high-velocity regions got longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The development of the shear layer starts closer to the jet inlet section and presents a smaller spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The number of sets of shocks and expansion waves increased and the visual of the 15 Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Abreu, Carlos Junqueira-Junior, Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Dauricio and Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Azevedo shocks and expansion waves are thinner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The numerical results showed that, in general, the increase in the resolution of the simulation, especially the number of degrees of freedom, produced better results when compared to experi- mental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This behavior is observed in the results of mean longitudinal velocity distribution and RMS of longitudinal velocity fluctuation distribution in the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' It is also observed in the results from the four spanwise planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The only results that do not follow this behavior are the mean longitudinal and RMS of longitudinal velocity fluctuation distributions along the jet lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' In these regions, with increased resolutions, the results are pushed away from exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The authors believe this behavior is related to the lack of resolution from the jet inlet boundary condition that could not reproduce the boundary layer and turbulent intensity from the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The analysis of the computational effort of the simulation showed that even utilizing a high- order method that costs more than a second-order method for the same number of degrees of freedom it was possible to reproduce a third-order simulation with only 5% more computational cost of total simulation by initializing the simulation with a smaller order of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The work reached is objective of identifying the guidelines for performing LES simulations of supersonic jet flows using a discontinuous Galerkin scheme with adequate results with a reasonable computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The open point on the jet inlet condition is the next step in the development of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors acknowledge the support for the present research provided by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 309985/2013- 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The work is also supported by the computational resources from the Center for Mathematical Sciences Applied to Industry, CeMEAI, funded by Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo, FAPESP, under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 2013/07375-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The authors further acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for pro- viding HPC resources of the SDumont supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This work was also granted access to the HPC resources of IDRIS under the allocation 2020-A0092A12067 / 2021-A0112A12067 made by GENCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The first author acknowledges authorization by his employer, Embraer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=', which has allowed his participation in the present research effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' The doctoral scholarship provide by FAPESP to the third author, under the Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 2018/05524-1, is thankfully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' Additional support to the fourth author under the FAPESP Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' 2013/07375- 0 is also gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' This study was financed in part by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} +page_content=' REFERENCES 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQfzv5V/content/2301.01773v1.pdf'} diff --git a/QNAzT4oBgHgl3EQfIvuz/content/tmp_files/2301.01068v1.pdf.txt b/QNAzT4oBgHgl3EQfIvuz/content/tmp_files/2301.01068v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..de62993d5dab21256746f192d3016f637d301b34 --- /dev/null +++ b/QNAzT4oBgHgl3EQfIvuz/content/tmp_files/2301.01068v1.pdf.txt @@ -0,0 +1,3705 @@ +1 +Fast Parallel Algorithms for Enumeration of +Simple, Temporal, and Hop-Constrained Cycles +Jovan Blanuˇsa, Kubilay Atasu, and Paolo Ienne +Abstract—Finding cycles in directed graphs enables important applications in various domains such as finance, biology, chemistry, +and network science. However, as the size of graph datasets continues to grow, it becomes increasingly difficult to discover cycles +within them, which necessitates more efficient algorithms and their parallel implementations. In this work, we propose scalable +parallelisation of state-of-the-art sequential algorithms for enumerating simple, temporal, and hop-constrained cycles. First, we focus +on the simple cycle enumeration problem and parallelise the algorithms by Johnson and by Read and Tarjan in a fine-grained manner. +We theoretically show that our resulting fine-grained parallel algorithms are scalable, with the fine-grained parallel Read-Tarjan +algorithm being strongly scalable. In contrast, we show that straightforward coarse-grained parallel versions of these simple cycle +enumeration algorithms that exploit edge- or vertex-level parallelism are not scalable. Next, we adapt our fine-grained approach to +enable scalable parallelisation of state-of-the-art algorithms for temporal and hop-constrained cycle enumeration. Our evaluation on a +cluster with 256 physical cores demonstrates a near-linear scalability of our fine-grained parallel algorithms when enumerating all the +aforementioned types of cycles. On the same cluster, our fine-grained parallel algorithms achieve, on average, one order of magnitude +speedup compared to the respective coarse-grained parallel versions of the state-of-the-art algorithms for cycle enumeration. The +performance gap between the fine-grained and the coarse-grained parallel algorithms increases as we use more CPU cores. +Index Terms—Cycle enumeration; Parallel graph algorithms; Graph pattern mining +! +1 +INTRODUCTION +A graph-based data representation is desirable when ana- +lyzing large and complex datasets because it exposes the +connectivity of the underlying data objects and enables +the discovery of complex relationships between them [1]. +Analysing graph-structured data has important applications +in many domains, such as finance [2], healthcare [3], cyber- +security [4], and advertising [5]. The existence in a graph +of certain patterns, such as cycles, cliques, and motifs, can +reveal nontrivial relationships between different graph ob- +jects [6]. As the volume of graph data continues to grow, the +discovery of such relationships becomes computationally +challenging, requiring more efficient parallel algorithms that +can exploit modern multi-core processors. +Cycle enumeration. This paper introduces efficient par- +allel algorithms for enumerating simple cycles of directed +graphs. A simple cycle is a sequence of edges that starts +and ends with the same vertex and visits other vertices +at most once. Enumerating simple cycles has important +applications in several domains. For example, in electronic +design automation, combinatorial loops in circuits are typ- +ically forbidden [7], [8], and such loops can be detected +© 2023 IEEE. Personal use of this material is permitted. Permission from +IEEE must be obtained for all other uses, in any current or future media, +including reprinting/republishing this material for advertising or promotional +purposes, creating new collective works, for resale or redistribution to servers +or lists, or reuse of any copyrighted component of this work in other works. +• +Jovan Blanuˇsa is with IBM Research Europe - Zurich and with the ´Ecole +Polytechnique F´ed´erale de Lausanne (EPFL) +E-mail: jov@zurich.ibm.com +• +Kubilay Atasu is with IBM Research Europe - Zurich +E-mail: kat@zurich.ibm.com +• +Paolo Ienne is with the ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL) +E-mail: paolo.ienne@epfl.ch +1 +64 +128 +192 +256 +Thread ID +0 +500 +1000 +1500 +Exec. time [s] +1 +64 +128 +192 +256 +Thread ID +0 +500 +1000 +1500 +Exec. time [s] +(a) +(b) +Fig. 1. Per-thread execution time of (a) the coarse-grained Johnson +algorithm vs. (b) our fine-grained Johnson algorithm using the WT graph +and a 12h time window. Thanks to a perfect load balancing, our fine- +grained method is 3× faster on a 64-core CPU executing 256 threads. +by enumerating simple cycles. In a software bug tracking +system, a dependency between two software bugs requires +one bug to be addressed before the other [9]. Circular +bug dependencies are undesirable and can be detected by +finding simple cycles. Other applications include detecting +feedback loops in biological networks [10], [11] and detect- +ing unstable relationships in social networks [12], [13]. +Furthermore, some graphs have their edges annotated +with timestamps, which we refer to as temporal graphs. +In such graphs one can also look for temporal cycles [14], +which are special cases of simple cycles, in which the edges +are ordered in time. For instance, in financial transaction +graphs, a temporal cycle represents a series of transac- +tions in which the money initially sent from one bank +account returns back to the same account; the existence of +such cycles is a strong indicator of financial fraud such as +money laundering, tax avoidance [15], [16], and credit card +fraud [17]. Finding temporal cycles in temporal graphs also +enables detecting circular trading, which can be used for +manipulating stock prices [18], [19], [20]. +Various other kinds of constraints are often imposed +on the cycles being searched because the search may be +computationally impossible otherwise [14], [17], [21]. The +arXiv:2301.01068v1 [cs.DS] 3 Jan 2023 + +2 +TABLE 1 +Our fine-grained parallel Read-Tarjan algorithm is the only solution that +is both work-efficient and scalable. +Parallel algorithm +Work-efficient +Scalable +Coarse-grained parallel algorithms + +Our fine-grained parallel Johnson + +Our fine-grained parallel Read-Tarjan + + +constraints may include hop constraints [17], [21], which +limit the length of paths explored during the search for +cycles, and time-window constraints [14], which restrict +the search to cycles that occur within a time window of a +given size. Imposing these constraints reduces the number +of paths explored during the search for cycles, making +the problem more tractable. For this reason, we focus on +searching for cycles under the aforementioned constraints. +Parallelisation challenges. We focus on parallelising the +algorithms by Johnson [22] and by Read and Tarjan [23] for +finding cycles because these algorithms achieve the lowest +time complexity bounds reported for directed graphs [24], +[25]. Both algorithms are recursively formulated and con- +struct a recursion tree in a depth-first fashion. However, +these algorithms employ different pruning techniques to +limit the amount of work they perform. In practice, the +Johnson algorithm is faster than the Read-Tarjan algorithm +due to more aggressive pruning techniques [24], [25]. Fur- +thermore, the state-of-the-art algorithms for temporal and +hop-constrained cycle enumeration are extensions of the +Johnson algorithm [14], [21]. Thus, parallelising the Johnson +algorithm also enables parallelisation of these temporal and +hop-constrained cycle enumeration algorithms. +The na¨ıve way of parallelising the Johnson and the Read- +Tarjan algorithms involves searching for cycles starting from +different vertices or edges in parallel, which we refer to +as the coarse-grained parallel methods. Such coarse-grained +parallel approaches are straightforward to implement using +the popular vertex-centric [26], [27] and edge-centric [28] +graph processing frameworks. However, real-world graphs +often exhibit a power-law or a log-normal distribution of +vertex degrees [29], [30]. In such graphs, the execution +time of coarse-grained parallel approaches is dominated by +searches that start from a small set of vertices or edges as +illustrated in Fig. 1a. This behaviour leads to a workload +imbalance and limits scalability of parallel implementations. +The shortcomings of coarse-grained parallel approaches +can be addressed by decomposing the search for cycles +starting from a given edge or vertex into finer-grained +tasks [31], [32], [33]. However, parallelising the Johnson +algorithm using the fine-grained approach is challenging +because the pruning efficiency of this algorithm depends on +a strictly sequential depth-first-search-based recursion tree +exploration. We demonstrate that the lesser-known Read- +Tarjan algorithm does not have such a requirement, and, +thus, it is easier to decompose into fine-grained tasks. +Contributions. This paper presents an extension of the +work by Blanuˇsa et al. [34], which introduced the following +contributions: +(i) Scalable fine-grained parallelisation of the Johnson and the +Read-Tarjan algorithms. To our knowledge, we are the first +ones to parallelise these asymptotically-optimal cycle enu- +meration algorithms in a fine-grained manner and achieve +an almost linear performance scaling on a system that +can execute up to a thousand concurrent software threads. +Such a scalability is enabled by our decomposition of long +sequential searches into fine-grained tasks, which are then +dynamically scheduled across CPU cores. To decompose +the Johnson algorithm into fine-grained tasks, we have re- +laxed its strictly depth-first-search-based exploration, which +enables this algorithm to perform multiple independent +depth-first searches in parallel. As a result, our fine-grained +parallel Johnson algorithm is able to achieve an ideal load +balancing as shown in Fig. 1b. +(ii) Theoretical analysis of the coarse- and fine-grained par- +allel algorithm. We theoretically show that both of our fine- +grained parallel algorithms are scalable, which is not the +case for the Johnson and the Read-Tarjan algorithms par- +allelised in a coarse-grained manner. Moreover, we show +that our fine-grained parallel Read-Tarjan algorithm per- +forms asymptotically the same amount of work as its serial +version, whereas our fine-grained parallel Johnson algo- +rithm does not. Therefore, our fine-grained parallel Read- +Tarjan algorithm is the only parallel algorithm based on an +asymptotically-optimal cycle enumeration algorithm that is +both work-efficient and scalable, as shown in Table 1. Inter- +estingly, despite not being work-efficient, our fine-grained +Johnson algorithm outperforms our fine-grained parallel +Read-Tarjan algorithm in most of our experiments. +In this paper, we extend our prior work [34] with the +following contributions: +(iii) General framework for parallelising temporal and hop- +constrained cycle enumeration. We show that our method for +parallelising the Johnson algorithm in a fine-grained manner +can be adapted to parallelise the state-of-the-art algorithms +for temporal and hop-constrained cycle enumeration. This +adaptation is possible because these state-of-the-art algo- +rithms, such as the 2SCENT algorithm for temporal cycle +enumeration [14] and the BC-DFS algorithm [21] for hop- +constrained cycle enumeration, are extensions of the John- +son algorithm. By parallelising these algorithms using our +fine-grained method, we were able to achieve speedups of +up to 40× and 61× compared to the coarse-grained parallel +versions of 2SCENT and BC-DFS, respectively. +(iv) Improvements to the pruning efficiency of the Read- +Tarjan algorithm. To make this algorithm competitive with +the Johnson algorithm, we have introduced several opti- +misations that enhance the pruning efficiency of the Read- +Tarjan algorithm. The optimisations reduce the amount of +unnecessary vertex visits that this algorithm performs. As a +result, our improved version of the Read-Tarjan algorithm is +up to 6.8× faster than the original version of this algorithm. +Paper structure. The remainder of this paper is organised +as follows. The related work and background are presented +in Section 2 and Section 3, respectively. Coarse-grained par- +allel versions of the Johnson and the Read-Tarjan algorithms +are covered in Section 4. Section 5 and Section 6 introduce +our fine-grained parallel versions of the Johnson and the +Read-Tarjan algorithms, respectively. Section 6 also includes +our optimisations for improving the pruning efficiency of +the Read-Tarjan algorithm. Our general framework for par- +allelising temporal and hop-constrained cycle enumeration +algorithms is presented in Section 7. In Section 8, we pro- +vide an experimental evaluation of our fine-grained parallel +algorithms. Finally, we conclude our work in Section 9. + +3 +TABLE 2 +Capabilities of the related work versus our own. Competing algorithms +either fail to exploit fine-grained parallelism or do it on top of +asymptotically inferior algorithms. +Related work +[14] +[17] +[21] +[46] +[47] +Ours +Fine-grained parallelism + + +Asymptotic optimality + + + + +Temporal cycles + + +Time-window constraints + + + +Hop constraints + + + + + +2 +RELATED WORK +Simple cycle enumeration algorithms. Enumeration of sim- +ple cycles of graphs is a classical computer science prob- +lem [22], [23], [24], [25], [35], [36], [37], [38], [39], [40]. The +backtracking-based algorithms by Johnson [22], Read and +Tarjan [23], and Szwarcfiter and Lauer [37] achieve the low- +est time complexity bounds for enumerating simple cycles +in directed graphs. These algorithms implement advanced +recursion tree pruning techniques to improve on the brute- +force Tiernan algorithm [35]. Section 3.4 covers such pruning +techniques in further detail. A cycle enumeration algorithm +that is asymptotically faster than the aforementioned al- +gorithms [22], [23], [37] has been proposed in Birmel´e et +al. [40], however, it is applicable only to undirected graphs. +Simple cycles can also be enumerated by computing the +powers of the adjacency matrix [41], [42], [43] or by using +circuit vector space algorithms [24], [44], [45], but the com- +plexity of such approaches grows exponentially with the +size of the cycles or the size of the input graphs. +Time-window, temporal ordering, and hop constraints. +It is common to search for cycles under some additional +constraints. For instance, in temporal graphs, it is common +to search for cycles within a sliding time window, such as +in Kumar and Calders [14] and Qiu et al [17]. In addition, +temporal ordering constraints can be imposed when search- +ing for cycles in temporal graphs, such as in Kumar and +Calders [14]. Furthermore, the maximum number of hops +in cycles or paths can be constrained, such as in Gupta +and Suzumura [47] and Peng et al. [21]. Note that hop- +constrained simple cycles can also be enumerated using +incremental algorithms, such as in Qiu et al. [17]. How- +ever, this algorithm is based on the brute-force Tiernan +algorithm [35], which makes it slower than nonincremental +algorithms that use recursion tree pruning techniques [21]. +Additionally, because incremental algorithms maintain aux- +iliary data structures, such as paths, to be able to construct +cycles incrementally, they are not as memory-efficient as +nonincremental algorithms [21]. Table 2 offers comparisons +between the capabilities of these methods and ours. +Parallel and distributed algorithms for cycle enumera- +tion. Cui et al. [48] proposed a multi-threaded algorithm for +detecting and removing simple cycles of a directed graph. +The algorithm divides the graph into its strongly-connected +components and each thread performs a depth-first search +on a different component to find cycles. However, sizes of +the strongly-connected components in real-world graphs +can vary significantly [49], which leads to a workload +imbalance. Rocha and Thatte [50] proposed a distributed +algorithm for simple cycle enumeration based on the bulk- +synchronous parallel model [51], but it searches for cycles in +TABLE 3 +Summary of the notation used in the paper. +Symbol +Description +G(V, E) +Graph with vertices V and edges E. +N(v) +The set of neighbours of the vertex v. +u → v +A directed edge connecting vertex u with v. +n, e +Number of vertices and edges in a graph. +δ +Size of a time window. +c +Number of simple cycles in a graph. +s +Number of maximal simple paths in a graph. +Π +Current simple path explored by an algorithm. +Blk +Set of blocked vertices. +Blist +Unblock list of the Johnson algorithm. +E +Path extension of the Read-Tarjan algorithm. +XTi +Data structure X is maintained by the thread Ti. +p +Number of threads used by parallel algorithms. +Tp(n) +Execution time of a parallel algorithm. +Wp(n) +Amount of work a parallel algorithm performs. +a brute-force manner. Qing et al. [46] introduced a parallel +algorithm for finding length-constrained simple cycles. It +is the only other fine-grained parallel algorithm we are +aware of in the sense that it can search for cycles starting +from the same vertex in parallel. However, the way this +algorithm searches for cycles is similar to the way the brute- +force Tiernan algorithm [35] works. To our knowledge, we +are the first ones to introduce fine-grained parallel versions +of asymptotically-optimal simple cycle enumeration algo- +rithms, which do not rely on a brute-force search, as we +show in Table 2. +3 +BACKGROUND +This section introduces the main theoretical concepts used in +this paper and provides an overview of the most prominent +simple cycle enumeration algorithms. The notation used is +given in Table 3. +3.1 +Preliminaries +We consider a directed graph G(V, E) having a set of vertices +V and a set of directed edges E = {u → v | u, v ∈ V}. +The set of neighbours of a given vertex v is defined as +N(v) = {w | +v → w ∈ E}. We refer to the vertex v of +an edge v → u as its source vertex and to the vertex u as +its destination vertex. An outgoing edge of a given vertex +v is defined as v → w and an incoming edge is defined +as u → v, where v → w, u → v ∈ E. A path between +the vertices v0 and vk, denoted as v0 → v1 . . . → vk, is a +sequence of vertices such that there exists an edge between +every two consecutive vertices of the sequence. A simple +path is a path with no repeated vertices. A simple path is +maximal if the last vertex of the path has no neighbours or +all of its neighbours are already in the path [52]. A cycle +is a path of non-zero length from a vertex v to the same +vertex v. A simple cycle is a cycle with no repeated vertices +except for the first and last vertex. The number of maximal +simple paths and the number of simple cycles in a graph +are denoted as s and c, respectively (see Table 3). Note that +s can be exponentially larger than c [36]. A path or a cycle +is said to satisfy a hop-constraint L if the number of edges +in that path or cycle is less than or equal to L. The goal of +simple cycle enumeration is to compute all simple cycles of +a directed graph G, ideally without computing all maximal +simple paths of it. + +4 +15 +11 +13 +10 +14 +7 +2 +5 +1 +6 +12 +(a) Time window [2 : 7] +15 +11 +13 +10 +14 +7 +2 +5 +6 +12 +1 +(b) Time window [10 : 15] +Fig. 2. Two snapshots of a temporal graph associated with two different +time windows of size δ = 5. The solid arrows indicate the edges that +belong to the respective time windows. +A temporal graph is a graph that has its edges annotated +with timestamps. [53]. In temporal graphs, a temporal cycle +is a simple cycle, in which the edges appear in the increasing +order of their timestamps. A simple cycle or a temporal cycle +of a temporal graph occurs within a time window [tw1 : tw2] +if every edge of that cycle has a timestamp ts such that +tw1 ≤ ts ≤ tw2. Fig. 2 shows the simple cycles of a temporal +graph that occur within two different time windows of size +δ = 5. This graph contains one simple cycle in the time +window [2 : 7] (Fig. 2a), which is also a temporal cycle, and +two simple cycles in the time window [10 : 15] (Fig. 2b), +neither being a temporal cycle. +3.2 +Task-level parallelism +The parallel algorithms described in this paper can be im- +plemented using shared-memory parallel processing frame- +works, such as TBB [54], Cilk [55], and OpenMP [56]. These +frameworks enable the decomposition of a program into +tasks that can be independently executed by different soft- +ware threads. In our setup, tasks are created and scheduled +dynamically. A parent task can spawn several child tasks. +The depth of a task is the number of its direct ancestors. A +dynamic task management system assigns the tasks created +to the work queues of the available threads. Furthermore, +a work-stealing scheduler [54], [55], [57] enables a thread +that is not executing a task to steal a task from the work +queue of another thread. Stealing tasks enables dynamic +load balancing and ensures full utilisation of the threads +when there are sufficiently many tasks. +3.3 +Work efficiency and scalability +We use the notions of work efficiency and scalability to analyse +parallel algorithms [58]. We refer to the time to execute a +parallel algorithm on a problem of size n using p threads +as Tp(n). The size of a graph is determined by the number +of vertices n as well as the number of edges e, but we will +refer only to n for simplicity. The depth of an algorithm is the +length of the longest sequence of dependent operations in +the algorithm. The time required to execute such a sequence +is equal to the execution time of the parallel algorithm using +an infinite number of threads, denoted by T∞. Furthermore, +work performed by a parallel algorithm on a problem of +size n using p threads, denoted as Wp(n), is the sum of the +execution times of the individual threads. The work efficiency +and the scalability are formally defined as follows. +Definition 1. (Work efficiency) A parallel algorithm is work- +efficient if and only if Wp(n) ∈ O(T1(n)). +Definition 2. (Scalability) A parallel algorithm is scalable if and +only if lim +n→∞ +� +lim +p→∞ +Tp(n) +T1(n) +� += 0. +Informally, a work-efficient parallel algorithm performs +the same amount of work as its serial version, within +a constant factor. Scalability implies that, for sufficiently +large inputs, increasing the number of threads increases the +speedup of the parallel algorithm with respect to its serial +version. +We also define the notion of strong scalability as fol- +lows [59]. +Definition +3. (Strong scalability) A parallel algorithm is +strongly scalable if and only if T1(n) +Tp(n) = Θ(p) for large enough n. +Whereas +Definition +2 +implies +that +the +speedup +T1(n)/Tp(n) achieved by a parallel algorithm with respect +to its serial execution is infinite when the number of threads +p is infinite, Definition 3 implies that the speedup is always +in the order of p. Another related concept is weak scalability, +which requires the speedup to be in the order of p when +the input size per thread is constant. Note that both strong +scalability and weak scalability imply scalability. +3.4 +Simple cycle enumeration algorithms +The following algorithms for simple cycle enumeration +perform recursive searches to incrementally update simple +paths that can lead to cycles. Each algorithm iterates the +vertices or edges of the graph and independently constructs +a recursion tree to enumerate all the cycles starting from that +vertex or edge. The difference between these algorithms is +to what extent they reduce the redundant work performed +during the recursive search, which we discuss next. +The Tiernan algorithm [35] enumerates simple cycles +using a brute-force search. It recursively extends a simple +path Π by appending a neighbour u of the last vertex v of +Π provided that u is not already in Π. A clear downside +of this algorithm is that it can repeatedly visit vertices that +can never lead to a cycle. When searching for cycles in the +graph shown in Fig. 3a starting from the vertex v0, this +algorithm would explore the path containing b1, . . . , bk 2m +times. From each vertex wi and ui, with i ∈ {1, . . . , m}, the +Tiernan algorithm would explore this path only to discover +that it cannot lead to a simple cycle. As noted by Tarjan [36], +the Tiernan algorithm explores every simple path and, con- +sequently, all maximal simple paths of a graph. Exploring +a maximal simple path takes O(e) time because it requires +visiting each edge of the graph in the worst case. Given a +graph with s maximal simple paths (see Table 3), the worst- +case time complexity of the Tiernan algorithm is O(se). +The Johnson algorithm [22] improves upon the Tiernan +algorithm by avoiding the vertices that cannot lead to sim- +ple cycles when appended to the current simple path Π. +For this purpose, the Johnson algorithm maintains a set of +blocked vertices Blk that are avoided during the search. In +addition, a list of vertices Blist[w] is stored for each blocked +vertex w. Whenever a vertex w is unblocked (i.e., removed +from Blk) by the Johnson algorithm, the vertices in Blist[w] +are also unblocked. This unblocking process is performed + +5 +𝑣1 +. . . +. . . +. . . +. . . +. . . +. . . +. . . +𝑣1 +𝑣2 +𝑤1 +𝑤2 +𝑤𝑚 +𝑢1 +𝑢2 +𝑢𝑚 +𝑏! +𝑏" +𝑏# +𝑏! +𝑏" +𝑏# +𝑣2 +𝑢1 +𝑢2 +𝑢𝑚 +𝑏! +𝑏" +𝑏# +𝑣0 +𝑣0 +Left subtree +Π +𝐸 +𝐸% +(a) Example graph +(b) Recursion tree +𝑣 +𝑢 Vertex 𝑣 in 𝐵𝑙𝑖𝑠𝑡[𝑢] +!0 +!0 +Vertex in !"# +Right subtree +!2 +!" +!1 +Fig. 3. +(a) An example graph and (b) the recursion tree constructed +when searching for cycles that start from v0. The nodes of the recursion +tree represent the recursive calls of the depth-first search. The dotted +path of the right subtree is explored only by the Read-Tarjan algorithm. +recursively until no more vertices can be unblocked, which +we refer to as the recursive unblocking procedure. +A vertex v is blocked (i.e., added to Blk) when visited by +the algorithm. If a cycle is found after recursively exploring +every neighbour of v that is not blocked, the vertex v is +unblocked. However, v is not immediately unblocked if no +cycles are found after exploring its neighbours. Instead, the +Blist data structure is updated to enable unblocking of v in a +later step by adding v to the list Blist[w] of every neighbour +w of v. This delayed unblocking of the vertices enables the +Johnson algorithm to discover each cycle in O(e) time in the +worst case. Because this algorithm requires O(n + e) time +to determine that there are no cycles, its worst-case time +complexity is O (n + e + ec) [37]. Note that because s can +be exponentially larger than c [36], the Johnson algorithm is +asymptotically faster than the Tiernan algorithm. +In the example shown in Fig. 3a, every simple path Π +that starts from v0 and contains vertices b1, . . . , bk is a max- +imal simple path, and, thus, it cannot lead to a simple cycle. +The Johnson algorithm would block b1, . . . , bk immediately +after visiting this sequence once and then keep these vertices +blocked until it finishes exploring the neighbours of v2. As +a result, the Johnson algorithm visits vertices b1, . . . , bk only +once, rather than 2m times the Tiernan algorithm would +visit them. Note that because these vertices get blocked +during the exploration of the left subtree of the recursion +tree, they are not going to be visited again during the +exploration of the right subtree. Effectively, a portion of +the right subtree is pruned (see the dotted path in Fig. 3b) +based on the updates made on Blk and Blist during the +exploration of the left subtree. This strictly sequential depth- +first exploration of the recursion tree is critically important +for achieving a high pruning efficiency, but it also makes +scalable parallelisation of the Johnson algorithm extremely +challenging, which we are going to cover in Section 5. +The Read-Tarjan algorithm [23] also has a worst-case +time complexity of O (n + e + ec). This algorithm main- +tains a current path Π between a starting vertex and a +frontier vertex. A recursive call of this algorithm iterates +the neighbours of the current frontier vertex and performs +a depth-first search (DFS). Assume that v0 is the starting +vertex and v1 is the frontier vertex of Π (see Fig. 3a). From +each neighbour y ∈ {v0, v2} of v1, a DFS tries to find a +path extension E back to v0 that would form a simple cycle +when appended to Π. In the example shown in Fig. 3a, the +algorithm finds two path extensions, one indicated as E and +one that consists of the edge v1 → v0. The algorithm then +explores each path extension by iteratively appending the +vertices from it to the path Π. For each vertex x added to Π, +the algorithm also searches for an alternate path extension +from that vertex x to v0 using a DFS. In the example given +in Fig. 3a, the algorithm iterates through the vertices of the +path extension E and finds an alternate path extension E′ +from the neighbour u1 of v2. If an alternate path extension +is found, a child recursive call is invoked with the updated +current path Π, which is v0 → v1 → v2 in our example. +Otherwise, if all the vertices in E have already been added +to the current path Π, Π is reported as a simple cycle. In our +example, the Read-Tarjan algorithm explores both E and E′ +path extensions, and each leads to the discovery of a cycle. +The Read-Tarjan algorithm also maintains a set of +blocked vertices Blk for recursion-tree pruning. However, +differently from the Johnson algorithm, Blk only keeps track +of the vertices that cannot lead to new cycles when explor- +ing the current path extension within the same recursive +call. The vertices in Blk are avoided while searching for +additional path extensions that branch from the current path +extension. For instance, the left subtree of the recursion tree +shown in Fig. 3b demonstrates the exploration of the path +extension E shown in Fig. 3a. During the exploration of E, +the vertices b1, . . . , bk are added to Blk immediately after +visiting w1, and they are not visited again while exploring +E. However, when exploring another path extension E′ in +the right subtree, the vertices b1, . . . , bk are visited once +again (see the dotted path of the right subtree). As a result, +the Read-Tarjan algorithm visits b1, . . . , bk twice instead of +just once. As we are going to show in Section 6, this draw- +back becomes an advantage when parallelising the Read- +Tarjan algorithm because it enables independent exploration +of different subtrees of the recursion tree. +4 +COARSE-GRAINED PARALLEL METHODS +The most straightforward way of parallelising the Johnson +and the Read-Tarjan algorithms is to search for cycles that +start from different vertices in parallel. Each such search +can then be executed by a different thread that explores its +own recursion tree. This approach is beneficial because it +is work-efficient and can be implemented using one of the +existing graph processing frameworks, such as Pregel [26], +in a manner similar to the method used by Rocha and +Thatte [50]. We refer to this parallelisation approach as the +coarse-grained parallel approach. +The coarse-grained approach can express more paral- +lelism if each thread performs a search for cycles that start +from a different edge rather than a different vertex. This +assumption is supported by the fact that graphs typically +have more edges than vertices. Nevertheless, the coarse- +grained approach is not scalable, which we prove here. +Proposition 1. The coarse-grained parallel Johnson and Read- +Tarjan algorithms are work-efficient. +The proof of Proposition 1 is trivial, and we omit it for +brevity. +Theorem 1. The coarse-grained parallel Johnson and Read- +Tarjan algorithms are not scalable. + +6 +. . . +𝑣! +𝑣"#! +𝑣$ +𝑣"#$ +⋮ +⋮ +⋮ +⋮ +⋮ +𝑣% +𝑣$ +𝑣& +𝑣' +𝑣( +𝑣& +𝑣' +𝑣( +𝑣' +𝑣( +𝑣( +𝑣( +𝑣' +𝑣( +𝑣( +𝑣( +𝑣! +Thread 1 +Thread 2 Thread 3 +(a) Example graph +(b) Recursion tree +Thread 4 +!! +Fig. 4. (a) A graph with an exponential number of simple cycles. (b) The +recursion tree of the Johnson algorithm for n = 6 constructed when the +algorithm starts from v0. Whereas a coarse-grained parallel algorithm +explores the complete recursion tree using a single thread, our fine- +grained parallel algorithms can explore different regions of the recursion +tree in parallel using several threads. +TABLE 4 +Work and depth of the coarse- and fine-grained parallel algorithms. +Parallel algorithm +Work +Depth +Coarse-grained algorithms +O (n + e + ec) +O (ec) +Fine-grained Johnson alg. +O (n + e + min{pce, se}) +O (e) +Fine-grained Read-Tarjan alg. +O (n + e + ec) +O (ne) +Proof. In this case, the depth T∞(n) represents the worst- +case execution time of a search for cycles that starts from +a single vertex or edge, and it depends on the number +of cycles found during this search. In the worst case, a +single recursive search can discover all cycles of a graph. +An example of such graph is given in Fig. 4a, where each +vertex vi, with i ∈ {1, . . . , n − 1}, is connected to v0 and to +every vertex vj such that j > i. In that graph, any subset of +vertices v2, . . . , vn−1 defines a different cycle. Therefore, the +total number of cycles in this graph is equal to the number +of all such subsets c = 2n−2. Before the search for cycles, +both the Johnson and the Read-Tarjan algorithm find all +vertices that start a cycle, which is only v0 in this case. +Therefore, the search for cycles will be performed only by +one thread. Because both the Johnson and the Read-Tarjan +algorithms require O(e) time to find each cycle, the depth +of the coarse-grained algorithms is T∞(n) ∈ O(ec). Because +lim +n→∞ T∞(n)/T1(n) ̸= 0, the coarse-grained algorithms are +not scalable based on Definition 2. +Theorem 1 shows that the main drawback of the coarse- +grained parallel algorithms is their limited scalability. This +limitation is apparent for the graph shown in Fig. 4a, which +has an exponential number of cycles in n. When using +a coarse-grained parallel algorithm on this graph, all the +cycles will be discovered by a single thread, and, thus, the +depth of this algorithm grows linearly with c, as shown in +Table 4. Because only one thread can be effectively utilised, +increasing the number of threads will not result in a re- +duction of the overall execution time of the coarse-grained +parallel algorithm. Fig. 1a shows the workload imbalance +exhibited by the coarse-grained parallel algorithms in prac- +tice. Section 8 demonstrates the limited scalability of coarse- +grained parallel algorithms in further detail. +5 +FINE-GRAINED PARALLEL JOHNSON +To address the load imbalance issues that manifest them- +selves in the coarse-grained parallel Johnson algorithm, +𝑣! +𝑏" +𝑏# +𝑏$ +𝑏! +⋮ +𝑢! 𝑢$ +𝑢# 𝑢% +𝑣$ +𝑣! +𝑢! +𝑢$ +𝑢# +𝑢% +𝑣$ +𝑣$ +𝑣$ +𝑣$ +𝑢% +𝑢% +𝑢% +Infeasible regions +explored by one thread +Infeasible regions explored +by multiple threads +(a) Example graph +(b) Recursion tree +!! +!! +Fig. 5. +(a) An example graph and (b) the recursion tree of our fine- +grained Johnson algorithm when enumerating cycles that start from v0. +Each thread of our fine-grained Johnson algorithm explores the vertices +b1, . . . , bm at most once. +we introduce the fine-grained parallel Johnson algorithm. The +main goal of our fine-grained algorithm is to enable several +threads to explore a recursion tree concurrently, as shown in +Fig. 4b, where each thread executes a subset of the recursive +calls of this tree. However, enabling concurrent exploration +of a recursion tree is in conflict with the sequential depth- +first exploration, required by the Johnson algorithm to +achieve a high pruning efficiency. +In this section, we first discuss the challenges that arise +when parallelising the exploration of a recursion tree of +the Johnson algorithm. Then, we introduce the copy-on-steal +mechanism used to address these challenges and present +our fine-grained parallel Johnson algorithm. Finally, we the- +oretically analyse our algorithm and show that it is scalable. +5.1 +Fine-grained parallelisation challenges +The requirement of the sequential depth-first exploration of +the Johnson algorithm makes it challenging to efficiently +parallelise this algorithm in a fine-grained manner. This +requirement is enforced by maintaining a set of blocked +vertices Blk throughout the exploration of a recursion tree. +If threads exploring the same recursion tree simply share +the same set of blocked vertices Blk, the parallel algorithm +could produce incorrect results. For example, considering +the graph given in Fig. 5a, a thread exploring the path +Π = v0 → v1 → u1 → v2 visit and block the vertex +u4 in this case because u4 cannot participate in a simple +cycle that begins with Π. Because the threads exploring this +graph share the blocked vertices, another thread attempting +to discover the cycle v0 → v1 → u4 → v2 → v0 would fail to +do so because u4 is blocked. Therefore, this approach might +not discover all cycles in a graph. +To enable several threads to correctly find all cycles while +exploring the same recursion tree, the algorithm could for- +ward a new copy of the Blk and Blist data structures when +invoking each child recursive call. However, this approach +would redundantly explore many paths in a graph. The +reason is that a recursive call would be unaware of the +vertices visited and blocked by other calls that precede it +in the depth-first order except for its direct ancestors in +the recursion tree. When enumerating the simple cycles of +the graph shown in Fig. 5a starting from v0, this approach +explores all 4 × 2m−1 + 3 maximal simple paths instead +of just seven, that the Johnson algorithm would explore. +Hence, this approach exhaustively explores all maximal + +7 +Algorithm 1: FGJ task (v, v0, d, T1) +Input: v - the current vertex, v0 - the starting vertex +d - the depth of this task +InOut: T1 - the thread that created this task +Output: true if a cycle was found +1 T2 = the thread executing this task; +// Check if this task is stolen +2 if T1 ̸= T2 then FGJ copyOnSteal(d, T1, T2); +3 MutexT2.lock(); +4 ΠT2.push(v); BlkT2 = BlkT2 ∪ {v}; +5 MutexT2.unlock(); +// Recursively explore the neighbours of v +6 found = false; +7 foreach u : N(v) s.t. u.id > v0.id do +8 +if u = v0 then +9 +report cycle ΠT2; +10 +found = true; +11 +else if u /∈ BlkT2 then +12 +f = spawn FGJ task(u, v0, d + 1, T2); +13 +found = found ∨ f; +14 wait for the spawned tasks; +15 MutexT2.lock(); +16 ΠT2.pop(); +// Unblock vertices if a cycle was found +17 if found then RecursiveUnblock(v, BlkT2, BlistT2); +18 else foreach u : N(v) do BlistT2[u] = BlistT2[u] ∪ {v}; +19 MutexT2.unlock(); +20 return found; +simple paths in the graph and is identical to the brute-force +solution of Tiernan (see Section 3.4). Next, we propose a +fine-grained parallel algorithm that addresses the aforemen- +tioned parallelisation challenges. +5.2 +Copy-on-steal +To enable different threads to concurrently explore the re- +cursion tree in a depth-first fashion while also taking ad- +vantage of the powerful pruning capabilities of the Johnson +algorithm, each thread executing our fine-grained parallel +Johnson algorithm maintains its own copy of the Π, Blk, +and Blist data structures. These data structures are copied +between threads only when these threads attempt to explore +the same recursion tree. To achieve this behaviour, our +fine-grained parallel Johnson algorithm implements each +recursive call of the Johnson algorithm as a separate task. +The pseudocode of this task is given in Algorithm 1, where +a data structure X, maintained by the thread Ti, is denoted +as XTi (see Table 3). If a child task and its parent task are +executed by the same thread Ti, the child task reuses the +ΠTi, Blk Ti, and BlistTi data structures of the parent task. +However, if a child task has been stolen—i.e., it is executed +by a thread other than the thread that created it, the child +task will allocate a new copy of these data structures (line 2 +of Algorithm 1). We refer to this mechanism as copy-on-steal. +The problem with copying data structures between dif- +ferent threads upon task stealing is that the thread that has +created the stolen task (i.e., the victim thread) can modify its +data structures before this task is stolen by another thread +(i.e., the stealing thread). This problem can be observed in +(a) Example graph +(b) Recursion tree +Victim +thread 𝑇! +Stealing +thread 𝑇" +The stolen task +Task +created +Task +stolen +𝑣0 +𝑣1 +𝑣2 +𝑣3 +𝑣4 +𝑣5 +𝑣2 +𝑣0 +𝑣1 +𝑣4 +𝑣5 +𝑣7 +𝑣4 +𝑣5 +𝑣3 +𝑣2 +𝑣3 +𝑣7 +∈ Blk!! +∈ Blk!" = B +⟺ 𝑣 ∈ Blist!![𝑢] +⟺ 𝑣 ∈ Blist!"[𝑢] +𝑣 +𝑢 +𝑣 +𝑢 +𝑣6 +𝑣0 +𝑣6 +𝑣6 +𝑣0 +Task that reports a simple cycle +∈ Π!! +∈ Π!" = $ +Fig. 6. +(a) An example graph and (b) the recursion tree of our fine- +grained Johnson algorithm when enumerating simple cycles that start +from v0. Here, XTi denotes a data structure X of the thread Ti. The +thread T2 can prune the dotted part of the tree by avoiding v5 and v6 +that the thread T1 has blocked after creating the task stolen by T2. +the example shown in Fig. 6. There, the victim thread +T1 and the stealing thread T2 explore the same recursion +tree given in Fig. 6b while searching for cycles that start +with P1 = v0 → v1 → v2 and P2 = v0 → v1 → v7, +respectively. In this case, T2 steals a task created by T1 +that explores v7, as indicated in Fig. 6b, and receives a +copy of the blocked vertices Blk T1 = {v4, v5, v6} discovered +by T1. The thread T1 blocked these vertices because they +cannot participate in any simple cycle that begins with P1. +If T2 simply uses a copy of these blocked vertices Blk T1 +without modifications, T2 will be unable to find the cycle +v0 → v1 → v7 → v4 → v2 → v3 → v0 because v4 is blocked. +Therefore, a method for unblocking vertices after copy-on- +steal is required to correctly find all cycles. +We explore two solutions for this problem: +(i) Copy-on-steal with complete unblocking. To enable +the threads of our algorithm to find cycles after performing +copy-on-steal, the stealing thread could unblock all vertices +that the victim thread had blocked after creating the stolen +task. In our example given in Fig. 6, the stealing thread T2 +unblocks all vertices Blk T1 = {v4, v5, v6} it received from +the victim thread T1. Although this approach enables T2 to +correctly find cycles, it also fails to take advantage of the +information collected by T1 to reduce the redundant work +of T2. For instance, in Fig. 6, T2 visits v5 and v6, even though +T1 already concluded that these vertices cannot participate +in any simple cycle that begins with P = v0 → v1, where P +is the largest common prefix of all the paths explored by T1 +and T2. As a result, T2 redundantly visits the dotted part of +the recursion tree given in Fig. 6b. +(ii) Copy-on-steal with recursive unblocking. In this +approach, the stealing thread capitalises on the information +already discovered by the victim thread. The stealing thread +T2 can reuse a subset B ⊂ Blk T1 of the blocked vertices +discovered by T1 if the vertices in B cannot participate in +simple cycles that begin with P, where P is the largest +common prefix of all the paths explored by T1 and T2. +Because any path discovered by T2 begins with P, T2 can +avoid visiting vertices from B. Thus, to correctly find simple +cycles, it is sufficient for T2 to unblock the vertices from +Blk T1 \ B. To achieve this behaviour, T2 invokes a recursive +unblocking procedure of the Johnson algorithm for every +vertex v ∈ ΠT1 \ P, as shown in Algorithm 2, where ΠT1 is +the path T1 is exploring during task stealing. The vertices in +B can only be unblocked by a recursive unblocking invoked + +8 +Algorithm 2: FGJ copyOnSteal (d, T1, T2) +Input: d - the depth of the task executing this function +InOut: T1 - the victim thread +T2 - the stealing thread +1 MutexT1.lock(); +2 {ΠT2, BlkT2, BlistT2} = copy ({ΠT1, BlkT1, BlistT1}); +3 MutexT1.unlock(); +4 while |ΠT2| ≥ d do +5 +u = ΠT2.pop(); +6 +RecursiveUnblock(u, BlkT2, BlistT2); +for v ∈ P; hence, the vertices in B remain blocked. In the +example given in Fig. 6, T2 invokes a recursive unblocking +procedure for ΠT1 \ P = {v2}, which results in unblocking +of v4. Thus, T2 is able to discover a cycle that contains v4. +The vertices B = {v5, v6} will not be unblocked because +they cannot take part in any simple cycle that begins with +P = v0 → v1. Therefore, thread T2 avoids visiting the +dotted part of the recursion tree given in Fig. 6b. +Without countermeasures, our algorithm can suffer from +race conditions because its data structures can be accessed +concurrently by different threads. For instance, a stealing +thread T2 can copy the data structures of a victim thread +T1 while T1 performs a recursive unblocking, in which +case T2 could receive the vertex set Blk T1 that is partially +unblocked. When using copy-on-steal with recursive un- +blocking, T2 may not be able to continue the interrupted +unblocking of Blk T1, causing the algorithm to miss certain +cycles. To avoid this problem, we define critical sections in +lines 15–19 of Algorithm 1 and in lines 1–3 of Algorithm 2 +using coarse-grained locking by maintaining a mutex per +thread. However, such a locking mechanism is not required +when using copy-on-steal with complete unblocking be- +cause T2 can correctly unblock vertices in Blk T1 simply by +removing all vertices from Blk T1 inserted after the stolen +task was created. Thus, it is sufficient to enable thread-safe +operations on Π, Blk, and Blist using fine-grained locking. +As a result, the critical sections are shorter when the copy- +on-steal with complete unblocking approach is used. +Nevertheless, we opt to use the copy-on-steal with recur- +sive unblocking approach in our fine-grained parallel John- +son algorithm because this approach leads to less redundant +work and rarely suffers from synchronisation overheads. +5.3 +Theoretical analysis +We now show that the fine-grained parallel Johnson algo- +rithm is scalable but not work-efficient. +Theorem 2. The fine-grained parallel Johnson algorithm is not +work-efficient. +Proof. According to Lemma 3 presented by Johnson [22], +a vertex cannot be unblocked more than once unless a +cycle is found, and once a vertex is visited, it can be +visited again only after being unblocked. Thus, the Johnson +algorithm visits each vertex and edge at most c times. In +the fine-grained parallel Johnson algorithm executed using +p threads, each thread maintains a separate set of data +structures used for managing blocked vertices. Because the +threads are unaware of each other’s blocked vertices, each +edge is visited at most pc times, c times by each thread. +Additionally, an edge cannot be visited more than s times +because each maximal simple path of a graph is explored by +a different thread in the worst case, and during each simple +path exploration, an edge is visited at most once. Therefore, +the maximum number of times an edge can be visited by +the fine-grained parallel Johnson algorithm is min {s, pc}. +Because the algorithm executes in O(n + e) time if there +does not exist a cycle or a path in the input graph, the work +performed by the fine-grained parallel Johnson algorithm is +Wp(n) ∈ O (n + e + min{pce, se}) . When c > 0, p > 1, +and s > c, the work performed by the fine-grained parallel +Johnson algorithm Wp(n) is greater than the execution +time T1(n) of the sequential Johnson algorithm. Thus, this +algorithm is not work-efficient. +The work inefficiency of our fine-grained parallel John- +son algorithm occurs if more than one thread performs +the work the sequential Johnson algorithm would perform +between the discovery of two cycles. This behaviour can +be illustrated using the graph from Fig. 5a, which contains +c = 4 cycles and s = c × 2m−1 + 3 maximal simple paths, +each starting from vertex v0. When discovering each cycle, +our fine-grained algorithm explores an infeasible region of +the recursion tree, as shown in Fig. 5b, in which the vertices +b1, . . . , bm are visited. If this infeasible region is explored +using a single thread, each vertex bi, with i ∈ {1, . . . , m}, +will be visited exactly once. However, if p threads are +exploring the same infeasible region of the recursion tree, +vertices b1, . . . , bm will be visited up to p times because the +threads are unaware of each other’s blocked vertices. In this +case, the fine-grained parallel Johnson algorithm performs +more work than necessary, and, thus, it is not work-efficient. +Additionally, each infeasible region of the recursion tree +that visits vertices b1, . . . , bk can be executed by at most +s/c = 2m−1 threads because there are 2m−1 maximal simple +paths that can be explored in each infeasible region. In this +case, each vertex bi, with i ∈ {1, . . . , m}, is visited up to s +times, and, thus, the fine-grained parallel Johnson algorithm +behaves as the Tiernan algorithm (see Section 3.4). +Lemma 1. The depth T∞(n) of the fine-grained parallel Johnson +algorithm is in O(e). +Proof. The worst-case depth of this algorithm occurs when +a thread performs copy-on-steal and explores a maximal +simple path. A thread explores such a path in O(e) time +because it visits at most e edges. As a result, Π and Blk +contain at most e vertices, and Blist contains at most e pairs +of vertices. Therefore, copy-on-steal requires O(e) time to +copy Π, Blk, and Blist, and to unblock vertices in Blk. As a +result, the depth of this algorithm is T∞(n) ∈ O(e). +Theorem 3. The fine-grained parallel Johnson algorithm is scal- +able when lim +n→∞ c = ∞. +Proof. For this algorithm, T1(n) ∈ O(n+e+ec) and T∞(n) ∈ +O(e) (see Lemma 1). Given e < n2 and our assumption that +lim +n→∞ c = ∞, we have lim +n→∞ +T∞(n) +T1(n) = lim +n→∞ +e +n + e + ec = 0. +Thus, this algorithm is scalable based on Definition 2. +For the fine-grained parallel Johnson algorithm to be +scalable, it is sufficient for c to increase sublinearly with +n. Even though this algorithm is scalable, a strong or weak + +9 +𝑣2 +𝑣0 +𝑣1 +𝑣5 +𝑣4 +𝑣0 +𝑣3 +𝑣4 +𝑣0 +𝑣6 +𝑣4 +𝑣0 +𝑣8 +𝑣7 +𝑣6 +𝑣6 +𝑣7 +𝑣7 +𝑣8 +𝑣8 +𝑣8 +(a) Example graph +(b) Recursion tree +𝑣0 +𝑣1 +𝑣2 +𝑣3 +𝑣4 +𝑣5 +𝑣6 +𝑣8 +Task +𝑣7 +𝐸1 +𝐸2 +𝐸3 +Path extension exploration +𝑡0 +𝑡1 +𝑡2 +𝑡3 +𝑡4 +Vertex blocked in 𝑡2, 𝑡3, and 𝑡4 +Vertex blocked in 𝑡3 +Vertex blocked in !1, !2, !3, and !4 +Simple cycle reported +Fig. 7. +(a) An example graph and (b) the recursion tree of our fine- +grained parallel Read-Tarjan algorithm when enumerating cycles that +start from v0. The nodes of the recursion tree represent the recursive +calls of the depth-first search. Tasks shown in (b) can be executed +independently of each other. +scalability is not guaranteed due to the work inefficiency +of this algorithm. Nevertheless, our experiments show that +this algorithm is strongly scalable in practice (see Fig. 16). +5.4 +Summary +Our +relaxation +of +the +strictly +depth-first-search-based +recursion-tree exploration reduces the pruning efficiency of +the Johnson algorithm. In the worst case, the fine-grained +parallel Johnson algorithm could perform as much work as +the brute-force Tiernan algorithm does—i.e., O(se). How- +ever, in practice, this worst-case scenario does not happen +(see Section 8). In addition, our fine-grained parallel Johnson +algorithm can suffer from synchronisation issues in some +rare cases (see Section 8) because our copy-on-steal mecha- +nism can lead to long critical sections. In the next section, we +introduce a fine-grained parallel algorithm that is scalable, +work-efficient, and less prone to synchronisation issues. +6 +FINE-GRAINED PARALLEL READ-TARJAN +In this section, we first introduce several optimisations +that reduce the number of unnecessary vertex visits per- +formed by the sequential Read-Tarjan algorithm. Then, we +present our fine-grained parallel Read-Tarjan algorithm that +includes these optimisations. Finally, we show that our +parallel algorithm is work-efficient and strongly-scalable. +6.1 +Improvements to the pruning efficiency +To improve the pruning efficiency of the sequential Read- +Tarjan algorithm, we include the following optimisations: +(i) Blocked vertex set forwarding enables a recursive call +of the Read-Tarjan algorithm to reuse vertices blocked by +its parent call, resulting in fewer vertex visits. The original +Read-Tarjan algorithm discards blocked vertices after each +recursive call [23], even though this information could be +reused later. In this optimisation, the algorithm forwards +the blocked vertices Blk of a recursive call to its child recur- +sive calls, preventing those child calls from unnecessarily +visiting the vertices in Blk again. For example, in Fig. 7, +the vertex v8 is blocked the first time the algorithm visits +v8 while exploring the path extension E1. This optimisa- +tion prevents the algorithm from visiting v8 again when +exploring the same extension E1 or another extension E3 +Algorithm 3: FGRT DFS(u, v0, Blk, Vis) +Input: u - the current vertex, v0 - the starting vertex +InOut: Blk - blocked vertices +Vis - vertices visited during the DFS +Output: E - the resulting path extension from u to v0 +1 if u = v0 then return u; +2 Vis = Vis ∪ {u}; +3 block = true; +4 foreach w : N(u) s.t. w.id > v0.id do +5 +if w = v0 then return u → w; +6 +else if w /∈ Blk ∧ w /∈ Vis then +7 +E = FGRT DFS(w, v0 Blk, Vis); +8 +if E ̸= ∅ then return E.push front(u); +9 +if w /∈ Blk then block = false; +10 if block then Blk = Blk ∪ {u}; +11 return ∅; +that branches from E1. As a result of this optimisation, the +algorithm can avoid the dotted part of the recursion tree. +(ii) Path extension forwarding prevents recomputation +of the path extension E found by a parent recursive call by +forwarding this path extension to its child recursive call. In +this way, each child recursive call performs one fewer DFS +invocation than the original Read-Tarjan algorithm [23]. +(iii) Blocking on a successful DFS is another mechanism +for discovering vertices to be blocked. As a reminder, the +Read-Tarjan algorithm searches for path extensions using a +DFS. In the original algorithm, a vertex is blocked only if +it is visited during an unsuccessful DFS invocation, which +fails to discover a path extension. However, successful DFS +invocations could also visit some vertices that have all +their neighbours blocked. Such vertices cannot lead to the +discovery of new cycles and, thus, can also be blocked. +The pseudocode of the DFS function that includes this +optimisation is given in Algorithm 3. In our example given +in Fig. 7, a successful DFS invoked from v3 finds a path +extension E3 and discovers that the only neighbour v8 of v7 +is blocked. The algorithm then blocks v7, which enables it to +avoid visiting v7 again when exploring E3. Therefore, fewer +vertices are visited during the execution of the algorithm. +6.2 +Fine-grained parallelisation +Although the optimisations presented in Section 6.1 elimi- +nate some of the redundant work performed by the Read- +Tarjan algorithm, this algorithm typically performs more +work than the Johnson algorithm (see Section 3.4). However, +this redundancy makes it possible to parallelise the Read- +Tarjan algorithm in a scalable and work-efficient manner. +Because the Read-Tarjan algorithm allocates a new Blk +set for each path extension exploration, a recursive call can +explore different path extensions in an arbitrary order. In +addition, discovery of a new path extension E results in +the invocation of a single recursive call, and these calls +can be executed in an arbitrary order. As a result, several +threads can concurrently explore different paths of the same +recursion tree constructed by the Read-Tarjan algorithm for +a given starting edge. There are neither data dependencies +nor ordering requirements between different calls, apart +from those that exist between a parent and a child. To + +10 +Algorithm 4: FGRT task(v, v0, E, d, T1) +Input: v - the current vertex, v0 - the starting vertex +E - the path extension from v to v0 +d - the depth of this task +InOut: T1 - the thread that created this task +1 T2 = the thread executing this task; +// Check if this task is stolen +2 if T1 ̸= T2 then {ΠT2, BlkT2} = copy ({ΠT1, BlkT1}); +// Operations on Π and Blk are thread-safe. +3 while ΠT2.back() ̸= v do ΠT2.pop(); +4 Remove vertices from BlkT2 inserted at depth d′ ≥ d; +5 found = false; +// Exploration of the path extension E +6 while E ̸= ∅ do +7 +v = E.pop front(); +8 +ΠT2 = ΠT2.push(v); BlkT2 = BlkT2 ∪ {v}; +9 +foreach u : N(v) s.t. u.id > v0.id do +10 +if u ̸= E.front() ∧ u /∈ BlkT2 then +11 +E′ = FGRT DFS(u, v0, BlkT2, Vis = ∅); +12 +if E′ ̸= ∅ then +13 +spawn FGRT task(v, v0, E′, d + 1, T2); +14 +found = true; +15 +else BlkT2 = BlkT2 ∪ Vis; +16 +if found then break; +17 if E = ∅ then report cycle ΠT2; +18 else spawn FGRT task(v, v0, E, d + 1, T2); +exploit the parallelism available during the recursion tree +exploration, we execute each path extension exploration in +each recursive call as a separate task, all of which can be +independently executed. Examples of such tasks are shown +in Fig. 7. We refer to the resulting algorithm as the fine- +grained parallel Read-Tarjan algorithm. +Our implementation shown in Algorithm 4 performs +only a single path extension exploration in a recursive call +and uses all the optimisations we introduced in Section 6.1. +We execute each such recursive call as a separate task using +a dynamic thread scheduling framework (see Section 3.2). +For each edge v0 → v, we execute a parallel for loop iteration +that uses Algorithm 3 to search for a path extension E from +v to v0. If such E exists, a task is created using v, v0, and E as +its input parameters. This task then recursively creates new +tasks, as shown in lines 13 and 18 of Algorithm 4, until all +cycles that start with the edge v0 → v have been discovered. +To prevent different threads from concurrently modify- +ing Π and Blk, each task allocates and maintains its own Π +and Blk sets. A task can receive a copy of Π and Blk directly +from its parent task at the time of task creation. However, +it is possible to minimise the copy overheads by copying +these sets only when a task is stolen. For this purpose, we +use the copy-on-steal with complete unblocking approach +described in Section 5.2, which has shorter critical sections +than the copy-on-steal with recursive unblocking approach +used by our fine-grained parallel Johnson algorithm. +6.3 +Theoretical analysis +We now show that the fine-grained parallel Read-Tarjan +algorithm is both work-efficient and strongly scalable. +Theorem 4. The fine-grained parallel Read-Tarjan algorithm is +work-efficient. +Proof. Because each task of our fine-grained parallel Read- +Tarjan algorithm either discovers a cycle or creates at least +two child tasks, our algorithm is executed using O(c) tasks. +Each task performs several unsuccessful DFS invocations +and one successful DFS per each child task it creates. All +unsuccessful DFS invocations explore at most e edges in +total because they share the same set of blocked vertices. +In the worst case, each edge is visited twice per task, once +by a successful DFS and once by one of the unsuccessful +DFS invocations. Thus, this algorithm performs O(e) work +per task. Because this algorithm performs O (n + e) work if +there are no cycles in the graph, the total amount of work +this algorithm performs is Wp(n) = O (n + e + ec). Hence, +this algorithm is work-efficient based on Definition 1. +The work-efficiency of our fine-grained parallel Read- +Tarjan algorithm can be demonstrated using the example +given in Fig. 5a. In this example, the threads of this algo- +rithm independently explore four different path extensions +Ei = v1 → ui → v2 → v0, with i ∈ {1 . . . 4}. A thread +exploring a path extension Ei invokes a DFS from v2, which +explores vertices b1, . . . , bm at most once and fails to find +any other path extension. Therefore, the amount of work the +fine-grained parallel Read-Tarjan algorithm performs does +not increase compared to its single-threaded execution. +Lemma 2. The depth T∞(n) of the fine-grained parallel Read- +Tarjan algorithm is in O(ne). +Proof. In the worst case, a thread executing this algorithm +creates a task for each vertex of its longest simple cycle, +which has a length of at most n. Before invoking its first +child task, a task executes a sequence of unsuccessful DFS +invocations in O(e) and a successful DFS invocation also in +O(e). Thus, the depth of this algorithm is O (ne). +The worst-case depth of our algorithm can be observed +when this algorithm is executed on the graph given in +Fig. 4a. This graph has c = 2n−2 cycles and the length of its +longest cycle v0 → . . . vn−1 → v0 is n. The algorithm creates +a task for each vertex of the cycle and performs a successful +DFS in each such call, which leads to T∞ ∈ O(ne). +Theorem 5. The fine-grained parallel Read-Tarjan algorithm is +strongly scalable when lim +n→∞ c/n = ∞. +Proof. Because the fine-grained parallel Read-Tarjan algo- +rithm is work-efficient, we can apply Brent’s rule [60]: +T1(n) +p +≤ Tp(n) ≤ T1(n) +p ++ T∞(n). +(1) +Substituting T1(n) with O(n+e+ec) and T∞(n) with O(ne) +(see Lemma 2), for a positive constant C0, it holds that +1 +��1 +p + C0 +n +c +� +< 1 +��1 +p + T∞(n) +T1(n) +� +≤ T1(n) +Tp(n) ≤ p. (2) +Given that lim +n→∞ c/n = ∞, there exist n0 > 0, C1 > 0 such +that if n > n0, then c/n > C1p. Thus, for every n > n0, it +holds that kp ≤ T1(n) +Tp(n) ≤ p, where k = C1/(C0 + C1) < 1. +As a result, T1(n) +Tp(n) = Θ(p), which, based on Definition 3, +completes the proof. + +11 +As shown in Table 4, our fine-grained parallel Read- +Tarjan algorithm has a higher depth than our fine-grained +parallel Johnson algorithm, introduced in Section 5. Nev- +ertheless, the former algorithm is strongly scalable when +c grows superlinearly with n, whereas strong scalability +cannot be guaranteed for the latter algorithm. +6.4 +Summary +The work of our fine-grained parallel Read-Tarjan algo- +rithm does not increase after fine-grained parallelisation. +This parallel algorithm performs Wp(n) ∈ O(n + e + ec) +work: the same as the work performed by its serial version. +Our optimisations introduced in Section 6.1 do not reduce +the work Wp(n) performed by our parallel algorithm in +the worst case. However, these optimisations significantly +improve its performance in practice (see Section 8.4). In +addition, the synchronisation overheads of the fine-grained +parallel Read-Tarjan algorithm are not as significant as those +of the fine-grained Johnson algorithm because of its shorter +critical sections. Furthermore, this algorithm is the only +asymptotically-optimal parallel algorithm for cycle enumer- +ation for which we are able to prove strong scalability. +7 +PARALLELISING CONSTRAINED CYCLE SEARCH +This section describes the methods for adapting our par- +allel algorithms to search for simple cycles under various +constraints. Because state-of-the-art algorithms for temporal +and hop-constrained cycle enumeration are extensions of +the Johnson algorithm [14], [21], our parallelisation ap- +proach described in Section 5 is also applicable to these algo- +rithms. In this section, we describe the changes to the fine- +grained parallel Johnson algorithm needed for enumeration +of temporal and hop-constrained cycles. We also introduce +modifications to the cycle enumeration algorithms required +for finding time-window-constrained cycles. +7.1 +Cycles in a time window +Cycle enumeration algorithms require minimal modifica- +tions to support time-window constraints. Such constraints +restrict the search for simple, temporal, and hop-constrained +cycles to those that occur within a time window of a +given size δ, as illustrated in Fig. 2. To find time-window- +constrained cycles that start with an edge that has times- +tamp t0, only the edges with timestamps that belong to +the time window [t0 : t0 + δ] are visited. To avoid reporting +the same cycle several times, another edge with the same +timestamp t0 is visited only if the source vertex of that edge +has an ID that is smaller than the ID of the vertex from +which the search for cycles was started. Overall, imposing +time-window constraints reduces the number of cycles dis- +covered, which results in a more tractable problem. +A strongly-connected component (SCC) can be used to +reduce the number of vertices visited during the search for +time-window-constrained cycles. The search for cycles that +start with the edge ε can be limited to use only the vertices +from the SCC that contains ε [22]. In the case of time- +window-constrained cycles, we compute an SCC for ε using +only the edges with timestamps that belong to [t0 : t0 + δ], +where t0 is the timestamp of ε. Because an SCC can be +Algorithm 5: CFGJ copyOnSteal(d, T1, T2) +Input: d - the depth of the task executing this function +InOut: T1 - the victim thread +T2 - the stealing thread +1 MutexT1.lock(); +// Blk contains closing times or barriers +// Blist is not used for hop-constrained cycles +2 {ΠT2, BlkT2, BlistT2} = copy ({ΠT1, BlkT1, BlistT1}); +3 {PrevLocksT2} = copy ({PrevLocksT1}); +4 MutexT1.unlock(); +5 while |ΠT2| > d do +6 +u = ΠT2.pop(); +7 +lock = PrevLocksT2.pop(); +8 +RecursiveUnblock(u, lock, BlkT2, BlistT2); +computed independently for each edge in O(e) time [61], +our fine-grained parallel algorithms remain scalable. +7.2 +Temporal cycles +To efficiently enumerate temporal cycles, the 2SCENT al- +gorithm [14] replaces the set of blocked vertices Blk in the +Johnson algorithm with closing times. The closing time ct of a +vertex v indicates that the outgoing temporal edges of v with +a timestamp greater than or equal to ct cannot participate +in a temporal cycle and are therefore blocked. Increasing the +closing time of v to a new value ct′ unblocks the blocked +outgoing edges of v that have timestamps smaller than ct′. +This operation triggers the recursive unblocking procedure +that unblocks the incoming edges of v with a timestamp +smaller than the maximal timestamp among the unblocked +outgoing edges of v. This process is repeated for every +vertex with unblocked outgoing edges. +Because the backtracking phase of 2SCENT is based on +the Johnson algorithm, it can be parallelised using our fine- +grained approach described in Section 5. For this purpose, +we use our copy-on-steal mechanism with recursive un- +blocking, introduced in Section 5.2, which enables a thread +to maintain its own set of data structures used for recur- +sion tree pruning. However, this mechanism is not directly +applicable in this case because the recursive unblocking +procedure of 2SCENT requires the new closing time for a +vertex as a parameter in addition to the vertex itself. For +this reason, an additional data structure called PrevLocks is +used alongside the current path Π that records the closing +time that each vertex v had before it was added to Π. Copy- +on-steal then performs the recursive unblocking procedure +for each vertex v removed from Π using the original closing +time of the vertex v obtained from PrevLocks, as shown in +Algorithm 5. We refer to the resulting algorithm as the fine- +grained parallel temporal Johnson algorithm. +The aforementioned modification to the copy-on-steal +with recursive unblocking approach also enables a thread +of our fine-grained parallel algorithm to reuse the edges +blocked by another thread. This behaviour can be observed +in the example shown in Fig. 8, where the thread T2 steals +the task indicated in Fig. 8b from the thread T1. Copy- +on-steal executed by T2 invokes recursive unblocking that +restores the closing time of v3 to its original value of 9 +obtained from PrevLocks. Note that this original closing + +12 +𝑣3 +3 +𝑣0 +1 +2 +4 +𝑣1 +𝑣2 +𝑣4 +𝑣3 +𝑣0 +𝑣1 +𝑣2 +𝑣3 +𝑣4 +𝑣6 +𝑣7 +𝑣8 +𝑣6 +𝑣7 +𝑣5 +𝑣3 +9 +7 +8 +5 +6 +8 +10 +2 +3 +𝑣5 +𝑣6 +𝑣7 +𝑣8 +Victim +thread 𝑇! +Stealing +thread 𝑇! +𝑣5 +(a) Example graph +(b) Recursion tree +The stolen task +𝑣0 → ∞ +𝑣1 → ∞ +𝑣2 → ∞ +𝑣3 → 9 +𝑣4 → ∞ +𝑃𝑟𝑒𝑣𝐿𝑜𝑐𝑘𝑠: +Task +created +Task +stolen +⟺ blocked veretx/edge of T' +⟺ blocked veretx/edge of T! +𝑣0 +𝑣0 +𝑣3 +Task that reports a temporal cycle +∈ path Π of T! +∈ path Π of T" +Fig. 8. (a) An example graph and (b) the recursion tree of our fine- +grained temporal Johnson algorithm when enumerating temporal cycles +that start from v0. The thread T2 can avoid the dotted part of the tree by +reusing the blocked edges v6 → v7 and v7 → v3 discovered by T1. +time of v3 was previously set by T1 while exploring the +path v0 → v1 → v3. The recursive unblocking that T2 +invokes for v3 unblocks only the edge v6 → v3 because +it is the only incoming edge of v3 with a timestamp smaller +than the closing time 9 of the vertex v3. Without recording +the previous closing times, T2 could instead unblock all +incoming edges of v3 by invoking recursive unblocking for +v3 with a closing time ∞, which also unblocks the edges +v6 → v7 and v7 → v3. However, because there is no +temporal cycle that contains these two edges and starts +with v0, T2 would unnecessarily visit them in this case. +Thus, restoring the closing time of v3 to its original value +9 prevents T2 from performing this redundant work. +We also adapt the Read-Tarjan algorithm and its fine- +grained and coarse-grained versions to enumerate temporal +cycles using closing times. The necessary changes to the al- +gorithm are trivial, and we omit discussing them for brevity. +To reduce the number of vertices visited during the +search for temporal cycles, we use a method similar to the +SCC-based technique discussed in Section 7.1. Instead of +computing an SCC for each edge ε, we compute a cycle-union +that represents an intersection of temporal ancestors and +temporal descendants of ε. The temporal descendants and +the temporal ancestors of ε are the vertices that belong to the +temporal paths in which ε is the first edge and the last edge, +respectively. Defined as such, a cycle-union contains only +the vertices that participate in temporal cycles that have ε +as their starting edge. Thus, the search for temporal cycles +that start with ε can be limited to only those vertices. +7.3 +Hop-constrained cycles +An efficient algorithm for enumerating hop-constrained +cycles and paths, called BC-DFS [21], replaces the set of +blocked vertices Blk in the Johnson algorithm with barriers. +A barrier value bar of a vertex v indicates that the starting +vertex v0 of a cycle cannot be reached within bar hops from +v. As a result, v is blocked if the length of the current path +Π when the algorithm attempts to visit v is greater than or +equal to L − bar, where L is the hop constraint. BC-DFS +modifies the recursive unblocking of the Johnson algorithm +to reduce the barrier bar of v to a specified value bar ′ < bar. +This procedure also sets the barrier of any vertex u that can +reach v in k hops to bar ′ + k if the previous barrier of u +was greater than bar ′ + k. Maintaining barriers in such a +𝑣0 +(a) Example graph +(b) Recursion tree +𝑣6 +𝑣0 +𝑣4 +𝑣0 → 0 +𝑣1 → 0 +𝑃𝑟𝑒𝑣𝐿𝑜𝑐𝑘𝑠: +𝑣1 +0,1,2,… Barrier values of T! +0,1,2,… Barrier values modified by +T" during copy-on-steal +𝑣2 +𝑣4 +𝑣7 +𝑣8 +𝑣2 +The stolen task +𝑣1 +2 +3 +4 +1 +𝑣6 +𝑣7 +𝑣8 +𝑣1 +𝑣3 +𝑣5 +𝑣7 +𝑣6 +𝑣2 +𝑣3 +Stealing +thread 𝑇# +𝑣0 +𝑣3 +𝑣0 +Victim +thread 𝑇* +Task +stolen +Task +created +𝑣9 +𝑣5 +1 +Task that reports a simple cycle +∈ path Π of T! +∈ path Π of T" +Fig. 9. (a) An example graph and (b) the recursion tree of our fine- +grained hop-constrained Johnson algorithm when enumerating cycles +of length L = 6 that start from v0. Barrier values of unmarked vertices +are 0. Copy-on-steal enables the thread T2 to reuse barriers discovered +by the thread T1 and to avoid exploring the dotted part of the tree. +way minimises redundant vertex visits when searching for +hop-constrained cycles. +To parallelise BC-DFS in a fine-grained manner, we use +the same technique as that used for fine-grained parallelisa- +tion of the Johnson algorithm (Section 5) and the 2SCENT +algorithm (Section 7.2). In this case, threads exploring a +recursion tree of BC-DFS maintain separate data structures, +such as the current path Π and barrier values for each ver- +tex, and use the copy-on-steal with the recursive unblocking +approach to copy these data structures among threads. +Similarly to our algorithm from Section 7.2, each thread +also maintains a data structure PrevLocks that records the +original barrier value of each vertex v from Π. When a +thread steals a task, it performs a recursive unblocking +procedure for each vertex v removed from Π using its +original barrier value obtained from PrevLocks, as shown +in Algorithm 5. This procedure reduces the barrier value of +the vertices that can reach v, enabling the stealing thread to +visit those vertices. We refer to the resulting algorithm as +the fine-grained parallel hop-constrained Johnson algorithm. +The modified copy-on-steal with recursive unblocking +approach given in Algorithm 5 enables a stealing thread of +the aforementioned fine-grained parallel algorithm to reuse +barriers discovered by other threads. This behaviour can be +observed in the example given in Fig. 9. In that example, the +thread T1 first visits the vertices v2, v6, v7, v8 and sets the +barrier value of each visited vertex to L − |Π| + 1 (values in +red shown in Fig. 9a) because it was not able to find a cycle +of length L = 6 [21]. Here, |Π| denotes the length of Π at +the moment of exploration of each vertex. When the thread +T2 steals the task indicated in Fig. 9b from T1, the copy- +on-steal mechanism executed by T2 performs a recursive +unblocking of the vertex v1 using the original barrier value +0 of v1 obtained from PrevLocks. This recursive unblocking +reduces the barrier value of v2 from 4 to 1, which enables +T2 to find the cycle that contains v2. The barrier values of +the vertices v6, v7, and v8 are not modified, and, thus, the +thread T2 avoids visiting these vertices unnecessarily. +7.4 +Summary +In this section, we described a method to adapt the cy- +cle enumeration algorithms, such as our fine-grained al- +gorithms introduced in Sections 5 and 6, to search for + +13 +TABLE 5 +Hardware platforms used in the experiments. Here, P, C/P, and T/C +represent the number of processors, the number of cores per +processor, and the number of hardware threads per core, respectively. +platform +Intel KNL [62] +Intel Xeon Skylake [63] +P × C/P × T/C +4 × 64 × 4 +5 × 48 × 2 +Total no. threads +1024 +480 +Frequency +1.3 GHz +2 GHz +Memory per proc. +110 GB +360 GB +L1d/L2/L3 cache +32 KB/512 KB/none +32 KB/1 MB/38.5 MB +cycles under time window constraints. In addition, we +introduced a modified version of our copy-on-steal with +recursive unblocking approach, introduced in Section 5, +that supports fine-grained parallelisation of temporal and +hop-constrained cycle enumeration algorithms [14], [21] +derived from the Johnson algorithm. As a result, our fine- +grained parallel algorithms can enumerate cycles under +time-window, temporal, and hop constraints. +8 +EXPERIMENTAL EVALUATION +This +section +evaluates +the +performance +of +our +fine- +grained parallel algorithms for simple, temporal, and hop- +constrained cycle enumeration1. As Table 2 shows, we are +the only ones to offer fine-grained parallel versions of the +asymptotically-optimal cycle enumeration algorithms, such +as the Johnson and the Read-Tarjan algorithms. However, +the methods covered in Table 2 can be parallelised using the +coarse-grained approach covered in Section 4. Thus, we use +the coarse-grained approach as our main comparison point. +The experiments are performed using two different clus- +ters: Intel2 KNL [62] and Intel Xeon Skylake [63]. The details +of these two clusters are given in Table 5. We developed our +code on the Intel KNL cluster and ran most of the analyses +there; yet, for completeness, we ran the comparisons to +competing implementations also on the Intel Xeon Skylake +cluster available in Google Cloud’s Compute Engine [63]. +Scalability experiments are conducted on the Intel KNL +cluster, where one thread per core is used if the number of +threads used is less than or equal to 256, and simultaneous +multithreading is enabled otherwise. +We use the Threading Building Blocks (TBB) [54] library +to parallelise the algorithms on a single processor. We +distribute the execution of the algorithms across multiple +processors using the Message Passing Interface (MPI) [64]. +When using distributed execution, each processor stores a +copy of the input graph in its main memory and searches +for cycles starting from a different set of graph edges. +The starting edges are divided among the processors such +that when the edges are ordered in the ascending order +of their timestamps, k consecutive edges in that order are +assigned to k different processors. Each processor then uses +its own dynamic scheduler to balance the workload across +its hardware threads. In this setup, workload imbalance +across processors may still occur, but its impact is limited +in our experiments because we use at most five processors. +1. The open-source implementations of our algorithms are main- +tained here: https://github.com/IBM/parallel-cycle-enumeration. +2. Intel and Intel Xeon are trademarks or registered trademarks of +Intel Corporation or its subsidiaries in the United States and other +countries. +TABLE 6 +Temporal graphs used in the experiments. Time span T is in days. +graph +abbr. +n +e +T +bitcoinalpha +BA +3.3 k +24 k +1901 +bitcoinotc +BO +4.8 k +36 k +1903 +CollegeMsg +CO +1.3 k +60 k +193 +email-Eu-core +EM +824 +332 k +803 +mathoverflow +MO +16 k +390 k +2350 +transactions +TR +83 k +530 k +1803 +higgs-activity +HG +278 k +555 k +6 +askubuntu +AU +102 k +727 k +2613 +superuser +SU +138 k +1.1 M +2773 +wiki-talk +WT +140 k +6.1 M +2277 +friends2008 +FR +481 k +12 M +1826 +wiki-dynamic-nl +NL +1 M +20 M +3602 +messages +MS +313 k +26 M +1880 +AML-Data +AML +10 M +34 M +30 +stackoverflow +SO +2.0 M +48 M +2774 +We perform the experiments using the graphs listed in +Table 6. The TR, FR, and MS graphs are from Harvard Data- +verse [65], the NL graph is from Konect [66], the AML graph +is from the AML-Data repository [67], and the rest are from +SNAP [68]. To make cycle enumeration problems tractable, +we use time-window constraints in all of our experiments. +The time window sizes used in our experiments are given in +the figures next to the graph names. We stop the execution +of an algorithm if it takes more than 24h on the Intel KNL +cluster or more than 6h on the Intel Xeon Skylake cluster. +8.1 +Temporal cycle enumeration +The goal of a temporal cycle enumeration problem is to +find all simple cycles with edges ordered in time. Here, +we evaluate the performance of our fine-grained parallel +algorithms for this problem introduced in Section 7.2. Our +main comparison points are the coarse-grained parallel ver- +sions of the temporal Johnson and temporal Read-Tarjan +algorithms. We refer to the backtracking phase of the state- +of-the-art 2SCENT algorithm [14] for temporal cycle enu- +meration as the temporal Johnson algorithm and parallelise +it in a coarse-grained manner for the experiments. We do +not parallelise the entire 2SCENT algorithm because the +preprocessing phase of 2SCENT is strictly sequential and +has a time complexity in the order of the complexity of +its backtracking phase. We also provide direct comparisons +with the 2SCENT algorithm. +Fig. 10 shows that our fine-grained parallel algorithms +achieve an order of magnitude speedup compared to the +coarse-grained algorithms on the Intel KNL cluster. For the +NL graph, this speedup reaches up to 40×. Because the +Intel Xeon Skylake cluster contains fewer physical cores +than the Intel KNL cluster, the speedup between our fine- +grained and the coarse-grained parallel Johnson algorithms +is smaller on the former cluster. As can be observed in +Fig. 11, this speedup increases as we increase the time +window size used in the algorithms. Note that enumerating +cycles in larger time windows is more challenging because +larger time windows contain a larger number of cycles. +The scalability evaluation of the parallel temporal cycle +enumeration algorithms is given in Fig. 12. We also report +the performance of the sequential 2SCENT algorithm in the +same figure. The performance of our fine-grained parallel +algorithms improves linearly until 256 threads, after which + +14 +1.9 +1.3 +1.4 +1.7 +1.4 +0.9 +1.4 +1.3 +1.5 +2.1 +2.0 +2.1 +2.0 +1.2 +1.8 +1.5 +32 +34 +3.1 +3.1 +10 +8.1 +14 +10 +9.2 +5.1 +7.6 +40 +7.7 +11 +6.6 +10 +67 +59 +5.0 +5.2 +15 +7.5 +22 +14 +18 +12 +15 +100 +20 +13 +7.6 +17 +10 +100 +1 k +10 k +100 k +BA 3000h BO 1000h +CO 96h +EM 144h MO 288h +TR 800h +HG 72h +AU 336h +SU 168h +WT 144h +FR 5h +NL 1000s +MS 4h +AML 720h +SO 66h +geomean +Execution +time [s] +Fine-grained parallel temp. Johnson +Fine-grained parallel temp. Read-Tarjan +Coarse-grained parallel temp. Johnson +Coarse-grained parallel temp. Read-Tarjan +(a) Performance of parallel algorithms for temporal cycle enumeration on the Intel KNL cluster using 1024 threads. +1.4 +1.4 +1.7 +2.2 +1.7 +0.9 +1.4 +1.4 +1.4 +2.3 +2.2 +1.3 +2.7 +1.3 +1.6 +1.6 +17 +30 +2.0 +2.4 +3.5 +7.0 +10 +3.7 +5.8 +2.6 +4.5 +>13 +3.1 +5.4 +>3.5 +>5.2 +27 +42 +7.5 +7.1 +9.3 +6.1 +16 +12 +13 +7.8 +13 +>13 +>45 +6.2 +>3.5 +>12 +10 +100 +1 k +10 k +100 k +BA 3000h BO 1000h +CO 96h +EM 144h MO 288h +TR 800h +HG 72h +AU 336h +SU 168h +WT 144h +FR 5h +NL 1000s +MS 4h +AML 720h +SO 66h +geomean +Execution +time [s] +(b) Performance of parallel algorithms for temporal cycle enumeration on the Intel Xeon Skylake cluster using 480 threads. +Fig. 10. Performance of parallel algorithms for temporal cycle enumeration on (a) the Intel KNL and (b) the Intel Xeon Skylake clusters. The numbers +above the bars show the execution time of each algorithm relative to that of our fine-grained parallel temporal Johnson for the same benchmark. +25 +18 +32 +13 +11 +34 +10 +6.4 +3.1 +2.9 +5 +3.1 +7.2 +8.3 +10 +20 +17 +8.1 +10 +15 +14 +6.1 +6.3 +10 +24 +17 +9.2 +1.6 +2.6 +5.1 +1.6 +16 +7.6 +1 +34 40 +4.5 +6.7 +7.7 +3 +5.9 +11 +1 +2.1 +6.6 +5.3 +8.9 +10 +1 +10 +100 +1 k +10 k +100 k +BA 2200h +BA 2600h +BA 3000h +BO 800h +BO 900h +BO 1000h +CO 72h +CO 84h +CO 96h +EM 96h +EM 120h +EM 144h +MO 192h +MO 240h +MO 288h +TR 600h +TR 700h +TR 800h +HG 48h +HG 60h +HG 72h +AU 240h +AU 288h +AU 336h +SU 120h +SU 144h +SU 168h +WT 96h +WT 120h +WT 144h +FR 3h +FR 4h +FR 5h +NL 10s +NL 100s +NL 1000s +MS 2h +MS 3h +MS 4h +AML 480h +AML 600h +AML 720h +SO 30h +SO 48h +SO 66h +geomean 1 +geomean 2 +geomean 3 +Fine-grained parallel temporal Johnson +Coarse-grained parallel temporal Johnson +Execution time [s] +Fig. 11. Larger time windows increase the performance gap between the algorithms. The algorithms are executed on the Intel KNL cluster using +1024 threads. The numbers above the bars show the execution times of the coarse-grained algorithm relative to that of the fine-grained algorithm. +0.11 +10 +100 +1000 +1 +4 +16 +64 +256 +1024 +Fine-grained parallel temporal Johnson +Fine-grained parallel temporal Read-Tarjan +Coarse-grained parallel temporal Johnson +2SCENT [14] +205 x +185 x +10 x +0.9 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(f) TR 600h +325 x +213 x +32 x +1.6 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(c) CO 72h +308 x +246 x +30 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(h) AU 336h +356 x +198 x +126 x +1.1 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 +1024 +Speedup +Number of threads +(d) EM 96h +285 x +170 x +21 x +1.0 x +0.1 +10 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(k) FR 4h +359 x +249 x +53 x +1.2 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(e) MO 192h +435 x +211 x +85 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(j) WT 144h +208 x +118 x +16 x +1.0 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(b) BO 800h +242 x +106 x +11 x +1.1 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(a) BA 2200h +315 x +173 x +14 x +1.3 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(i) SU 120h +20 x +17 x +1.9 x +0.1 +1 +10 +100 +16 +64 +256 +1024 +Speedup +Number of threads +(n) AML 720 h +27 x +13 x +4.1 x +0.1 +1 +10 +100 +16 +64 +256 +1024 +Speedup +Number of threads +(m) MS 4h +242 x +198 x +7.6 x +1.6 x +0.1 +10 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(l) NL 100s +14 x +7.9 x +2.2 x +0.1 +1 +10 +100 +32 +64 +128 +256 +512 +1024 +Speedup +Number of threads +(o) SO 66h +218 x +137 x +23 x +0.5 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(g) HG 48h +Fig. 12. Scalability evaluation of parallel temporal cycle enumeration algorithms executed on the Intel KNL cluster. The baseline is our fine-grained +parallel temporal Johnson algorithm. The relative performance of 2SCENT [14] is shown when it completes in 24 hours. Note that the 2SCENT +implementation is single-threaded and the single-threaded execution results are not available for all graphs. +it becomes sublinear due to simultaneous multithreading. +As a result, our fine-grained versions of the Johnson and +the Read-Tarjan algorithms reach 435× and 470× speedups, +respectively, compared to their serial versions. Addition- +ally, when using 1024 threads, our fine-grained Johnson +algorithm is on average 260× faster than 2SCENT when +2SCENT completes in 24 hours. On the other hand, the +coarse-grained Johnson algorithm does not scale as well as +the fine-grained algorithms. As a result, the performance +gap between the fine-grained and the coarse-grained algo- +rithms increases as we increase the number of threads. +Overall, the fastest algorithm for temporal cycle enu- +meration that we tested is our fine-grained Johnson al- +gorithm, which is, on average, 60% faster than our fine- +grained Read-Tarjan algorithm. When using 1024 threads, +both fine-grained algorithms are an order of magnitude +faster than their coarse-grained counterparts. Moreover, our +fine-grained parallel algorithms, executed on the Intel KNL +cluster using 1024 threads, are two orders of magnitude +faster than the state-of-the-art algorithm 2SCENT [14]. +8.2 +Hop-constrained cycle enumeration +In hop-constrained cycle enumeration, we search for all +simple cycles in a graph that are shorter than the spec- +ified hop constraint. We here compare our fine-grained +parallel hop-constrained Johnson algorithm, introduced in +Section 7.3, with the state-of-the-art algorithms BC-DFS +and JOIN [21] for this problem. For this evaluation, we +parallelised BC-DFS and JOIN in the coarse-grained manner. +Because adapting the Read-Tarjan algorithm to enumerate +hop-constrained cycles is not trivial, we do not report the +performance of the fine-grained and coarse-grained versions +of this algorithm. We also omit the performance results for +the MS graph because our fine-grained algorithm did not +finish under 12h when using the smallest time window size. + +15 +2 +10 +14 +3 +10 +15 +2 +8 +9 +1 +5 +13 +4 16 15 +1 +4 +8 +4 +22 +26 +5 +53 +60 +2 +38 +61 +2 +17 20 +1 +5 +20 +44 51 51 +2 +18 32 +1 +1 +3 +3 +12 +18 +2 +3 +24 +2 +3 +16 +2 +1 +3 +2 +3 +3 +3 +4 +26 +2 +3 +4 +1 +1 +1 +4 +17 +300 +4 +5 +5 +8 +9 +199 +5 +5 +4 +21 +>1.7k >1.7k +2 +233 +> 1.5k +4 +5 +3 +3 +8 +0.01 +1 +100 +10 k +Exec. time [s] +BA 120h +BO 140h +CO 5h +EM 5h +MO 30h +TR 110h +HG 1h +AU 20h +SU 10h +WT 12h +FR 1500s +NL 25s +AML 72h +SO 6h +geomean +12 +10 +15 +20 +Fine-grained par. hop-constrained Johnson +Coarse-grained par. BC-DFS [21] +hop constraint: +Coarse-grained par. JOIN [21] +(a) Performance of parallel algorithms for finding hop-constrained simple cycles on the Intel KNL cluster using 1024 threads. +2 +9 +11 +2 +9 +11 +2 +11 +9 +1 +4 +16 +3 +8 +8 +1 +6 +8 +4 +19 +20 +4 +30 36 +1 +26 +38 +1 +8 11 +1 +4 +10 +26 36 36 +1 +14 +25 +1 +1 +2 +2 +10 +13 +2 +3 +18 +1 +2 +11 +1 +1 +3 +2 +3 +5 +2 +3 +9 +2 +2 +2 +1 +1 +1 +4 +10 +126 +3 +4 +3 +4 +5 +72 +4 +4 +5 +9 +>1.4k >1.4k +1 +64 +> 1k +1 +2 +1 +2 +5 +0.01 +1 +100 +10 k +Exec. time [s] +BA 120h +BO 140h +CO 5h +EM 5h +MO 30h +TR 110h +HG 1h +AU 20h +SU 10h +WT 12h +FR 1500s +NL 25s +AML 72h +SO 6h +geomean +10 +(b) Performance of parallel algorithms for finding hop-constrained simple cycles on the Intel Xeon Skylake cluster using 480 threads. +Fig. 13. Performance of parallel algorithms for hop-constrained simple cycle enumeration on (a) the Intel KNL and (b) the Intel Xeon Skylake +clusters. The numbers above the bars show the execution time of the coarse-grained parallel algorithms relative to that of our fine-grained parallel +algorithm. Larger hop constraints increase the performance gap between the two algorithms. +0.11 +10 +100 +1000 +1 +4 +16 +64 +256 +1024 +Fine-grained parallel hop-constrained Johnson +Coarse-grained parallel BC-DFS [21] +368 x +6.9 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(c) AU 20h +315 x +71 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(b) TR 110h +338 x +68 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(e) FR 1500s +38… +23 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(d) WT 12h +329 x +40 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(a) CO 5h +Fig. 14. Scalability evaluation of parallel hop-constrained cycle enumeration algorithms executed on the Intel KNL cluster using the hop constraint +of 15. The speedup values are relative to the single-threaded execution of BC-DFS. Evaluation on other graphs is omitted due to space constraints. +Fig. 13 shows that our fine-grained parallel algorithm +is, on average, more than 10× faster than the coarse- +grained parallel BC-DFS algorithm for the two largest hop +constraints tested. When using the hop-constraint that is +less than or equal to ten, the coarse-grained parallelisation +approach is able to achieve workload balance across cores, +and, thus, the performance of this approach is similar to that +of our fine-grained approach in this case. As we increase +the hop constraint, the probability of encountering deeper +recursion trees also increases. Exploring such trees using the +coarse-grained approach leads to workload imbalance (see +Section 4). Our fine-grained algorithm is designed to resolve +this problem by exploring a recursion tree using several +threads. Therefore, increasing the hop constraint increases +the speedup of our fine-grained algorithm with respect to +the coarse-grained algorithm. +When the hop constraint is set to 20, our fine-grained +parallel algorithm is, on average, 10× faster than the coarse- +grained parallel JOIN algorithm, as shown in Fig. 13. Al- +though the latter algorithm can be competitive with our fine- +grained algorithm, it can also suffer from long execution +times, such as in the cases of the AU, NL, and AML graphs. +The reason for these long execution times is the fact that +the JOIN algorithm might temporarily construct many non- +simple cycles while searching for simple cycles. Because this +algorithm constructs cycles by combining simple paths, it is +not guaranteed that each combination results in a simple +cycle. The overhead of combining paths can dominate the +execution time of JOIN if this algorithm constructs orders +of magnitude more non-simple cycles than simple cycles. +For instance, this situation occurs in the case of AU and +hop constraint of 20, where JOIN discovers 600× more non- +simple cycles than simple cycles. As a result, the speedup of +our fine-grained algorithm compared to the coarse-grained +JOIN algorithm can reach up to three orders of magnitude. +Fig. 14 shows that the speedup of our fine-grained par- +allel Johnson algorithm with respect to the coarse-grained +parallel BC-DFS can be increased by using more threads. +The performance of our fine-grained parallel algorithm +scales linearly with the number of threads, whereas the +scaling of the coarse-grained parallel BC-DFS eventually +slows down. Thus, in addition to being, on average, an +order of magnitude faster than the coarse-grained parallel +BC-DFS, our fine-grained algorithm is also more scalable. +8.3 +Simple cycle enumeration +Here, we evaluate our fine-grained parallel algorithms for +simple cycle enumeration. The computational complexity of +simple cycle enumeration is higher than the complexity of +temporal and hop-constrained cycle enumeration because +simple cycle enumeration does not impose temporal order- +ing or hop constraints. The only constraint we impose is the +time-window constraint. Because the complexity of enumer- +ating simple cycles is higher, we use smaller time windows +compared to the cases of temporal and hop-constrained cy- +cle enumeration. We use the coarse-grained parallel versions +of the Johnson and the Read-Tarjan algorithms as our main +comparison points. We do not report the results for the MS +graph because our algorithms did not finish in 12h even if +we set the time window to one second. +As we can see in Fig. 15, our fine-grained parallel al- +gorithms show an order of magnitude average speedup +compared to coarse-grained parallel algorithms on two +different platforms. The reason for this speedup is better +scalability of our fine-grained algorithms, which we demon- +strate in Fig. 16. Similarly to the cases of temporal and +hop-constrained cycle enumeration (see Figs. 12 and 14), +our fine-grained parallel algorithms scale linearly with the +number of physical cores used whereas the coarse-grained +parallel Johnson algorithm does not scale as well. Thus, the +speedup between the fine-grained and the coarse-grained +algorithms increases by utilising more threads. + +16 +0.9 +0.8 +1.0 +1.3 +1.1 +1.3 +1.4 +1.2 +1.1 +1.1 +1.3 +2.6 +0.8 +1.2 +1.2 +30 +9 +30 +30 +19 +11 +24 +30 +35 +7.7 +11 +15 +4.0 +>20 +>16 +29 +9 +33 +39 +23 +18 +31 +36 +42 +17 +18 +>32 +5.3 +>20 +>22 +0.1 +10 +1 k +100 k +BA 71h +BO 75h +CO 3h +EM 4h +MO 30h +TR 72h +HG 3000s +AU 20h +SU 5h +WT 12h +FR 1300s +NL 29s +AML 48h +SO 3h +geomean +Fine-grained parallel Johnson +Fine-grained parallel Read-Tarjan +Coarse-grained parallel Johnson +Coarse-grained parallel Read-Tarjan +Execution +time [s] +(a) Performance of parallel algorithms for simple cycle enumeration on the Intel KNL cluster using 1024 threads. +1.0 +1.0 +1.2 +1.3 +1.2 +1.4 +1.6 +1.3 +1.2 +1.0 +1.4 +2.8 +0.7 +1.1 +1.2 +10 +11 +17 +22 +8.8 +9.6 +12 +26 +27 +7.5 +16 +>31 +2.1 +>17 +>16 +11 +11 +21 +28 +21 +15 +16 +31 +33 +11 +14 +>31 +2.1 +>17 +>16 +0.1 +10 +1 k +100 k +BA 71h +BO 75h +CO 3h +EM 4h +MO 30h +TR 72h +HG 3000s +AU 20h +SU 5h +WT 12h +FR 1300s +NL 29s +AML 48h +SO 3h +geomean +Execution +time [s] +(b) Performance of parallel algorithms for simple cycle enumeration on the Intel Xeon Skylake cluster using 480 threads. +Fig. 15. Performance of parallel algorithms for simple cycle enumeration on (a) the Intel KNL and (b) the Intel Xeon Skylake clusters. The numbers +above the bars show the execution time of each algorithm relative to that of our fine-grained parallel Johnson algorithm for the same benchmark. +0.11 +10 +100 +1000 +1 +4 +16 +64 +256 +1024 +Fine-grained parallel Johnson +Fine-grained parallel Read-Tarjan +Coarse-grained parallel Johnson +239 x +208 x +31 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(d) WT 12h +180 x +140 x +16 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 +1024 +Speedup +Number of threads +(b) TR 72h +224 x +167 x +20 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(e) FR 1300s +154 x +133 x +5 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(c) AU 20h +271 x +273 x +9 x +0.1 +1 +10 +100 +1000 +1 +4 +16 +64 +256 1024 +Speedup +Number of threads +(a) CO 3h +Fig. 16. Scalability evaluation of parallel simple cycle enumeration algorithms executed on the Intel KNL cluster. The speedup values are relative to +the single-threaded execution of the Johnson algorithm. Evaluation on other graphs is omitted due to space constraints. +2.0 +1.8 +2.1 +2.6 +2.0 +2.2 +2.2 +1.8 +2.1 +1.7 +2.2 +1.6 +1.6 +1.8 +2.0 +0 +1 +2 +3 +BA 71h +BO 75h +CO 3h +EM 4h +MO 30h +TR 72h +HG 3000s +AU 20h +SU 5h +WT 12h +FR 1300s +NL 29s +AML 48h +SO 3h +geomean +Normalized +exec. time +no optimisations +path ext. and blk set forwarding +blk set forwarding +all optimisations +6.8 +3.7 +3.9 +3.8 +4.5 +2.6 +3.4 +3.5 +4.1 +3.1 +3.4 +4.1 +2.1 +2.6 +1.8 +3.4 +0 +2 +4 +6 +8 +BA 2600h +BO 900h +CO 72h +EM 120h +MO 240h +TR 700h +HG 60h +AU 288h +SU 144h +WT 120h +FR 4h +NL 100s +MS 3h +AML 600h +SO 48h +geomean +Normalized +exec. time +(a) +(b) +Fig. 17. Effect of the pruning improvements to our fine-grained parallel +Read-Tarjan algorithm for (a) simple and (b) temporal cycle enumer- +ation. Execution times are normalised to the case that includes all +optimisations. Our optimisations accelerate this algorithm by up to 6.8×. +The synchronisation overheads caused by recursive un- +blocking of our fine-grained parallel Johnson algorithm (see +Section 5.2) are visible only in the case of AML. In this case, +the fine-grained parallel Johnson algorithm performs 60% +fewer edge visits than the fine-grained parallel Read-Tarjan; +however, it is 25% slower. These synchronisation overheads +can be explained by a very low cycle-to-vertex ratio. Because +a vertex is blocked if it cannot take part in a cycle, the +probability of a vertex being blocked is higher when the +cycle-to-vertex ratio is lower. In consequence, more vertices +are unblocked during the recursive unblocking of the fine- +grained parallel Johnson algorithm, leading to longer critical +sections and more contention on the locks. Nevertheless, +our fine-grained parallel Johnson algorithm achieves a good +trade-off between pruning efficiency and lock contention in +most cases. +Overall, our fine-grained parallel Johnson and fine- +grained parallel Read-Tarjan algorithms have comparable +performance, as shown in Fig. 15. Although the former +algorithm is slightly faster, it can suffer from synchronisa- +tion overheads in some cases. Nevertheless, both parallel +algorithms achieve linear scaling with the number of phys- +ical cores used and achieve, on average, more than 10× +speedup with respect to coarse-grained parallel versions of +the algorithms. These conclusions also hold in the cases of +temporal and hop-constrained cycle enumeration. +8.4 +Improvements to the Read-Tarjan algorithm +Fig. 17 shows the effect of our pruning improvements, intro- +duced in Section 6.1, on the performance of our fine-grained +Read-Tarjan algorithm. The experiments are performed us- +ing a single Intel KNL processor using 256 threads. Note +that using one processor instead of the entire cluster results +in longer execution times, but it enables us to eliminate +the effect of workload imbalance across processors in this +experiment. The execution time of the fine-grained parallel +Read-Tarjan algorithm decreases after activating each opti- +misation because fewer redundant vertex and edge visits +are performed during the execution of this algorithm. When +all optimisations are enabled, the average speedup of our +algorithm for simple cycle enumeration compared to its +unoptimised version is 2×. In the case of temporal cycle +enumeration, the average speedup increases to 3.4×. As a +result, our pruning improvements enable the fine-grained +parallel Read-Tarjan algorithm to be competitive with the +fine-grained parallel Johnson algorithm. +9 +CONCLUSIONS +This work has made three contributions to the area of par- +allel cycle enumeration. First, we have introduced scalable +fine-grained parallel versions of the state-of-the-art Johnson +and Read-Tarjan algorithms for enumerating simple cycles. +In addition, we have shown that the novel fine-grained +parallel approach we contributed for parallelising the John- +son algorithm can be adapted to support the enumeration + +17 +of temporal and hop-constrained cycles as well. Our fine- +grained parallel algorithms for enumerating the aforemen- +tioned types of cycles achieve a near-linear performance +scaling on a compute cluster with a total number of 256 CPU +cores that can execute 1024 simultaneous software threads. +Secondly, we have shown that our fine-grained parallel +cycle enumeration algorithms are scalable both in theory +and in practice. In contrast, their coarse-grained parallel ver- +sions do not share this property. When using 1024 software +threads, our fine-grained parallel algorithms are on aver- +age an order of magnitude faster than their coarse-grained +counterparts. In addition, the performance gap between the +fine-grained and coarse-grained parallel algorithms widens +as we use more physical CPU cores. This performance gap +also widens when increasing the time window in the case +of temporal cycle enumeration and when increasing the hop +constraint in the case of hop-constrained cycle enumeration. +Thirdly, we have shown that, whereas our fine-grained +parallel Read-Tarjan algorithm is work efficient, our fine- +grained parallel Johnson algorithm is not. In general, the +former is competitive against the latter because of the new +pruning methods we introduced, yet the latter outperforms +the former in most experiments. In some rare cases, our fine- +grained parallel Johnson algorithm can suffer from synchro- +nisation overheads. In such cases, our fine-grained parallel +Read-Tarjan algorithm offers a more scalable alternative. +ACKNOWLEDGMENTS +The support of Swiss National Science Foundation (project +number 172610) for this work is gratefully acknowledged. +REFERENCES +[1] +M. 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Kunegis, “KONECT: the Koblenz network collection,” in Pro- +ceedings of the 22nd International Conference on World Wide Web - +WWW ’13 Companion. +Rio de Janeiro, Brazil: ACM Press, 2013, +pp. 1343–1350, doi: 10.1145/2487788.2488173. +[67] E. Altman, “Aml-data,” Available online: https://github.com/ +IBM/AML-Data, 2021, accessed: 2022-05-30. +[68] J. Leskovec and A. Krevl, “SNAP Datasets: Stanford large network +dataset collection,” Available online: https://snap.stanford.edu/ +data, Jun. 2014, accessed: 2022-05-30. +Jovan Blanuˇsa received the BSc. degree in +electrical engineering and computer science +from the University of Belgrade, Belgrade, Serbia +and the MSc. degree in electrical and electronic +engineering from EPFL, Lausanne, Switzerland. +He is currently working towards the Ph.D. de- +gree in computer science at EPFL, Lausanne, +Switzerland. Mr. Blanuˇsa is also working as a +predoctoral researcher at IBM Research Europe, +Zurich, Switzerland. His research interests in- +clude acceleration of graph mining algorithms +and their applications. He is a Student Member of the ACM. +Kubilay Atasu received his BSc. and Ph.D. de- +grees in computer engineering from Bo˘gazic¸i +University, Istanbul, Turkey. He also holds an +MEng. degree in embedded system design from +University of Lugano. Dr. Atasu is currently a +research staff member at IBM Research Europe, +Zurich, Switzerland. He was the recipient of a +best paper award at DAC in 2003 and at ASAP in +2008. He was also a best-paper award nominee +at FPL in 2016. He served as a program co-chair +of the ASAP 2013 conference and as a general +chair of the ASAP 2014 conference. Dr. Atasu is currently serving in the +program committee of the DAC, FCCM and ASAP conferences. In the +past, he also served in the program committees of the DATE, ICPP, CF, +FPL and FPT conferences. He is a Senior Member of the IEEE. + +19 +Paolo Ienne received the laurea degree in elec- +trical engineering from Politecnico di Milano, Mi- +lan, Italy, in 1991, and the Ph.D. degree in com- +puter science from EPFL, Lausanne, Switzer- +land, in 1996. Since 2000, he has been a Pro- +fessor with the School of Computer and Commu- +nication Sciences, EPFL. His research interests +include computer and processor architecture, +FPGAs and reconfigurable computing, electronic +design automation, and computer arithmetic. +Some of his articles have received the Best Pa- +per Awards at prestigious venues (including at the FPGA, FPL, CASES, +and DAC conferences), and several others have been nominated. Prof. +Ienne has served as general, program, and topic chair of renown inter- +national conferences, serves on the steering committee of the ARITH, +FPL, and FPGA conferences, and is regularly a member of several +program committees. He was an associate editor of the ACM TODAES +and is an associate editor of ACM CSUR and ACM TACO. He is a Senior +Member of the IEEE and a Member of the ACM. + diff --git a/QNAzT4oBgHgl3EQfIvuz/content/tmp_files/load_file.txt b/QNAzT4oBgHgl3EQfIvuz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ccda5494b51bea1ba04ed87bbace7b3f4ea27874 --- /dev/null +++ b/QNAzT4oBgHgl3EQfIvuz/content/tmp_files/load_file.txt @@ -0,0 +1,1961 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf,len=1960 +page_content='1 Fast Parallel Algorithms for Enumeration of Simple, Temporal, and Hop-Constrained Cycles Jovan Blanuˇsa, Kubilay Atasu, and Paolo Ienne Abstract—Finding cycles in directed graphs enables important applications in various domains such as finance, biology, chemistry, and network science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, as the size of graph datasets continues to grow, it becomes increasingly difficult to discover cycles within them, which necessitates more efficient algorithms and their parallel implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this work, we propose scalable parallelisation of state-of-the-art sequential algorithms for enumerating simple, temporal, and hop-constrained cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' First, we focus on the simple cycle enumeration problem and parallelise the algorithms by Johnson and by Read and Tarjan in a fine-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We theoretically show that our resulting fine-grained parallel algorithms are scalable, with the fine-grained parallel Read-Tarjan algorithm being strongly scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In contrast, we show that straightforward coarse-grained parallel versions of these simple cycle enumeration algorithms that exploit edge- or vertex-level parallelism are not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Next, we adapt our fine-grained approach to enable scalable parallelisation of state-of-the-art algorithms for temporal and hop-constrained cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our evaluation on a cluster with 256 physical cores demonstrates a near-linear scalability of our fine-grained parallel algorithms when enumerating all the aforementioned types of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' On the same cluster, our fine-grained parallel algorithms achieve, on average, one order of magnitude speedup compared to the respective coarse-grained parallel versions of the state-of-the-art algorithms for cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The performance gap between the fine-grained and the coarse-grained parallel algorithms increases as we use more CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Index Terms—Cycle enumeration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Parallel graph algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Graph pattern mining !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1 INTRODUCTION A graph-based data representation is desirable when ana- lyzing large and complex datasets because it exposes the connectivity of the underlying data objects and enables the discovery of complex relationships between them [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Analysing graph-structured data has important applications in many domains, such as finance [2], healthcare [3], cyber- security [4], and advertising [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The existence in a graph of certain patterns, such as cycles, cliques, and motifs, can reveal nontrivial relationships between different graph ob- jects [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As the volume of graph data continues to grow, the discovery of such relationships becomes computationally challenging, requiring more efficient parallel algorithms that can exploit modern multi-core processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This paper introduces efficient par- allel algorithms for enumerating simple cycles of directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A simple cycle is a sequence of edges that starts and ends with the same vertex and visits other vertices at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Enumerating simple cycles has important applications in several domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For example, in electronic design automation, combinatorial loops in circuits are typ- ically forbidden [7], [8], and such loops can be detected © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Jovan Blanuˇsa is with IBM Research Europe - Zurich and with the ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL) E-mail: jov@zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='com Kubilay Atasu is with IBM Research Europe - Zurich E-mail: kat@zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='com Paolo Ienne is with the ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL) E-mail: paolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='ienne@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='ch 1 64 128 192 256 Thread ID 0 500 1000 1500 Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time [s] 1 64 128 192 256 Thread ID 0 500 1000 1500 Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time [s] (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Per-thread execution time of (a) the coarse-grained Johnson algorithm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (b) our fine-grained Johnson algorithm using the WT graph and a 12h time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thanks to a perfect load balancing, our fine- grained method is 3× faster on a 64-core CPU executing 256 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' by enumerating simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In a software bug tracking system, a dependency between two software bugs requires one bug to be addressed before the other [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Circular bug dependencies are undesirable and can be detected by finding simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Other applications include detecting feedback loops in biological networks [10], [11] and detect- ing unstable relationships in social networks [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Furthermore, some graphs have their edges annotated with timestamps, which we refer to as temporal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In such graphs one can also look for temporal cycles [14], which are special cases of simple cycles, in which the edges are ordered in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For instance, in financial transaction graphs, a temporal cycle represents a series of transac- tions in which the money initially sent from one bank account returns back to the same account;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' the existence of such cycles is a strong indicator of financial fraud such as money laundering, tax avoidance [15], [16], and credit card fraud [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Finding temporal cycles in temporal graphs also enables detecting circular trading, which can be used for manipulating stock prices [18], [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Various other kinds of constraints are often imposed on the cycles being searched because the search may be computationally impossible otherwise [14], [17], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='01068v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='DS] 3 Jan 2023 2 TABLE 1 Our fine-grained parallel Read-Tarjan algorithm is the only solution that is both work-efficient and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Parallel algorithm Work-efficient Scalable Coarse-grained parallel algorithms \x13 Our fine-grained parallel Johnson \x13 Our fine-grained parallel Read-Tarjan \x13 \x13 constraints may include hop constraints [17], [21], which limit the length of paths explored during the search for cycles, and time-window constraints [14], which restrict the search to cycles that occur within a time window of a given size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Imposing these constraints reduces the number of paths explored during the search for cycles, making the problem more tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this reason, we focus on searching for cycles under the aforementioned constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Parallelisation challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We focus on parallelising the algorithms by Johnson [22] and by Read and Tarjan [23] for finding cycles because these algorithms achieve the lowest time complexity bounds reported for directed graphs [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Both algorithms are recursively formulated and con- struct a recursion tree in a depth-first fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, these algorithms employ different pruning techniques to limit the amount of work they perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In practice, the Johnson algorithm is faster than the Read-Tarjan algorithm due to more aggressive pruning techniques [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fur- thermore, the state-of-the-art algorithms for temporal and hop-constrained cycle enumeration are extensions of the Johnson algorithm [14], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, parallelising the Johnson algorithm also enables parallelisation of these temporal and hop-constrained cycle enumeration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The na¨ıve way of parallelising the Johnson and the Read- Tarjan algorithms involves searching for cycles starting from different vertices or edges in parallel, which we refer to as the coarse-grained parallel methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Such coarse-grained parallel approaches are straightforward to implement using the popular vertex-centric [26], [27] and edge-centric [28] graph processing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, real-world graphs often exhibit a power-law or a log-normal distribution of vertex degrees [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In such graphs, the execution time of coarse-grained parallel approaches is dominated by searches that start from a small set of vertices or edges as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This behaviour leads to a workload imbalance and limits scalability of parallel implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The shortcomings of coarse-grained parallel approaches can be addressed by decomposing the search for cycles starting from a given edge or vertex into finer-grained tasks [31], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, parallelising the Johnson algorithm using the fine-grained approach is challenging because the pruning efficiency of this algorithm depends on a strictly sequential depth-first-search-based recursion tree exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We demonstrate that the lesser-known Read- Tarjan algorithm does not have such a requirement, and, thus, it is easier to decompose into fine-grained tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This paper presents an extension of the work by Blanuˇsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [34], which introduced the following contributions: (i) Scalable fine-grained parallelisation of the Johnson and the Read-Tarjan algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To our knowledge, we are the first ones to parallelise these asymptotically-optimal cycle enu- meration algorithms in a fine-grained manner and achieve an almost linear performance scaling on a system that can execute up to a thousand concurrent software threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Such a scalability is enabled by our decomposition of long sequential searches into fine-grained tasks, which are then dynamically scheduled across CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To decompose the Johnson algorithm into fine-grained tasks, we have re- laxed its strictly depth-first-search-based exploration, which enables this algorithm to perform multiple independent depth-first searches in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, our fine-grained parallel Johnson algorithm is able to achieve an ideal load balancing as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (ii) Theoretical analysis of the coarse- and fine-grained par- allel algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We theoretically show that both of our fine- grained parallel algorithms are scalable, which is not the case for the Johnson and the Read-Tarjan algorithms par- allelised in a coarse-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Moreover, we show that our fine-grained parallel Read-Tarjan algorithm per- forms asymptotically the same amount of work as its serial version, whereas our fine-grained parallel Johnson algo- rithm does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, our fine-grained parallel Read- Tarjan algorithm is the only parallel algorithm based on an asymptotically-optimal cycle enumeration algorithm that is both work-efficient and scalable, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Inter- estingly, despite not being work-efficient, our fine-grained Johnson algorithm outperforms our fine-grained parallel Read-Tarjan algorithm in most of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this paper, we extend our prior work [34] with the following contributions: (iii) General framework for parallelising temporal and hop- constrained cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We show that our method for parallelising the Johnson algorithm in a fine-grained manner can be adapted to parallelise the state-of-the-art algorithms for temporal and hop-constrained cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This adaptation is possible because these state-of-the-art algo- rithms, such as the 2SCENT algorithm for temporal cycle enumeration [14] and the BC-DFS algorithm [21] for hop- constrained cycle enumeration, are extensions of the John- son algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' By parallelising these algorithms using our fine-grained method, we were able to achieve speedups of up to 40× and 61× compared to the coarse-grained parallel versions of 2SCENT and BC-DFS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (iv) Improvements to the pruning efficiency of the Read- Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To make this algorithm competitive with the Johnson algorithm, we have introduced several opti- misations that enhance the pruning efficiency of the Read- Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The optimisations reduce the amount of unnecessary vertex visits that this algorithm performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, our improved version of the Read-Tarjan algorithm is up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8× faster than the original version of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Paper structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The remainder of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The related work and background are presented in Section 2 and Section 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Coarse-grained par- allel versions of the Johnson and the Read-Tarjan algorithms are covered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Section 5 and Section 6 introduce our fine-grained parallel versions of the Johnson and the Read-Tarjan algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Section 6 also includes our optimisations for improving the pruning efficiency of the Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our general framework for par- allelising temporal and hop-constrained cycle enumeration algorithms is presented in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In Section 8, we pro- vide an experimental evaluation of our fine-grained parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Finally, we conclude our work in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 TABLE 2 Capabilities of the related work versus our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Competing algorithms either fail to exploit fine-grained parallelism or do it on top of asymptotically inferior algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Related work [14] [17] [21] [46] [47] Ours Fine-grained parallelism \x13 \x13 Asymptotic optimality \x13 \x13 \x13 \x13 Temporal cycles \x13 \x13 Time-window constraints \x13 \x13 \x13 Hop constraints \x13 \x13 \x13 \x13 \x13 2 RELATED WORK Simple cycle enumeration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Enumeration of sim- ple cycles of graphs is a classical computer science prob- lem [22], [23], [24], [25], [35], [36], [37], [38], [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The backtracking-based algorithms by Johnson [22], Read and Tarjan [23], and Szwarcfiter and Lauer [37] achieve the low- est time complexity bounds for enumerating simple cycles in directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' These algorithms implement advanced recursion tree pruning techniques to improve on the brute- force Tiernan algorithm [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 covers such pruning techniques in further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A cycle enumeration algorithm that is asymptotically faster than the aforementioned al- gorithms [22], [23], [37] has been proposed in Birmel´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [40], however, it is applicable only to undirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Simple cycles can also be enumerated by computing the powers of the adjacency matrix [41], [42], [43] or by using circuit vector space algorithms [24], [44], [45], but the com- plexity of such approaches grows exponentially with the size of the cycles or the size of the input graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Time-window, temporal ordering, and hop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' It is common to search for cycles under some additional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For instance, in temporal graphs, it is common to search for cycles within a sliding time window, such as in Kumar and Calders [14] and Qiu et al [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, temporal ordering constraints can be imposed when search- ing for cycles in temporal graphs, such as in Kumar and Calders [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Furthermore, the maximum number of hops in cycles or paths can be constrained, such as in Gupta and Suzumura [47] and Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that hop- constrained simple cycles can also be enumerated using incremental algorithms, such as in Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' How- ever, this algorithm is based on the brute-force Tiernan algorithm [35], which makes it slower than nonincremental algorithms that use recursion tree pruning techniques [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Additionally, because incremental algorithms maintain aux- iliary data structures, such as paths, to be able to construct cycles incrementally, they are not as memory-efficient as nonincremental algorithms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Table 2 offers comparisons between the capabilities of these methods and ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Parallel and distributed algorithms for cycle enumera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [48] proposed a multi-threaded algorithm for detecting and removing simple cycles of a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The algorithm divides the graph into its strongly-connected components and each thread performs a depth-first search on a different component to find cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, sizes of the strongly-connected components in real-world graphs can vary significantly [49], which leads to a workload imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Rocha and Thatte [50] proposed a distributed algorithm for simple cycle enumeration based on the bulk- synchronous parallel model [51], but it searches for cycles in TABLE 3 Summary of the notation used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Symbol Description G(V, E) Graph with vertices V and edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' N(v) The set of neighbours of the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' u → v A directed edge connecting vertex u with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' n, e Number of vertices and edges in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' δ Size of a time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' c Number of simple cycles in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' s Number of maximal simple paths in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Π Current simple path explored by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Blk Set of blocked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Blist Unblock list of the Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' E Path extension of the Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' XTi Data structure X is maintained by the thread Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' p Number of threads used by parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Tp(n) Execution time of a parallel algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Wp(n) Amount of work a parallel algorithm performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' a brute-force manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Qing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [46] introduced a parallel algorithm for finding length-constrained simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' It is the only other fine-grained parallel algorithm we are aware of in the sense that it can search for cycles starting from the same vertex in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, the way this algorithm searches for cycles is similar to the way the brute- force Tiernan algorithm [35] works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To our knowledge, we are the first ones to introduce fine-grained parallel versions of asymptotically-optimal simple cycle enumeration algo- rithms, which do not rely on a brute-force search, as we show in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 BACKGROUND This section introduces the main theoretical concepts used in this paper and provides an overview of the most prominent simple cycle enumeration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The notation used is given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 Preliminaries We consider a directed graph G(V, E) having a set of vertices V and a set of directed edges E = {u → v | u, v ∈ V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The set of neighbours of a given vertex v is defined as N(v) = {w | v → w ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to the vertex v of an edge v → u as its source vertex and to the vertex u as its destination vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' An outgoing edge of a given vertex v is defined as v → w and an incoming edge is defined as u → v, where v → w, u → v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A path between the vertices v0 and vk, denoted as v0 → v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' → vk, is a sequence of vertices such that there exists an edge between every two consecutive vertices of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A simple path is a path with no repeated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A simple path is maximal if the last vertex of the path has no neighbours or all of its neighbours are already in the path [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A cycle is a path of non-zero length from a vertex v to the same vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A simple cycle is a cycle with no repeated vertices except for the first and last vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The number of maximal simple paths and the number of simple cycles in a graph are denoted as s and c, respectively (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that s can be exponentially larger than c [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A path or a cycle is said to satisfy a hop-constraint L if the number of edges in that path or cycle is less than or equal to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The goal of simple cycle enumeration is to compute all simple cycles of a directed graph G, ideally without computing all maximal simple paths of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 15 11 13 10 14 7 2 5 1 6 12 (a) Time window [2 : 7] 15 11 13 10 14 7 2 5 6 12 1 (b) Time window [10 : 15] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Two snapshots of a temporal graph associated with two different time windows of size δ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The solid arrows indicate the edges that belong to the respective time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A temporal graph is a graph that has its edges annotated with timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In temporal graphs, a temporal cycle is a simple cycle, in which the edges appear in the increasing order of their timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A simple cycle or a temporal cycle of a temporal graph occurs within a time window [tw1 : tw2] if every edge of that cycle has a timestamp ts such that tw1 ≤ ts ≤ tw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2 shows the simple cycles of a temporal graph that occur within two different time windows of size δ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This graph contains one simple cycle in the time window [2 : 7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2a), which is also a temporal cycle, and two simple cycles in the time window [10 : 15] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2b), neither being a temporal cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 Task-level parallelism The parallel algorithms described in this paper can be im- plemented using shared-memory parallel processing frame- works, such as TBB [54], Cilk [55], and OpenMP [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' These frameworks enable the decomposition of a program into tasks that can be independently executed by different soft- ware threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In our setup, tasks are created and scheduled dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A parent task can spawn several child tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The depth of a task is the number of its direct ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A dynamic task management system assigns the tasks created to the work queues of the available threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Furthermore, a work-stealing scheduler [54], [55], [57] enables a thread that is not executing a task to steal a task from the work queue of another thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Stealing tasks enables dynamic load balancing and ensures full utilisation of the threads when there are sufficiently many tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 Work efficiency and scalability We use the notions of work efficiency and scalability to analyse parallel algorithms [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to the time to execute a parallel algorithm on a problem of size n using p threads as Tp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The size of a graph is determined by the number of vertices n as well as the number of edges e, but we will refer only to n for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The depth of an algorithm is the length of the longest sequence of dependent operations in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The time required to execute such a sequence is equal to the execution time of the parallel algorithm using an infinite number of threads, denoted by T∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Furthermore, work performed by a parallel algorithm on a problem of size n using p threads, denoted as Wp(n), is the sum of the execution times of the individual threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The work efficiency and the scalability are formally defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (Work efficiency) A parallel algorithm is work- efficient if and only if Wp(n) ∈ O(T1(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (Scalability) A parallel algorithm is scalable if and only if lim n→∞ � lim p→∞ Tp(n) T1(n) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Informally, a work-efficient parallel algorithm performs the same amount of work as its serial version, within a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Scalability implies that, for sufficiently large inputs, increasing the number of threads increases the speedup of the parallel algorithm with respect to its serial version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We also define the notion of strong scalability as fol- lows [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (Strong scalability) A parallel algorithm is strongly scalable if and only if T1(n) Tp(n) = Θ(p) for large enough n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Whereas Definition 2 implies that the speedup T1(n)/Tp(n) achieved by a parallel algorithm with respect to its serial execution is infinite when the number of threads p is infinite, Definition 3 implies that the speedup is always in the order of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Another related concept is weak scalability, which requires the speedup to be in the order of p when the input size per thread is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that both strong scalability and weak scalability imply scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 Simple cycle enumeration algorithms The following algorithms for simple cycle enumeration perform recursive searches to incrementally update simple paths that can lead to cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Each algorithm iterates the vertices or edges of the graph and independently constructs a recursion tree to enumerate all the cycles starting from that vertex or edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The difference between these algorithms is to what extent they reduce the redundant work performed during the recursive search, which we discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The Tiernan algorithm [35] enumerates simple cycles using a brute-force search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' It recursively extends a simple path Π by appending a neighbour u of the last vertex v of Π provided that u is not already in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A clear downside of this algorithm is that it can repeatedly visit vertices that can never lead to a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When searching for cycles in the graph shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3a starting from the vertex v0, this algorithm would explore the path containing b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk 2m times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' From each vertex wi and ui, with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , m}, the Tiernan algorithm would explore this path only to discover that it cannot lead to a simple cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As noted by Tarjan [36], the Tiernan algorithm explores every simple path and, con- sequently, all maximal simple paths of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Exploring a maximal simple path takes O(e) time because it requires visiting each edge of the graph in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Given a graph with s maximal simple paths (see Table 3), the worst- case time complexity of the Tiernan algorithm is O(se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The Johnson algorithm [22] improves upon the Tiernan algorithm by avoiding the vertices that cannot lead to sim- ple cycles when appended to the current simple path Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this purpose, the Johnson algorithm maintains a set of blocked vertices Blk that are avoided during the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, a list of vertices Blist[w] is stored for each blocked vertex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Whenever a vertex w is unblocked (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=', removed from Blk) by the Johnson algorithm, the vertices in Blist[w] are also unblocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This unblocking process is performed 5 𝑣1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑣1 𝑣2 𝑤1 𝑤2 𝑤𝑚 𝑢1 𝑢2 𝑢𝑚 𝑏!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑏" 𝑏# 𝑏!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑏" 𝑏# 𝑣2 𝑢1 𝑢2 𝑢𝑚 𝑏!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑏" 𝑏# 𝑣0 𝑣0 Left subtree Π 𝐸 𝐸% (a) Example graph (b) Recursion tree 𝑣 𝑢 Vertex 𝑣 in 𝐵𝑙𝑖𝑠𝑡[𝑢] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 Vertex in !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' "# Right subtree !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) An example graph and (b) the recursion tree constructed when searching for cycles that start from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The nodes of the recursion tree represent the recursive calls of the depth-first search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The dotted path of the right subtree is explored only by the Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' recursively until no more vertices can be unblocked, which we refer to as the recursive unblocking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A vertex v is blocked (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=', added to Blk) when visited by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If a cycle is found after recursively exploring every neighbour of v that is not blocked, the vertex v is unblocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, v is not immediately unblocked if no cycles are found after exploring its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Instead, the Blist data structure is updated to enable unblocking of v in a later step by adding v to the list Blist[w] of every neighbour w of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This delayed unblocking of the vertices enables the Johnson algorithm to discover each cycle in O(e) time in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because this algorithm requires O(n + e) time to determine that there are no cycles, its worst-case time complexity is O (n + e + ec) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that because s can be exponentially larger than c [36], the Johnson algorithm is asymptotically faster than the Tiernan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3a, every simple path Π that starts from v0 and contains vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk is a max- imal simple path, and, thus, it cannot lead to a simple cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The Johnson algorithm would block b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk immediately after visiting this sequence once and then keep these vertices blocked until it finishes exploring the neighbours of v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, the Johnson algorithm visits vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk only once, rather than 2m times the Tiernan algorithm would visit them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that because these vertices get blocked during the exploration of the left subtree of the recursion tree, they are not going to be visited again during the exploration of the right subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Effectively, a portion of the right subtree is pruned (see the dotted path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3b) based on the updates made on Blk and Blist during the exploration of the left subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This strictly sequential depth- first exploration of the recursion tree is critically important for achieving a high pruning efficiency, but it also makes scalable parallelisation of the Johnson algorithm extremely challenging, which we are going to cover in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The Read-Tarjan algorithm [23] also has a worst-case time complexity of O (n + e + ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This algorithm main- tains a current path Π between a starting vertex and a frontier vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A recursive call of this algorithm iterates the neighbours of the current frontier vertex and performs a depth-first search (DFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Assume that v0 is the starting vertex and v1 is the frontier vertex of Π (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' From each neighbour y ∈ {v0, v2} of v1, a DFS tries to find a path extension E back to v0 that would form a simple cycle when appended to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3a, the algorithm finds two path extensions, one indicated as E and one that consists of the edge v1 → v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The algorithm then explores each path extension by iteratively appending the vertices from it to the path Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For each vertex x added to Π, the algorithm also searches for an alternate path extension from that vertex x to v0 using a DFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3a, the algorithm iterates through the vertices of the path extension E and finds an alternate path extension E′ from the neighbour u1 of v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If an alternate path extension is found, a child recursive call is invoked with the updated current path Π, which is v0 → v1 → v2 in our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Otherwise, if all the vertices in E have already been added to the current path Π, Π is reported as a simple cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In our example, the Read-Tarjan algorithm explores both E and E′ path extensions, and each leads to the discovery of a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The Read-Tarjan algorithm also maintains a set of blocked vertices Blk for recursion-tree pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, differently from the Johnson algorithm, Blk only keeps track of the vertices that cannot lead to new cycles when explor- ing the current path extension within the same recursive call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The vertices in Blk are avoided while searching for additional path extensions that branch from the current path extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For instance, the left subtree of the recursion tree shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3b demonstrates the exploration of the path extension E shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' During the exploration of E, the vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk are added to Blk immediately after visiting w1, and they are not visited again while exploring E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, when exploring another path extension E′ in the right subtree, the vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk are visited once again (see the dotted path of the right subtree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, the Read-Tarjan algorithm visits b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk twice instead of just once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As we are going to show in Section 6, this draw- back becomes an advantage when parallelising the Read- Tarjan algorithm because it enables independent exploration of different subtrees of the recursion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 COARSE-GRAINED PARALLEL METHODS The most straightforward way of parallelising the Johnson and the Read-Tarjan algorithms is to search for cycles that start from different vertices in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Each such search can then be executed by a different thread that explores its own recursion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This approach is beneficial because it is work-efficient and can be implemented using one of the existing graph processing frameworks, such as Pregel [26], in a manner similar to the method used by Rocha and Thatte [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to this parallelisation approach as the coarse-grained parallel approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The coarse-grained approach can express more paral- lelism if each thread performs a search for cycles that start from a different edge rather than a different vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This assumption is supported by the fact that graphs typically have more edges than vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Nevertheless, the coarse- grained approach is not scalable, which we prove here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The coarse-grained parallel Johnson and Read- Tarjan algorithms are work-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The proof of Proposition 1 is trivial, and we omit it for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The coarse-grained parallel Johnson and Read- Tarjan algorithms are not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑣"#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑣$ 𝑣"#$ ⋮ ⋮ ⋮ ⋮ ⋮ 𝑣% 𝑣$ 𝑣& 𝑣\' 𝑣( 𝑣& 𝑣\' 𝑣( 𝑣\' 𝑣( 𝑣( 𝑣( 𝑣\' 𝑣( 𝑣( 𝑣( 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thread 1 Thread 2 Thread 3 (a) Example graph (b) Recursion tree Thread 4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) A graph with an exponential number of simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (b) The recursion tree of the Johnson algorithm for n = 6 constructed when the algorithm starts from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Whereas a coarse-grained parallel algorithm explores the complete recursion tree using a single thread, our fine- grained parallel algorithms can explore different regions of the recursion tree in parallel using several threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' TABLE 4 Work and depth of the coarse- and fine-grained parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Parallel algorithm Work Depth Coarse-grained algorithms O (n + e + ec) O (ec) Fine-grained Johnson alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' O (n + e + min{pce, se}) O (e) Fine-grained Read-Tarjan alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' O (n + e + ec) O (ne) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this case, the depth T∞(n) represents the worst- case execution time of a search for cycles that starts from a single vertex or edge, and it depends on the number of cycles found during this search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the worst case, a single recursive search can discover all cycles of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' An example of such graph is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4a, where each vertex vi, with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , n − 1}, is connected to v0 and to every vertex vj such that j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In that graph, any subset of vertices v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , vn−1 defines a different cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, the total number of cycles in this graph is equal to the number of all such subsets c = 2n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Before the search for cycles, both the Johnson and the Read-Tarjan algorithm find all vertices that start a cycle, which is only v0 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, the search for cycles will be performed only by one thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because both the Johnson and the Read-Tarjan algorithms require O(e) time to find each cycle, the depth of the coarse-grained algorithms is T∞(n) ∈ O(ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because lim n→∞ T∞(n)/T1(n) ̸= 0, the coarse-grained algorithms are not scalable based on Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Theorem 1 shows that the main drawback of the coarse- grained parallel algorithms is their limited scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This limitation is apparent for the graph shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4a, which has an exponential number of cycles in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When using a coarse-grained parallel algorithm on this graph, all the cycles will be discovered by a single thread, and, thus, the depth of this algorithm grows linearly with c, as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because only one thread can be effectively utilised, increasing the number of threads will not result in a re- duction of the overall execution time of the coarse-grained parallel algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1a shows the workload imbalance exhibited by the coarse-grained parallel algorithms in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Section 8 demonstrates the limited scalability of coarse- grained parallel algorithms in further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5 FINE-GRAINED PARALLEL JOHNSON To address the load imbalance issues that manifest them- selves in the coarse-grained parallel Johnson algorithm, 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑏" 𝑏# 𝑏$ 𝑏!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' ⋮ 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑢$ 𝑢# 𝑢% 𝑣$ 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑢$ 𝑢# 𝑢% 𝑣$ 𝑣$ 𝑣$ 𝑣$ 𝑢% 𝑢% 𝑢% Infeasible regions explored by one thread Infeasible regions explored by multiple threads (a) Example graph (b) Recursion tree !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) An example graph and (b) the recursion tree of our fine- grained Johnson algorithm when enumerating cycles that start from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Each thread of our fine-grained Johnson algorithm explores the vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bm at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' we introduce the fine-grained parallel Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The main goal of our fine-grained algorithm is to enable several threads to explore a recursion tree concurrently, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4b, where each thread executes a subset of the recursive calls of this tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, enabling concurrent exploration of a recursion tree is in conflict with the sequential depth- first exploration, required by the Johnson algorithm to achieve a high pruning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this section, we first discuss the challenges that arise when parallelising the exploration of a recursion tree of the Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Then, we introduce the copy-on-steal mechanism used to address these challenges and present our fine-grained parallel Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Finally, we the- oretically analyse our algorithm and show that it is scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 Fine-grained parallelisation challenges The requirement of the sequential depth-first exploration of the Johnson algorithm makes it challenging to efficiently parallelise this algorithm in a fine-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This requirement is enforced by maintaining a set of blocked vertices Blk throughout the exploration of a recursion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If threads exploring the same recursion tree simply share the same set of blocked vertices Blk, the parallel algorithm could produce incorrect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For example, considering the graph given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5a, a thread exploring the path Π = v0 → v1 → u1 → v2 visit and block the vertex u4 in this case because u4 cannot participate in a simple cycle that begins with Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the threads exploring this graph share the blocked vertices, another thread attempting to discover the cycle v0 → v1 → u4 → v2 → v0 would fail to do so because u4 is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, this approach might not discover all cycles in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To enable several threads to correctly find all cycles while exploring the same recursion tree, the algorithm could for- ward a new copy of the Blk and Blist data structures when invoking each child recursive call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, this approach would redundantly explore many paths in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The reason is that a recursive call would be unaware of the vertices visited and blocked by other calls that precede it in the depth-first order except for its direct ancestors in the recursion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When enumerating the simple cycles of the graph shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5a starting from v0, this approach explores all 4 × 2m−1 + 3 maximal simple paths instead of just seven, that the Johnson algorithm would explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Hence, this approach exhaustively explores all maximal 7 Algorithm 1: FGJ task (v, v0, d, T1) Input: v - the current vertex, v0 - the starting vertex d - the depth of this task InOut: T1 - the thread that created this task Output: true if a cycle was found 1 T2 = the thread executing this task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Check if this task is stolen 2 if T1 ̸= T2 then FGJ copyOnSteal(d, T1, T2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 MutexT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='lock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='push(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' BlkT2 = BlkT2 ∪ {v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5 MutexT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='unlock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Recursively explore the neighbours of v 6 found = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7 foreach u : N(v) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='id > v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='id do 8 if u = v0 then 9 report cycle ΠT2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 10 found = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 11 else if u /∈ BlkT2 then 12 f = spawn FGJ task(u, v0, d + 1, T2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 13 found = found ∨ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 14 wait for the spawned tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 15 MutexT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='lock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 16 ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Unblock vertices if a cycle was found 17 if found then RecursiveUnblock(v, BlkT2, BlistT2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 18 else foreach u : N(v) do BlistT2[u] = BlistT2[u] ∪ {v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 19 MutexT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='unlock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 20 return found;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' simple paths in the graph and is identical to the brute-force solution of Tiernan (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Next, we propose a fine-grained parallel algorithm that addresses the aforemen- tioned parallelisation challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 Copy-on-steal To enable different threads to concurrently explore the re- cursion tree in a depth-first fashion while also taking ad- vantage of the powerful pruning capabilities of the Johnson algorithm, each thread executing our fine-grained parallel Johnson algorithm maintains its own copy of the Π, Blk, and Blist data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' These data structures are copied between threads only when these threads attempt to explore the same recursion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To achieve this behaviour, our fine-grained parallel Johnson algorithm implements each recursive call of the Johnson algorithm as a separate task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The pseudocode of this task is given in Algorithm 1, where a data structure X, maintained by the thread Ti, is denoted as XTi (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If a child task and its parent task are executed by the same thread Ti, the child task reuses the ΠTi, Blk Ti, and BlistTi data structures of the parent task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, if a child task has been stolen—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=', it is executed by a thread other than the thread that created it, the child task will allocate a new copy of these data structures (line 2 of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to this mechanism as copy-on-steal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The problem with copying data structures between dif- ferent threads upon task stealing is that the thread that has created the stolen task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=', the victim thread) can modify its data structures before this task is stolen by another thread (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=', the stealing thread).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This problem can be observed in (a) Example graph (b) Recursion tree Victim thread 𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Stealing thread 𝑇" The stolen task Task created Task stolen 𝑣0 𝑣1 𝑣2 𝑣3 𝑣4 𝑣5 𝑣2 𝑣0 𝑣1 𝑣4 𝑣5 𝑣7 𝑣4 𝑣5 𝑣3 𝑣2 𝑣3 𝑣7 ∈ Blk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' ∈ Blk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='" = B ⟺ 𝑣 ∈ Blist!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [𝑢] ⟺ 𝑣 ∈ Blist!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' "[𝑢] 𝑣 𝑢 𝑣 𝑢 𝑣6 𝑣0 𝑣6 𝑣6 𝑣0 Task that reports a simple cycle ∈ Π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' ∈ Π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='" = $ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) An example graph and (b) the recursion tree of our fine- grained Johnson algorithm when enumerating simple cycles that start from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Here, XTi denotes a data structure X of the thread Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The thread T2 can prune the dotted part of the tree by avoiding v5 and v6 that the thread T1 has blocked after creating the task stolen by T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' There, the victim thread T1 and the stealing thread T2 explore the same recursion tree given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6b while searching for cycles that start with P1 = v0 → v1 → v2 and P2 = v0 → v1 → v7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this case, T2 steals a task created by T1 that explores v7, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6b, and receives a copy of the blocked vertices Blk T1 = {v4, v5, v6} discovered by T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The thread T1 blocked these vertices because they cannot participate in any simple cycle that begins with P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If T2 simply uses a copy of these blocked vertices Blk T1 without modifications, T2 will be unable to find the cycle v0 → v1 → v7 → v4 → v2 → v3 → v0 because v4 is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, a method for unblocking vertices after copy-on- steal is required to correctly find all cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We explore two solutions for this problem: (i) Copy-on-steal with complete unblocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To enable the threads of our algorithm to find cycles after performing copy-on-steal, the stealing thread could unblock all vertices that the victim thread had blocked after creating the stolen task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In our example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6, the stealing thread T2 unblocks all vertices Blk T1 = {v4, v5, v6} it received from the victim thread T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Although this approach enables T2 to correctly find cycles, it also fails to take advantage of the information collected by T1 to reduce the redundant work of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6, T2 visits v5 and v6, even though T1 already concluded that these vertices cannot participate in any simple cycle that begins with P = v0 → v1, where P is the largest common prefix of all the paths explored by T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, T2 redundantly visits the dotted part of the recursion tree given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (ii) Copy-on-steal with recursive unblocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this approach, the stealing thread capitalises on the information already discovered by the victim thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The stealing thread T2 can reuse a subset B ⊂ Blk T1 of the blocked vertices discovered by T1 if the vertices in B cannot participate in simple cycles that begin with P, where P is the largest common prefix of all the paths explored by T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because any path discovered by T2 begins with P, T2 can avoid visiting vertices from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, to correctly find simple cycles, it is sufficient for T2 to unblock the vertices from Blk T1 \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To achieve this behaviour, T2 invokes a recursive unblocking procedure of the Johnson algorithm for every vertex v ∈ ΠT1 \\ P, as shown in Algorithm 2, where ΠT1 is the path T1 is exploring during task stealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The vertices in B can only be unblocked by a recursive unblocking invoked 8 Algorithm 2: FGJ copyOnSteal (d, T1, T2) Input: d - the depth of the task executing this function InOut: T1 - the victim thread T2 - the stealing thread 1 MutexT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='lock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2 {ΠT2, BlkT2, BlistT2} = copy ({ΠT1, BlkT1, BlistT1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 MutexT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='unlock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 while |ΠT2| ≥ d do 5 u = ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6 RecursiveUnblock(u, BlkT2, BlistT2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' for v ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' hence, the vertices in B remain blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6, T2 invokes a recursive unblocking procedure for ΠT1 \\ P = {v2}, which results in unblocking of v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, T2 is able to discover a cycle that contains v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The vertices B = {v5, v6} will not be unblocked because they cannot take part in any simple cycle that begins with P = v0 → v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, thread T2 avoids visiting the dotted part of the recursion tree given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Without countermeasures, our algorithm can suffer from race conditions because its data structures can be accessed concurrently by different threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For instance, a stealing thread T2 can copy the data structures of a victim thread T1 while T1 performs a recursive unblocking, in which case T2 could receive the vertex set Blk T1 that is partially unblocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When using copy-on-steal with recursive un- blocking, T2 may not be able to continue the interrupted unblocking of Blk T1, causing the algorithm to miss certain cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To avoid this problem, we define critical sections in lines 15–19 of Algorithm 1 and in lines 1–3 of Algorithm 2 using coarse-grained locking by maintaining a mutex per thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, such a locking mechanism is not required when using copy-on-steal with complete unblocking be- cause T2 can correctly unblock vertices in Blk T1 simply by removing all vertices from Blk T1 inserted after the stolen task was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, it is sufficient to enable thread-safe operations on Π, Blk, and Blist using fine-grained locking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, the critical sections are shorter when the copy- on-steal with complete unblocking approach is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Nevertheless, we opt to use the copy-on-steal with recur- sive unblocking approach in our fine-grained parallel John- son algorithm because this approach leads to less redundant work and rarely suffers from synchronisation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 Theoretical analysis We now show that the fine-grained parallel Johnson algo- rithm is scalable but not work-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The fine-grained parallel Johnson algorithm is not work-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' According to Lemma 3 presented by Johnson [22], a vertex cannot be unblocked more than once unless a cycle is found, and once a vertex is visited, it can be visited again only after being unblocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, the Johnson algorithm visits each vertex and edge at most c times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the fine-grained parallel Johnson algorithm executed using p threads, each thread maintains a separate set of data structures used for managing blocked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the threads are unaware of each other’s blocked vertices, each edge is visited at most pc times, c times by each thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Additionally, an edge cannot be visited more than s times because each maximal simple path of a graph is explored by a different thread in the worst case, and during each simple path exploration, an edge is visited at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, the maximum number of times an edge can be visited by the fine-grained parallel Johnson algorithm is min {s, pc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the algorithm executes in O(n + e) time if there does not exist a cycle or a path in the input graph, the work performed by the fine-grained parallel Johnson algorithm is Wp(n) ∈ O (n + e + min{pce, se}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When c > 0, p > 1, and s > c, the work performed by the fine-grained parallel Johnson algorithm Wp(n) is greater than the execution time T1(n) of the sequential Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, this algorithm is not work-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The work inefficiency of our fine-grained parallel John- son algorithm occurs if more than one thread performs the work the sequential Johnson algorithm would perform between the discovery of two cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This behaviour can be illustrated using the graph from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5a, which contains c = 4 cycles and s = c × 2m−1 + 3 maximal simple paths, each starting from vertex v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When discovering each cycle, our fine-grained algorithm explores an infeasible region of the recursion tree, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5b, in which the vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bm are visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If this infeasible region is explored using a single thread, each vertex bi, with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , m}, will be visited exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, if p threads are exploring the same infeasible region of the recursion tree, vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bm will be visited up to p times because the threads are unaware of each other’s blocked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this case, the fine-grained parallel Johnson algorithm performs more work than necessary, and, thus, it is not work-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Additionally, each infeasible region of the recursion tree that visits vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bk can be executed by at most s/c = 2m−1 threads because there are 2m−1 maximal simple paths that can be explored in each infeasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this case, each vertex bi, with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , m}, is visited up to s times, and, thus, the fine-grained parallel Johnson algorithm behaves as the Tiernan algorithm (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The depth T∞(n) of the fine-grained parallel Johnson algorithm is in O(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The worst-case depth of this algorithm occurs when a thread performs copy-on-steal and explores a maximal simple path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A thread explores such a path in O(e) time because it visits at most e edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, Π and Blk contain at most e vertices, and Blist contains at most e pairs of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, copy-on-steal requires O(e) time to copy Π, Blk, and Blist, and to unblock vertices in Blk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, the depth of this algorithm is T∞(n) ∈ O(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The fine-grained parallel Johnson algorithm is scal- able when lim n→∞ c = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this algorithm, T1(n) ∈ O(n+e+ec) and T∞(n) ∈ O(e) (see Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Given e < n2 and our assumption that lim n→∞ c = ∞, we have lim n→∞ T∞(n) T1(n) = lim n→∞ e n + e + ec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, this algorithm is scalable based on Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For the fine-grained parallel Johnson algorithm to be scalable, it is sufficient for c to increase sublinearly with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Even though this algorithm is scalable, a strong or weak 9 𝑣2 𝑣0 𝑣1 𝑣5 𝑣4 𝑣0 𝑣3 𝑣4 𝑣0 𝑣6 𝑣4 𝑣0 𝑣8 𝑣7 𝑣6 𝑣6 𝑣7 𝑣7 𝑣8 𝑣8 𝑣8 (a) Example graph (b) Recursion tree 𝑣0 𝑣1 𝑣2 𝑣3 𝑣4 𝑣5 𝑣6 𝑣8 Task 𝑣7 𝐸1 𝐸2 𝐸3 Path extension exploration 𝑡0 𝑡1 𝑡2 𝑡3 𝑡4 Vertex blocked in 𝑡2, 𝑡3, and 𝑡4 Vertex blocked in 𝑡3 Vertex blocked in !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3, and !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 Simple cycle reported Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) An example graph and (b) the recursion tree of our fine- grained parallel Read-Tarjan algorithm when enumerating cycles that start from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The nodes of the recursion tree represent the recursive calls of the depth-first search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Tasks shown in (b) can be executed independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' scalability is not guaranteed due to the work inefficiency of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Nevertheless, our experiments show that this algorithm is strongly scalable in practice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 Summary Our relaxation of the strictly depth-first-search-based recursion-tree exploration reduces the pruning efficiency of the Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the worst case, the fine-grained parallel Johnson algorithm could perform as much work as the brute-force Tiernan algorithm does—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=', O(se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' How- ever, in practice, this worst-case scenario does not happen (see Section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, our fine-grained parallel Johnson algorithm can suffer from synchronisation issues in some rare cases (see Section 8) because our copy-on-steal mecha- nism can lead to long critical sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the next section, we introduce a fine-grained parallel algorithm that is scalable, work-efficient, and less prone to synchronisation issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6 FINE-GRAINED PARALLEL READ-TARJAN In this section, we first introduce several optimisations that reduce the number of unnecessary vertex visits per- formed by the sequential Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Then, we present our fine-grained parallel Read-Tarjan algorithm that includes these optimisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Finally, we show that our parallel algorithm is work-efficient and strongly-scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 Improvements to the pruning efficiency To improve the pruning efficiency of the sequential Read- Tarjan algorithm, we include the following optimisations: (i) Blocked vertex set forwarding enables a recursive call of the Read-Tarjan algorithm to reuse vertices blocked by its parent call, resulting in fewer vertex visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The original Read-Tarjan algorithm discards blocked vertices after each recursive call [23], even though this information could be reused later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this optimisation, the algorithm forwards the blocked vertices Blk of a recursive call to its child recur- sive calls, preventing those child calls from unnecessarily visiting the vertices in Blk again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7, the vertex v8 is blocked the first time the algorithm visits v8 while exploring the path extension E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This optimisa- tion prevents the algorithm from visiting v8 again when exploring the same extension E1 or another extension E3 Algorithm 3: FGRT DFS(u, v0, Blk, Vis) Input: u - the current vertex, v0 - the starting vertex InOut: Blk - blocked vertices Vis - vertices visited during the DFS Output: E - the resulting path extension from u to v0 1 if u = v0 then return u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2 Vis = Vis ∪ {u};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 block = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 foreach w : N(u) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='id > v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='id do 5 if w = v0 then return u → w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6 else if w /∈ Blk ∧ w /∈ Vis then 7 E = FGRT DFS(w, v0 Blk, Vis);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8 if E ̸= ∅ then return E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='push front(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9 if w /∈ Blk then block = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 10 if block then Blk = Blk ∪ {u};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 11 return ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' that branches from E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result of this optimisation, the algorithm can avoid the dotted part of the recursion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (ii) Path extension forwarding prevents recomputation of the path extension E found by a parent recursive call by forwarding this path extension to its child recursive call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this way, each child recursive call performs one fewer DFS invocation than the original Read-Tarjan algorithm [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (iii) Blocking on a successful DFS is another mechanism for discovering vertices to be blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a reminder, the Read-Tarjan algorithm searches for path extensions using a DFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the original algorithm, a vertex is blocked only if it is visited during an unsuccessful DFS invocation, which fails to discover a path extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, successful DFS invocations could also visit some vertices that have all their neighbours blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Such vertices cannot lead to the discovery of new cycles and, thus, can also be blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The pseudocode of the DFS function that includes this optimisation is given in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In our example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7, a successful DFS invoked from v3 finds a path extension E3 and discovers that the only neighbour v8 of v7 is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The algorithm then blocks v7, which enables it to avoid visiting v7 again when exploring E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, fewer vertices are visited during the execution of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 Fine-grained parallelisation Although the optimisations presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 elimi- nate some of the redundant work performed by the Read- Tarjan algorithm, this algorithm typically performs more work than the Johnson algorithm (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, this redundancy makes it possible to parallelise the Read- Tarjan algorithm in a scalable and work-efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the Read-Tarjan algorithm allocates a new Blk set for each path extension exploration, a recursive call can explore different path extensions in an arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, discovery of a new path extension E results in the invocation of a single recursive call, and these calls can be executed in an arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, several threads can concurrently explore different paths of the same recursion tree constructed by the Read-Tarjan algorithm for a given starting edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' There are neither data dependencies nor ordering requirements between different calls, apart from those that exist between a parent and a child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To 10 Algorithm 4: FGRT task(v, v0, E, d, T1) Input: v - the current vertex, v0 - the starting vertex E - the path extension from v to v0 d - the depth of this task InOut: T1 - the thread that created this task 1 T2 = the thread executing this task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Check if this task is stolen 2 if T1 ̸= T2 then {ΠT2, BlkT2} = copy ({ΠT1, BlkT1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Operations on Π and Blk are thread-safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 while ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='back() ̸= v do ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 Remove vertices from BlkT2 inserted at depth d′ ≥ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5 found = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Exploration of the path extension E 6 while E ̸= ∅ do 7 v = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='pop front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8 ΠT2 = ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='push(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' BlkT2 = BlkT2 ∪ {v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9 foreach u : N(v) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='id > v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='id do 10 if u ̸= E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='front() ∧ u /∈ BlkT2 then 11 E′ = FGRT DFS(u, v0, BlkT2, Vis = ∅);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 12 if E′ ̸= ∅ then 13 spawn FGRT task(v, v0, E′, d + 1, T2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 14 found = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 15 else BlkT2 = BlkT2 ∪ Vis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 16 if found then break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 17 if E = ∅ then report cycle ΠT2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 18 else spawn FGRT task(v, v0, E, d + 1, T2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' exploit the parallelism available during the recursion tree exploration, we execute each path extension exploration in each recursive call as a separate task, all of which can be independently executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Examples of such tasks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to the resulting algorithm as the fine- grained parallel Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our implementation shown in Algorithm 4 performs only a single path extension exploration in a recursive call and uses all the optimisations we introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We execute each such recursive call as a separate task using a dynamic thread scheduling framework (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For each edge v0 → v, we execute a parallel for loop iteration that uses Algorithm 3 to search for a path extension E from v to v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' If such E exists, a task is created using v, v0, and E as its input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This task then recursively creates new tasks, as shown in lines 13 and 18 of Algorithm 4, until all cycles that start with the edge v0 → v have been discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To prevent different threads from concurrently modify- ing Π and Blk, each task allocates and maintains its own Π and Blk sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A task can receive a copy of Π and Blk directly from its parent task at the time of task creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, it is possible to minimise the copy overheads by copying these sets only when a task is stolen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this purpose, we use the copy-on-steal with complete unblocking approach described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2, which has shorter critical sections than the copy-on-steal with recursive unblocking approach used by our fine-grained parallel Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 Theoretical analysis We now show that the fine-grained parallel Read-Tarjan algorithm is both work-efficient and strongly scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The fine-grained parallel Read-Tarjan algorithm is work-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because each task of our fine-grained parallel Read- Tarjan algorithm either discovers a cycle or creates at least two child tasks, our algorithm is executed using O(c) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Each task performs several unsuccessful DFS invocations and one successful DFS per each child task it creates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' All unsuccessful DFS invocations explore at most e edges in total because they share the same set of blocked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the worst case, each edge is visited twice per task, once by a successful DFS and once by one of the unsuccessful DFS invocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, this algorithm performs O(e) work per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because this algorithm performs O (n + e) work if there are no cycles in the graph, the total amount of work this algorithm performs is Wp(n) = O (n + e + ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Hence, this algorithm is work-efficient based on Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The work-efficiency of our fine-grained parallel Read- Tarjan algorithm can be demonstrated using the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this example, the threads of this algo- rithm independently explore four different path extensions Ei = v1 → ui → v2 → v0, with i ∈ {1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A thread exploring a path extension Ei invokes a DFS from v2, which explores vertices b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' , bm at most once and fails to find any other path extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, the amount of work the fine-grained parallel Read-Tarjan algorithm performs does not increase compared to its single-threaded execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The depth T∞(n) of the fine-grained parallel Read- Tarjan algorithm is in O(ne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the worst case, a thread executing this algorithm creates a task for each vertex of its longest simple cycle, which has a length of at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Before invoking its first child task, a task executes a sequence of unsuccessful DFS invocations in O(e) and a successful DFS invocation also in O(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, the depth of this algorithm is O (ne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The worst-case depth of our algorithm can be observed when this algorithm is executed on the graph given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This graph has c = 2n−2 cycles and the length of its longest cycle v0 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' vn−1 → v0 is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The algorithm creates a task for each vertex of the cycle and performs a successful DFS in each such call, which leads to T∞ ∈ O(ne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The fine-grained parallel Read-Tarjan algorithm is strongly scalable when lim n→∞ c/n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the fine-grained parallel Read-Tarjan algo- rithm is work-efficient, we can apply Brent’s rule [60]: T1(n) p ≤ Tp(n) ≤ T1(n) p + T∞(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (1) Substituting T1(n) with O(n+e+ec) and T∞(n) with O(ne) (see Lemma 2), for a positive constant C0, it holds that 1 ��1 p + C0 n c � < 1 ��1 p + T∞(n) T1(n) � ≤ T1(n) Tp(n) ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (2) Given that lim n→∞ c/n = ∞, there exist n0 > 0, C1 > 0 such that if n > n0, then c/n > C1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, for every n > n0, it holds that kp ≤ T1(n) Tp(n) ≤ p, where k = C1/(C0 + C1) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, T1(n) Tp(n) = Θ(p), which, based on Definition 3, completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 11 As shown in Table 4, our fine-grained parallel Read- Tarjan algorithm has a higher depth than our fine-grained parallel Johnson algorithm, introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Nev- ertheless, the former algorithm is strongly scalable when c grows superlinearly with n, whereas strong scalability cannot be guaranteed for the latter algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 Summary The work of our fine-grained parallel Read-Tarjan algo- rithm does not increase after fine-grained parallelisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This parallel algorithm performs Wp(n) ∈ O(n + e + ec) work: the same as the work performed by its serial version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our optimisations introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 do not reduce the work Wp(n) performed by our parallel algorithm in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, these optimisations significantly improve its performance in practice (see Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, the synchronisation overheads of the fine-grained parallel Read-Tarjan algorithm are not as significant as those of the fine-grained Johnson algorithm because of its shorter critical sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Furthermore, this algorithm is the only asymptotically-optimal parallel algorithm for cycle enumer- ation for which we are able to prove strong scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7 PARALLELISING CONSTRAINED CYCLE SEARCH This section describes the methods for adapting our par- allel algorithms to search for simple cycles under various constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because state-of-the-art algorithms for temporal and hop-constrained cycle enumeration are extensions of the Johnson algorithm [14], [21], our parallelisation ap- proach described in Section 5 is also applicable to these algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this section, we describe the changes to the fine- grained parallel Johnson algorithm needed for enumeration of temporal and hop-constrained cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We also introduce modifications to the cycle enumeration algorithms required for finding time-window-constrained cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 Cycles in a time window Cycle enumeration algorithms require minimal modifica- tions to support time-window constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Such constraints restrict the search for simple, temporal, and hop-constrained cycles to those that occur within a time window of a given size δ, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To find time-window- constrained cycles that start with an edge that has times- tamp t0, only the edges with timestamps that belong to the time window [t0 : t0 + δ] are visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To avoid reporting the same cycle several times, another edge with the same timestamp t0 is visited only if the source vertex of that edge has an ID that is smaller than the ID of the vertex from which the search for cycles was started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Overall, imposing time-window constraints reduces the number of cycles dis- covered, which results in a more tractable problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A strongly-connected component (SCC) can be used to reduce the number of vertices visited during the search for time-window-constrained cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The search for cycles that start with the edge ε can be limited to use only the vertices from the SCC that contains ε [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the case of time- window-constrained cycles, we compute an SCC for ε using only the edges with timestamps that belong to [t0 : t0 + δ], where t0 is the timestamp of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because an SCC can be Algorithm 5: CFGJ copyOnSteal(d, T1, T2) Input: d - the depth of the task executing this function InOut: T1 - the victim thread T2 - the stealing thread 1 MutexT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='lock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' // Blk contains closing times or barriers // Blist is not used for hop-constrained cycles 2 {ΠT2, BlkT2, BlistT2} = copy ({ΠT1, BlkT1, BlistT1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 3 {PrevLocksT2} = copy ({PrevLocksT1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 4 MutexT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='unlock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 5 while |ΠT2| > d do 6 u = ΠT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7 lock = PrevLocksT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8 RecursiveUnblock(u, lock, BlkT2, BlistT2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' computed independently for each edge in O(e) time [61], our fine-grained parallel algorithms remain scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 Temporal cycles To efficiently enumerate temporal cycles, the 2SCENT al- gorithm [14] replaces the set of blocked vertices Blk in the Johnson algorithm with closing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The closing time ct of a vertex v indicates that the outgoing temporal edges of v with a timestamp greater than or equal to ct cannot participate in a temporal cycle and are therefore blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Increasing the closing time of v to a new value ct′ unblocks the blocked outgoing edges of v that have timestamps smaller than ct′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This operation triggers the recursive unblocking procedure that unblocks the incoming edges of v with a timestamp smaller than the maximal timestamp among the unblocked outgoing edges of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This process is repeated for every vertex with unblocked outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the backtracking phase of 2SCENT is based on the Johnson algorithm, it can be parallelised using our fine- grained approach described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this purpose, we use our copy-on-steal mechanism with recursive un- blocking, introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2, which enables a thread to maintain its own set of data structures used for recur- sion tree pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, this mechanism is not directly applicable in this case because the recursive unblocking procedure of 2SCENT requires the new closing time for a vertex as a parameter in addition to the vertex itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this reason, an additional data structure called PrevLocks is used alongside the current path Π that records the closing time that each vertex v had before it was added to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Copy- on-steal then performs the recursive unblocking procedure for each vertex v removed from Π using the original closing time of the vertex v obtained from PrevLocks, as shown in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to the resulting algorithm as the fine- grained parallel temporal Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The aforementioned modification to the copy-on-steal with recursive unblocking approach also enables a thread of our fine-grained parallel algorithm to reuse the edges blocked by another thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This behaviour can be observed in the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8, where the thread T2 steals the task indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8b from the thread T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Copy- on-steal executed by T2 invokes recursive unblocking that restores the closing time of v3 to its original value of 9 obtained from PrevLocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that this original closing 12 𝑣3 3 𝑣0 1 2 4 𝑣1 𝑣2 𝑣4 𝑣3 𝑣0 𝑣1 𝑣2 𝑣3 𝑣4 𝑣6 𝑣7 𝑣8 𝑣6 𝑣7 𝑣5 𝑣3 9 7 8 5 6 8 10 2 3 𝑣5 𝑣6 𝑣7 𝑣8 Victim thread 𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Stealing thread 𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=" 𝑣5 (a) Example graph (b) Recursion tree The stolen task 𝑣0 → ∞ 𝑣1 → ∞ 𝑣2 → ∞ 𝑣3 → 9 𝑣4 → ∞ 𝑃𝑟𝑒𝑣𝐿𝑜𝑐𝑘𝑠: Task created Task stolen ⟺ blocked veretx/edge of T' ⟺ blocked veretx/edge of T!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 𝑣0 𝑣0 𝑣3 Task that reports a temporal cycle ∈ path Π of T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' ∈ path Π of T" Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) An example graph and (b) the recursion tree of our fine- grained temporal Johnson algorithm when enumerating temporal cycles that start from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The thread T2 can avoid the dotted part of the tree by reusing the blocked edges v6 → v7 and v7 → v3 discovered by T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time of v3 was previously set by T1 while exploring the path v0 → v1 → v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The recursive unblocking that T2 invokes for v3 unblocks only the edge v6 → v3 because it is the only incoming edge of v3 with a timestamp smaller than the closing time 9 of the vertex v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Without recording the previous closing times, T2 could instead unblock all incoming edges of v3 by invoking recursive unblocking for v3 with a closing time ∞, which also unblocks the edges v6 → v7 and v7 → v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, because there is no temporal cycle that contains these two edges and starts with v0, T2 would unnecessarily visit them in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, restoring the closing time of v3 to its original value 9 prevents T2 from performing this redundant work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We also adapt the Read-Tarjan algorithm and its fine- grained and coarse-grained versions to enumerate temporal cycles using closing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The necessary changes to the al- gorithm are trivial, and we omit discussing them for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To reduce the number of vertices visited during the search for temporal cycles, we use a method similar to the SCC-based technique discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Instead of computing an SCC for each edge ε, we compute a cycle-union that represents an intersection of temporal ancestors and temporal descendants of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The temporal descendants and the temporal ancestors of ε are the vertices that belong to the temporal paths in which ε is the first edge and the last edge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Defined as such, a cycle-union contains only the vertices that participate in temporal cycles that have ε as their starting edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, the search for temporal cycles that start with ε can be limited to only those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 Hop-constrained cycles An efficient algorithm for enumerating hop-constrained cycles and paths, called BC-DFS [21], replaces the set of blocked vertices Blk in the Johnson algorithm with barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' A barrier value bar of a vertex v indicates that the starting vertex v0 of a cycle cannot be reached within bar hops from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, v is blocked if the length of the current path Π when the algorithm attempts to visit v is greater than or equal to L − bar, where L is the hop constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' BC-DFS modifies the recursive unblocking of the Johnson algorithm to reduce the barrier bar of v to a specified value bar ′ < bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This procedure also sets the barrier of any vertex u that can reach v in k hops to bar ′ + k if the previous barrier of u was greater than bar ′ + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Maintaining barriers in such a 𝑣0 (a) Example graph (b) Recursion tree 𝑣6 𝑣0 𝑣4 𝑣0 → 0 𝑣1 → 0 𝑃𝑟𝑒𝑣𝐿𝑜𝑐𝑘𝑠: 𝑣1 0,1,2,… Barrier values of T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 0,1,2,… Barrier values modified by T" during copy-on-steal 𝑣2 𝑣4 𝑣7 𝑣8 𝑣2 The stolen task 𝑣1 2 3 4 1 𝑣6 𝑣7 𝑣8 𝑣1 𝑣3 𝑣5 𝑣7 𝑣6 𝑣2 𝑣3 Stealing thread 𝑇# 𝑣0 𝑣3 𝑣0 Victim thread 𝑇* Task stolen Task created 𝑣9 𝑣5 1 Task that reports a simple cycle ∈ path Π of T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' ∈ path Π of T" Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' (a) An example graph and (b) the recursion tree of our fine- grained hop-constrained Johnson algorithm when enumerating cycles of length L = 6 that start from v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Barrier values of unmarked vertices are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Copy-on-steal enables the thread T2 to reuse barriers discovered by the thread T1 and to avoid exploring the dotted part of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' way minimises redundant vertex visits when searching for hop-constrained cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To parallelise BC-DFS in a fine-grained manner, we use the same technique as that used for fine-grained parallelisa- tion of the Johnson algorithm (Section 5) and the 2SCENT algorithm (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this case, threads exploring a recursion tree of BC-DFS maintain separate data structures, such as the current path Π and barrier values for each ver- tex, and use the copy-on-steal with the recursive unblocking approach to copy these data structures among threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Similarly to our algorithm from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2, each thread also maintains a data structure PrevLocks that records the original barrier value of each vertex v from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When a thread steals a task, it performs a recursive unblocking procedure for each vertex v removed from Π using its original barrier value obtained from PrevLocks, as shown in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This procedure reduces the barrier value of the vertices that can reach v, enabling the stealing thread to visit those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to the resulting algorithm as the fine-grained parallel hop-constrained Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The modified copy-on-steal with recursive unblocking approach given in Algorithm 5 enables a stealing thread of the aforementioned fine-grained parallel algorithm to reuse barriers discovered by other threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This behaviour can be observed in the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In that example, the thread T1 first visits the vertices v2, v6, v7, v8 and sets the barrier value of each visited vertex to L − |Π| + 1 (values in red shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9a) because it was not able to find a cycle of length L = 6 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Here, |Π| denotes the length of Π at the moment of exploration of each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When the thread T2 steals the task indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9b from T1, the copy- on-steal mechanism executed by T2 performs a recursive unblocking of the vertex v1 using the original barrier value 0 of v1 obtained from PrevLocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This recursive unblocking reduces the barrier value of v2 from 4 to 1, which enables T2 to find the cycle that contains v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The barrier values of the vertices v6, v7, and v8 are not modified, and, thus, the thread T2 avoids visiting these vertices unnecessarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 Summary In this section, we described a method to adapt the cy- cle enumeration algorithms, such as our fine-grained al- gorithms introduced in Sections 5 and 6, to search for 13 TABLE 5 Hardware platforms used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Here, P, C/P, and T/C represent the number of processors, the number of cores per processor, and the number of hardware threads per core, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' platform Intel KNL [62] Intel Xeon Skylake [63] P × C/P × T/C 4 × 64 × 4 5 × 48 × 2 Total no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' threads 1024 480 Frequency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 GHz 2 GHz Memory per proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 110 GB 360 GB L1d/L2/L3 cache 32 KB/512 KB/none 32 KB/1 MB/38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='5 MB cycles under time window constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, we introduced a modified version of our copy-on-steal with recursive unblocking approach, introduced in Section 5, that supports fine-grained parallelisation of temporal and hop-constrained cycle enumeration algorithms [14], [21] derived from the Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, our fine- grained parallel algorithms can enumerate cycles under time-window, temporal, and hop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8 EXPERIMENTAL EVALUATION This section evaluates the performance of our fine- grained parallel algorithms for simple, temporal, and hop- constrained cycle enumeration1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As Table 2 shows, we are the only ones to offer fine-grained parallel versions of the asymptotically-optimal cycle enumeration algorithms, such as the Johnson and the Read-Tarjan algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' However, the methods covered in Table 2 can be parallelised using the coarse-grained approach covered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, we use the coarse-grained approach as our main comparison point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The experiments are performed using two different clus- ters: Intel2 KNL [62] and Intel Xeon Skylake [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The details of these two clusters are given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We developed our code on the Intel KNL cluster and ran most of the analyses there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' yet, for completeness, we ran the comparisons to competing implementations also on the Intel Xeon Skylake cluster available in Google Cloud’s Compute Engine [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Scalability experiments are conducted on the Intel KNL cluster, where one thread per core is used if the number of threads used is less than or equal to 256, and simultaneous multithreading is enabled otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We use the Threading Building Blocks (TBB) [54] library to parallelise the algorithms on a single processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We distribute the execution of the algorithms across multiple processors using the Message Passing Interface (MPI) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When using distributed execution, each processor stores a copy of the input graph in its main memory and searches for cycles starting from a different set of graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The starting edges are divided among the processors such that when the edges are ordered in the ascending order of their timestamps, k consecutive edges in that order are assigned to k different processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Each processor then uses its own dynamic scheduler to balance the workload across its hardware threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this setup, workload imbalance across processors may still occur, but its impact is limited in our experiments because we use at most five processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The open-source implementations of our algorithms are main- tained here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='com/IBM/parallel-cycle-enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Intel and Intel Xeon are trademarks or registered trademarks of Intel Corporation or its subsidiaries in the United States and other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' TABLE 6 Temporal graphs used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Time span T is in days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' graph abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' n e T bitcoinalpha BA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 k 24 k 1901 bitcoinotc BO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 k 36 k 1903 CollegeMsg CO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 k 60 k 193 email-Eu-core EM 824 332 k 803 mathoverflow MO 16 k 390 k 2350 transactions TR 83 k 530 k 1803 higgs-activity HG 278 k 555 k 6 askubuntu AU 102 k 727 k 2613 superuser SU 138 k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 M 2773 wiki-talk WT 140 k 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 M 2277 friends2008 FR 481 k 12 M 1826 wiki-dynamic-nl NL 1 M 20 M 3602 messages MS 313 k 26 M 1880 AML-Data AML 10 M 34 M 30 stackoverflow SO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 M 48 M 2774 We perform the experiments using the graphs listed in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The TR, FR, and MS graphs are from Harvard Data- verse [65], the NL graph is from Konect [66], the AML graph is from the AML-Data repository [67], and the rest are from SNAP [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' To make cycle enumeration problems tractable, we use time-window constraints in all of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The time window sizes used in our experiments are given in the figures next to the graph names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We stop the execution of an algorithm if it takes more than 24h on the Intel KNL cluster or more than 6h on the Intel Xeon Skylake cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 Temporal cycle enumeration The goal of a temporal cycle enumeration problem is to find all simple cycles with edges ordered in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Here, we evaluate the performance of our fine-grained parallel algorithms for this problem introduced in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our main comparison points are the coarse-grained parallel ver- sions of the temporal Johnson and temporal Read-Tarjan algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We refer to the backtracking phase of the state- of-the-art 2SCENT algorithm [14] for temporal cycle enu- meration as the temporal Johnson algorithm and parallelise it in a coarse-grained manner for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We do not parallelise the entire 2SCENT algorithm because the preprocessing phase of 2SCENT is strictly sequential and has a time complexity in the order of the complexity of its backtracking phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We also provide direct comparisons with the 2SCENT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 10 shows that our fine-grained parallel algorithms achieve an order of magnitude speedup compared to the coarse-grained algorithms on the Intel KNL cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For the NL graph, this speedup reaches up to 40×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the Intel Xeon Skylake cluster contains fewer physical cores than the Intel KNL cluster, the speedup between our fine- grained and the coarse-grained parallel Johnson algorithms is smaller on the former cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 11, this speedup increases as we increase the time window size used in the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that enumerating cycles in larger time windows is more challenging because larger time windows contain a larger number of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The scalability evaluation of the parallel temporal cycle enumeration algorithms is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We also report the performance of the sequential 2SCENT algorithm in the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The performance of our fine-grained parallel algorithms improves linearly until 256 threads, after which 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='5 22 14 18 12 15 100 20 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 17 10 100 1 k 10 k 100 k BA 3000h BO 1000h CO 96h EM 144h MO 288h TR 800h HG 72h AU 336h SU 168h WT 144h FR 5h NL 1000s MS 4h AML 720h SO 66h geomean Execution time [s] Fine-grained parallel temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Johnson Fine-grained parallel temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Read-Tarjan Coarse-grained parallel temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Johnson Coarse-grained parallel temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Read-Tarjan (a) Performance of parallel algorithms for temporal cycle enumeration on the Intel KNL cluster using 1024 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='7 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Performance of parallel algorithms for temporal cycle enumeration on (a) the Intel KNL and (b) the Intel Xeon Skylake clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The numbers above the bars show the execution time of each algorithm relative to that of our fine-grained parallel temporal Johnson for the same benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 25 18 32 13 11 34 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 3.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='AML 720h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='SO 30h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='SO 48h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='SO 66h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='geomean 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='geomean 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='geomean 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='Fine-grained parallel temporal Johnson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='Coarse-grained parallel temporal Johnson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='Execution time [s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Larger time windows increase the performance gap between the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The algorithms are executed on the Intel KNL cluster using 1024 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The numbers above the bars show the execution times of the coarse-grained algorithm relative to that of the fine-grained algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='11 10 100 1000 1 4 16 64 256 1024 Fine-grained parallel temporal Johnson Fine-grained parallel temporal Read-Tarjan Coarse-grained parallel temporal Johnson 2SCENT [14] 205 x 185 x 10 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='9 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (f) TR 600h 325 x 213 x 32 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (c) CO 72h 308 x 246 x 30 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (h) AU 336h 356 x 198 x 126 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (d) EM 96h 285 x 170 x 21 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 10 1000 1 4 16 64 256 1024 Speedup Number of threads (k) FR 4h 359 x 249 x 53 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (e) MO 192h 435 x 211 x 85 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (j) WT 144h 208 x 118 x 16 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (b) BO 800h 242 x 106 x 11 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (a) BA 2200h 315 x 173 x 14 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (i) SU 120h 20 x 17 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='9 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 16 64 256 1024 Speedup Number of threads (n) AML 720 h 27 x 13 x 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 16 64 256 1024 Speedup Number of threads (m) MS 4h 242 x 198 x 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 10 1000 1 4 16 64 256 1024 Speedup Number of threads (l) NL 100s 14 x 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='9 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 32 64 128 256 512 1024 Speedup Number of threads (o) SO 66h 218 x 137 x 23 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='5 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (g) HG 48h Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Scalability evaluation of parallel temporal cycle enumeration algorithms executed on the Intel KNL cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The baseline is our fine-grained parallel temporal Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The relative performance of 2SCENT [14] is shown when it completes in 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that the 2SCENT implementation is single-threaded and the single-threaded execution results are not available for all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' it becomes sublinear due to simultaneous multithreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, our fine-grained versions of the Johnson and the Read-Tarjan algorithms reach 435× and 470× speedups, respectively, compared to their serial versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Addition- ally, when using 1024 threads, our fine-grained Johnson algorithm is on average 260× faster than 2SCENT when 2SCENT completes in 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' On the other hand, the coarse-grained Johnson algorithm does not scale as well as the fine-grained algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, the performance gap between the fine-grained and the coarse-grained algo- rithms increases as we increase the number of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Overall, the fastest algorithm for temporal cycle enu- meration that we tested is our fine-grained Johnson al- gorithm, which is, on average, 60% faster than our fine- grained Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When using 1024 threads, both fine-grained algorithms are an order of magnitude faster than their coarse-grained counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Moreover, our fine-grained parallel algorithms, executed on the Intel KNL cluster using 1024 threads, are two orders of magnitude faster than the state-of-the-art algorithm 2SCENT [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 Hop-constrained cycle enumeration In hop-constrained cycle enumeration, we search for all simple cycles in a graph that are shorter than the spec- ified hop constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We here compare our fine-grained parallel hop-constrained Johnson algorithm, introduced in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3, with the state-of-the-art algorithms BC-DFS and JOIN [21] for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For this evaluation, we parallelised BC-DFS and JOIN in the coarse-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because adapting the Read-Tarjan algorithm to enumerate hop-constrained cycles is not trivial, we do not report the performance of the fine-grained and coarse-grained versions of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We also omit the performance results for the MS graph because our fine-grained algorithm did not finish under 12h when using the smallest time window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 15 2 10 14 3 10 15 2 8 9 1 5 13 4 16 15 1 4 8 4 22 26 5 53 60 2 38 61 2 17 20 1 5 20 44 51 51 2 18 32 1 1 3 3 12 18 2 3 24 2 3 16 2 1 3 2 3 3 3 4 26 2 3 4 1 1 1 4 17 300 4 5 5 8 9 199 5 5 4 21 >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='7k >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='7k 2 233 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='5k 4 5 3 3 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='01 1 100 10 k Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time [s] BA 120h BO 140h CO 5h EM 5h MO 30h TR 110h HG 1h AU 20h SU 10h WT 12h FR 1500s NL 25s AML 72h SO 6h geomean 12 10 15 20 Fine-grained par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' hop-constrained Johnson Coarse-grained par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' BC-DFS [21] hop constraint: Coarse-grained par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' JOIN [21] (a) Performance of parallel algorithms for finding hop-constrained simple cycles on the Intel KNL cluster using 1024 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2 9 11 2 9 11 2 11 9 1 4 16 3 8 8 1 6 8 4 19 20 4 30 36 1 26 38 1 8 11 1 4 10 26 36 36 1 14 25 1 1 2 2 10 13 2 3 18 1 2 11 1 1 3 2 3 5 2 3 9 2 2 2 1 1 1 4 10 126 3 4 3 4 5 72 4 4 5 9 >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4k >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4k 1 64 > 1k 1 2 1 2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='01 1 100 10 k Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time [s] BA 120h BO 140h CO 5h EM 5h MO 30h TR 110h HG 1h AU 20h SU 10h WT 12h FR 1500s NL 25s AML 72h SO 6h geomean 10 (b) Performance of parallel algorithms for finding hop-constrained simple cycles on the Intel Xeon Skylake cluster using 480 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Performance of parallel algorithms for hop-constrained simple cycle enumeration on (a) the Intel KNL and (b) the Intel Xeon Skylake clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The numbers above the bars show the execution time of the coarse-grained parallel algorithms relative to that of our fine-grained parallel algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Larger hop constraints increase the performance gap between the two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='11 10 100 1000 1 4 16 64 256 1024 Fine-grained parallel hop-constrained Johnson Coarse-grained parallel BC-DFS [21] 368 x 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='9 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (c) AU 20h 315 x 71 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (b) TR 110h 338 x 68 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (e) FR 1500s 38… 23 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (d) WT 12h 329 x 40 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (a) CO 5h Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Scalability evaluation of parallel hop-constrained cycle enumeration algorithms executed on the Intel KNL cluster using the hop constraint of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The speedup values are relative to the single-threaded execution of BC-DFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Evaluation on other graphs is omitted due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 13 shows that our fine-grained parallel algorithm is, on average, more than 10× faster than the coarse- grained parallel BC-DFS algorithm for the two largest hop constraints tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When using the hop-constraint that is less than or equal to ten, the coarse-grained parallelisation approach is able to achieve workload balance across cores, and, thus, the performance of this approach is similar to that of our fine-grained approach in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As we increase the hop constraint, the probability of encountering deeper recursion trees also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Exploring such trees using the coarse-grained approach leads to workload imbalance (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our fine-grained algorithm is designed to resolve this problem by exploring a recursion tree using several threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Therefore, increasing the hop constraint increases the speedup of our fine-grained algorithm with respect to the coarse-grained algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When the hop constraint is set to 20, our fine-grained parallel algorithm is, on average, 10× faster than the coarse- grained parallel JOIN algorithm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Al- though the latter algorithm can be competitive with our fine- grained algorithm, it can also suffer from long execution times, such as in the cases of the AU, NL, and AML graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The reason for these long execution times is the fact that the JOIN algorithm might temporarily construct many non- simple cycles while searching for simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because this algorithm constructs cycles by combining simple paths, it is not guaranteed that each combination results in a simple cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The overhead of combining paths can dominate the execution time of JOIN if this algorithm constructs orders of magnitude more non-simple cycles than simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' For instance, this situation occurs in the case of AU and hop constraint of 20, where JOIN discovers 600× more non- simple cycles than simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, the speedup of our fine-grained algorithm compared to the coarse-grained JOIN algorithm can reach up to three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 14 shows that the speedup of our fine-grained par- allel Johnson algorithm with respect to the coarse-grained parallel BC-DFS can be increased by using more threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The performance of our fine-grained parallel algorithm scales linearly with the number of threads, whereas the scaling of the coarse-grained parallel BC-DFS eventually slows down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, in addition to being, on average, an order of magnitude faster than the coarse-grained parallel BC-DFS, our fine-grained algorithm is also more scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 Simple cycle enumeration Here, we evaluate our fine-grained parallel algorithms for simple cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The computational complexity of simple cycle enumeration is higher than the complexity of temporal and hop-constrained cycle enumeration because simple cycle enumeration does not impose temporal order- ing or hop constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The only constraint we impose is the time-window constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because the complexity of enumer- ating simple cycles is higher, we use smaller time windows compared to the cases of temporal and hop-constrained cy- cle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We use the coarse-grained parallel versions of the Johnson and the Read-Tarjan algorithms as our main comparison points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' We do not report the results for the MS graph because our algorithms did not finish in 12h even if we set the time window to one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 15, our fine-grained parallel al- gorithms show an order of magnitude average speedup compared to coarse-grained parallel algorithms on two different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The reason for this speedup is better scalability of our fine-grained algorithms, which we demon- strate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Similarly to the cases of temporal and hop-constrained cycle enumeration (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 12 and 14), our fine-grained parallel algorithms scale linearly with the number of physical cores used whereas the coarse-grained parallel Johnson algorithm does not scale as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thus, the speedup between the fine-grained and the coarse-grained algorithms increases by utilising more threads.' metadata={'source': 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geomean Fine-grained parallel Johnson Fine-grained parallel Read-Tarjan Coarse-grained parallel Johnson Coarse-grained parallel Read-Tarjan Execution time [s] (a) Performance of parallel algorithms for simple cycle enumeration on the Intel KNL cluster using 1024 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='3 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 >17 >16 11 11 21 28 21 15 16 31 33 11 14 >31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 >17 >16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 10 1 k 100 k BA 71h BO 75h CO 3h EM 4h MO 30h TR 72h HG 3000s AU 20h SU 5h WT 12h FR 1300s NL 29s AML 48h SO 3h geomean Execution time [s] (b) Performance of parallel algorithms for simple cycle enumeration on the Intel Xeon Skylake cluster using 480 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Performance of parallel algorithms for simple cycle enumeration on (a) the Intel KNL and (b) the Intel Xeon Skylake clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The numbers above the bars show the execution time of each algorithm relative to that of our fine-grained parallel Johnson algorithm for the same benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='11 10 100 1000 1 4 16 64 256 1024 Fine-grained parallel Johnson Fine-grained parallel Read-Tarjan Coarse-grained parallel Johnson 239 x 208 x 31 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (d) WT 12h 180 x 140 x 16 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (b) TR 72h 224 x 167 x 20 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (e) FR 1300s 154 x 133 x 5 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (c) AU 20h 271 x 273 x 9 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1 10 100 1000 1 4 16 64 256 1024 Speedup Number of threads (a) CO 3h Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Scalability evaluation of parallel simple cycle enumeration algorithms executed on the Intel KNL cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The speedup values are relative to the single-threaded execution of the Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Evaluation on other graphs is omitted due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='0 0 1 2 3 BA 71h BO 75h CO 3h EM 4h MO 30h TR 72h HG 3000s AU 20h SU 5h WT 12h FR 1300s NL 29s AML 48h SO 3h geomean Normalized exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time no optimisations path ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' and blk set forwarding blk set forwarding all optimisations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 0 2 4 6 8 BA 2600h BO 900h CO 72h EM 120h MO 240h TR 700h HG 60h AU 288h SU 144h WT 120h FR 4h NL 100s MS 3h AML 600h SO 48h geomean Normalized exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' time (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Effect of the pruning improvements to our fine-grained parallel Read-Tarjan algorithm for (a) simple and (b) temporal cycle enumer- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Execution times are normalised to the case that includes all optimisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our optimisations accelerate this algorithm by up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='8×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The synchronisation overheads caused by recursive un- blocking of our fine-grained parallel Johnson algorithm (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='2) are visible only in the case of AML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In this case, the fine-grained parallel Johnson algorithm performs 60% fewer edge visits than the fine-grained parallel Read-Tarjan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' however, it is 25% slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' These synchronisation overheads can be explained by a very low cycle-to-vertex ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Because a vertex is blocked if it cannot take part in a cycle, the probability of a vertex being blocked is higher when the cycle-to-vertex ratio is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In consequence, more vertices are unblocked during the recursive unblocking of the fine- grained parallel Johnson algorithm, leading to longer critical sections and more contention on the locks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Nevertheless, our fine-grained parallel Johnson algorithm achieves a good trade-off between pruning efficiency and lock contention in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Overall, our fine-grained parallel Johnson and fine- grained parallel Read-Tarjan algorithms have comparable performance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Although the former algorithm is slightly faster, it can suffer from synchronisa- tion overheads in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Nevertheless, both parallel algorithms achieve linear scaling with the number of phys- ical cores used and achieve, on average, more than 10× speedup with respect to coarse-grained parallel versions of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' These conclusions also hold in the cases of temporal and hop-constrained cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4 Improvements to the Read-Tarjan algorithm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 17 shows the effect of our pruning improvements, intro- duced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='1, on the performance of our fine-grained Read-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The experiments are performed us- ing a single Intel KNL processor using 256 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Note that using one processor instead of the entire cluster results in longer execution times, but it enables us to eliminate the effect of workload imbalance across processors in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' The execution time of the fine-grained parallel Read-Tarjan algorithm decreases after activating each opti- misation because fewer redundant vertex and edge visits are performed during the execution of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When all optimisations are enabled, the average speedup of our algorithm for simple cycle enumeration compared to its unoptimised version is 2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the case of temporal cycle enumeration, the average speedup increases to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='4×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' As a result, our pruning improvements enable the fine-grained parallel Read-Tarjan algorithm to be competitive with the fine-grained parallel Johnson algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 9 CONCLUSIONS This work has made three contributions to the area of par- allel cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' First, we have introduced scalable fine-grained parallel versions of the state-of-the-art Johnson and Read-Tarjan algorithms for enumerating simple cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, we have shown that the novel fine-grained parallel approach we contributed for parallelising the John- son algorithm can be adapted to support the enumeration 17 of temporal and hop-constrained cycles as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Our fine- grained parallel algorithms for enumerating the aforemen- tioned types of cycles achieve a near-linear performance scaling on a compute cluster with a total number of 256 CPU cores that can execute 1024 simultaneous software threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Secondly, we have shown that our fine-grained parallel cycle enumeration algorithms are scalable both in theory and in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In contrast, their coarse-grained parallel ver- sions do not share this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' When using 1024 software threads, our fine-grained parallel algorithms are on aver- age an order of magnitude faster than their coarse-grained counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In addition, the performance gap between the fine-grained and coarse-grained parallel algorithms widens as we use more physical CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' This performance gap also widens when increasing the time window in the case of temporal cycle enumeration and when increasing the hop constraint in the case of hop-constrained cycle enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Thirdly, we have shown that, whereas our fine-grained parallel Read-Tarjan algorithm is work efficient, our fine- grained parallel Johnson algorithm is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In general, the former is competitive against the latter because of the new pruning methods we introduced, yet the latter outperforms the former in most experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In some rare cases, our fine- grained parallel Johnson algorithm can suffer from synchro- nisation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In such cases, our fine-grained parallel Read-Tarjan algorithm offers a more scalable alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' ACKNOWLEDGMENTS The support of Swiss National Science Foundation (project number 172610) for this work is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Needham, Graph algorithms : practical examples in Apache Spark and Neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Beijing: O’Reilly, 2019, isbn: 978-1492047681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' [2] N.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='edu/ data, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 2014, accessed: 2022-05-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Jovan Blanuˇsa received the BSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' degree in electrical engineering and computer science from the University of Belgrade, Belgrade, Serbia and the MSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' degree in electrical and electronic engineering from EPFL, Lausanne, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He is currently working towards the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' de- gree in computer science at EPFL, Lausanne, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Blanuˇsa is also working as a predoctoral researcher at IBM Research Europe, Zurich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' His research interests in- clude acceleration of graph mining algorithms and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He is a Student Member of the ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Kubilay Atasu received his BSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' de- grees in computer engineering from Bo˘gazic¸i University, Istanbul, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He also holds an MEng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' degree in embedded system design from University of Lugano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Atasu is currently a research staff member at IBM Research Europe, Zurich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He was the recipient of a best paper award at DAC in 2003 and at ASAP in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He was also a best-paper award nominee at FPL in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He served as a program co-chair of the ASAP 2013 conference and as a general chair of the ASAP 2014 conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Atasu is currently serving in the program committee of the DAC, FCCM and ASAP conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' In the past, he also served in the program committees of the DATE, ICPP, CF, FPL and FPT conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He is a Senior Member of the IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' 19 Paolo Ienne received the laurea degree in elec- trical engineering from Politecnico di Milano, Mi- lan, Italy, in 1991, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' degree in com- puter science from EPFL, Lausanne, Switzer- land, in 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Since 2000, he has been a Pro- fessor with the School of Computer and Commu- nication Sciences, EPFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' His research interests include computer and processor architecture, FPGAs and reconfigurable computing, electronic design automation, and computer arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Some of his articles have received the Best Pa- per Awards at prestigious venues (including at the FPGA, FPL, CASES, and DAC conferences), and several others have been nominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' Ienne has served as general, program, and topic chair of renown inter- national conferences, serves on the steering committee of the ARITH, FPL, and FPGA conferences, and is regularly a member of several program committees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He was an associate editor of the ACM TODAES and is an associate editor of ACM CSUR and ACM TACO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} +page_content=' He is a Senior Member of the IEEE and a Member of the ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf'} diff --git a/RtFRT4oBgHgl3EQfLTfA/content/tmp_files/2301.13502v1.pdf.txt b/RtFRT4oBgHgl3EQfLTfA/content/tmp_files/2301.13502v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33339ef8df22959af3c1e9a68f1351df19642661 --- /dev/null +++ b/RtFRT4oBgHgl3EQfLTfA/content/tmp_files/2301.13502v1.pdf.txt @@ -0,0 +1,1654 @@ +Parity-violation in bouncing cosmology +Mian Zhu1,2∗ and Yong Cai1† +1 School of Physics and Microelectronics, +Zhengzhou University, Zhengzhou, Henan 450001, China and +2 Faculty of Physics, Astronomy and Applied Computer Science, +Jagiellonian University, 30-348 Krakow, Poland +Abstract +We investigate the possibility of the enhancement of parity-violation signal in bouncing cosmol- +ogy. Specifically, we are interested in deciding which phase should generate the most significant +parity-violation signals. We find that the dominant contribution comes from the bouncing phase, +while the contraction phase has a smaller contribution. Therefore, bouncing cosmology can en- +hance the parity-violation signals during the bouncing phase. Moreover, since the bouncing phase +has the highest energy scale in bouncing cosmology, we can also probe new physics at this scale by +studying the parity-violation effect. +PACS numbers: +∗ Corresponding author: mzhuan@connect.ust.hk +† Corresponding author: yongcai phy@outlook.com +1 +arXiv:2301.13502v1 [gr-qc] 31 Jan 2023 + +Contents +I. Introduction +2 +II. Model +5 +A. Action +5 +B. Bouncing background +5 +C. The effective action on high energy scale +8 +III. Tensor perturbation +9 +A. Formalism +9 +B. Dynamics of tensor perturbation +10 +C. Parity violation signal +12 +D. Comment on the resulting signal +13 +E. Semi-analytic investigation +15 +IV. Conclusion and Outlook +18 +V. Acknowledgement +19 +References +19 +I. +INTRODUCTION +The primordial gravitational waves (GWs) might encode rich information about the very +early universe, which may help distinguish between different scenarios of the primordial +universe, including inflation and its alternatives. Chirality is a distinct characteristic of +GWs, which could be manifested in parity-violating theories of gravity. Recently, it is found +that the polarization data of Planck and WMAP [1–3] may be a hint of parity-violating +physics in the cosmic microwave background, though further confirmations are required. +The explorations of parity-violating primordial GWs have aroused a lot of interest, see e.g. +[4–36], see also [37–43]. +In single-field slow-roll inflation models where inflaton is non-minimally coupled to a +parity-violating term, such as the gravitational Chern-Simons (gCS) term [44, 45], the effect +2 + +of parity-violation should be suppressed by the slow-roll condition. However, in the modifi- +cations or alternatives to the single field slow-roll inflation, the slow-roll condition could be +violated at least for some moment. As a result, the effect of parity-violation could be en- +hanced due to the dynamical coupling of the scalar field to the parity-violating term, see e.g. +[46] for the enhanced parity-violating GWs caused by violation of the null energy condition +(NEC) [47] during inflation. Therefore, observations of a parity-violating GW background +might provide us with a new way to identify physics beyond single-field slow-roll inflation. +Bouncing cosmology as a possible solution to the initial cosmological singularity problem +of inflation and the Big Bang cosmology has attracted a lot of interest [48–74]. +In the +bouncing scenario, the universe originates from a contracting phase and enters an expanding +phase after going through a non-singular bounce, where the NEC is violated. One important +issue is the ghost and gradient instabilities in the bouncing phase, which is a generic feature +in a large class of theories [75–77]. +To acquire a healthy bouncing model, new physics +effective at bouncing phase is introduced [78–95]. +In principle we may explore the new physics by studying their phenomenological predic- +tions. Unfortunately, the signals from the new physics, which is generically effective only in +the bouncing phase, are suppressed by the short duration of the bounce. For example, in +[90], it is found that the new physics at the bouncing phase has negligible contribution to +the power spectrum. Consequently, in many studies of non-singular cosmology [96–103], the +signals from the bouncing phase are small, the main phenomenological contribution comes +from the contraction phase, so it’s difficult to probe the new physics in bouncing phase 1. +Specifically, in previous literature addressing the parity-violation effect in bouncing cosmol- +ogy [10], the bouncing phase is directly replaced by a simple junction condition so there is +only a contraction and an expansion in their scenario. +It is then interesting to study if the parity-violation effect could be generated in bouncing +cosmology, especially during the bouncing phase. Intuitively, the derivative of the scalar +field is non-trivial around the bouncing phase, which may be able to amplify the effect of +parity-violation, as long as the scalar field is non-minimally coupled to a parity-violating +term. +Additionally, the effective sound horizon of the primordial GW mode could also +1 Some counterexamples comes from the quantum bounce models [104–106]. However, this is beyond our +scope since we consider purely classical bouncing cosmology. +3 + +be nontrivial during the bouncing phase2, especially when chirality of the GW mode is +considered. Therefore, we expect the non-trivial parity-violation signals to come from the +bouncing phase. +In this paper, we investigate the parity-violation effect in a toy bouncing model, where +the source term is taken to be a gCS action coupled to the scalar field. We are especially +interested in the following question: which phase, the contraction or the bouncing phase, +contributes to the enhancement of the parity-violation effect dominantly? As we will see +in section III C, the bouncing phase can generate non-trivial parity-violation signals, while +the contraction phase has negligible effect. Moreover, the enhancement is sensitive to the +detailed physics during the bouncing phase, so in principle, we can probe the new physics +during bouncing through parity-violation. Therefore, our result is twofold: we can not only +explain the possibly observed parity-violation signal in the framework of bouncing cosmology, +but also provide a possible way to probe new physics at the bouncing phase by studying +their imprint on parity-violation signals. +The paper is organized as follows. In section II we briefly introduce our model. After +the basic formalism for tensor perturbation in III A, we numerically evaluate the dynamics +of tensor perturbation in section III B and the parity-violation signal in section III C. We +comment on some conceptual issues about our result in section III D and explain our numer- +ical result in a semi-analytical way in section III E. From the semi-analytical argument, we +find that our numerical result should be qualitatively valid for a large variety of bouncing +models, although the numerics are taken in a toy bouncing model. We finally conclude in +section IV. +Throughout this paper, we take the sign of the metric to be (−, +, +, +). We will take +ℏ = 1, c = 1, M 2 +p = (8πG)−1 = 1, so that all quantities are in Planck units. The canonical +kinetic term is defined as X ≡ −∇µφ∇µφ/2, such that X = ˙φ2/2 at the background level. +2 The bouncing phase is defined by dH/dt ≥ 0, where H is the Hubble parameter. +4 + +II. +MODEL +A. +Action +We take the action to be +S = +� +d4x√−g +�M 2 +p +2 R + LH + LG + LHE +� +. +(1) +The term LH is responsible for setting the background evolution, where we set +LH = M 2 +pf1(φ)X + f2(φ)X2 − M 4 +pV (φ) , +(2) +which is eligible for the background dynamics. In the next section II B, we will use specific +coupling functions f1 and f2 to construct a cosmological bouncing model. +The LG term is the gravitational CS term, with +LG = f3(φ) +8 +R ∧ R = f3(φ) +8 +ϵαβρσRαβµνR +µν +ρσ , +(3) +and ϵαβρσ to be four-dimensional Levi-Civita symbol with ϵ0123 = −1/√−g. +Finally, the term LHE represents the action effective at some high energy scale. Since +there will be ghost or gradient instability problems in the generic bouncing models [75, 76, +99], such terms are obligated to eliminate such instabilities. We will discuss in details of this +term in section II C. +We mention that in (1) we scale the scalar field φ to be dimensionless so that the coupling +functions fi are dimensionless. +B. +Bouncing background +It is well-known that the gCS term will not contribute to the background dynamics. We +shall assume that the correction term LHE also satisfies this criterion. Therefore, Fried- +mann’s equations are totally determined by the Einstein-Hilbert action and the LH term. +In a flat FLRW background +ds2 = −dt2 + a2(t)d⃗x2 , +(4) +we have +3M 2 +pH2 = M 2 +p +2 f1 ˙φ2 + 3 +4f2 ˙φ4 + M 4 +pV (φ) , +(5) +5 + +− 2M 2 +p ˙H = M 2 +pf1 ˙φ2 + f2 ˙φ4 , +(6) +or in terms of the scalar field φ: +� +M 2 +pf1 + 3β ˙φ2� +¨φ + 3H ˙φ +� +M 2 +pf1 + β ˙φ2� ++ M 4 +p +dV +dφ + M 2 +p +2 +df1 +dφ +˙φ2 = 0 . +(7) +Now we choose a similar ansatz as that from [85, 90]: +f1(φ) = 1 − +g +cosh ω1φ , f2 = β ≡ const , V (φ) = − +V0 +cosh ωV φ , +(8) +where the background dynamics are well-studied. In the initial state of the universe where +˙φ → 0 and φ → −∞, the universe undergoes an Ekpyrotic contraction [107] +φ ≃ − 1 +ωV +ln ω4 +V V0t2 +ω2 +V − 6 , a(t) = a− +� t − tc +t− − tc +� +2 +ω2 +V +. +(9) +The Ekpyrotic phase makes us free from conceptual issues of bouncing cosmology [108], at +the cost of requiring ω2 +V > 6. Note that we set t = 0 to be the bouncing point, i.e. the stage +where the scale factor is minimal, so need an integration constant tc to correctly describe a. +We also use the minus sign to denote the end of the Ekpyrotic phase, e.g. a− is the scale +factor at the end of the Ekpyrotic contraction. +When |φ| → 1, the hyperbolic function approaches 1, and if we take g > 1, the f1X +term inverses sign and NEC can be violated. The non-singular bounce phase starts when +the NEC is violated, and the universe transit from contraction to expansion. The dynamics +during the bouncing phase are generically complicated, but for a short bounce, i.e. bouncing +phase with short enough time, the following parameterization can be valid +H = γM 2 +pt , γ = const. > 0 → a = a0e +1 +2 γM2 +pt2 , +(10) +where we have set a0 = a(0), which is the scale factor at the bouncing point. +After the bouncing phase, the universe comes to an expansion phase, where the scale +factor behaves as +a(t) = a+ +� t − te +t+ − te +� 1 +3 +, H(t) = +1 +3(t − te) , +(11) +where we similarly use the “+” sign to denote the end of the bouncing phase, and te is +another integration constant. +We shall comment more on the expansion phase. Notice that, the factor aH from (11) is +proportional to (t − te)− 2 +3. Hence, for any wave mode that is initially sub-horizon at t = t+, +6 + +-10000 +-5000 +0 +5000 +10000 +-0.0002 +-0.0001 +0.0000 +0.0001 +0.0002 +0.0003 +t +H +-1 +-0.5 +0 +0.5 +1 +0 +0.001 +0.002 +0.003 +t +H +-10000 +-5000 +0 +5000 +10000 +0 +2. × 10-8 +4. × 10-8 +6. × 10-8 +8. × 10-8 +1. × 10-7 +t +ρ +-1-0.500.51 +0 +0.00001 +0.00002 +0.00003 +t +ρ +-10000 +-5000 +0 +5000 +10000 +-2 +-1 +0 +1 +2 +3 +4 +t +ϕ +-10000 +-5000 +0 +5000 +10000 +0.0001 +0.0002 +0.0003 +0.0004 +0.0005 +0.0006 +0.0007 +t +ϕ +-1-0.5 0 0.5 1 +0.0 +0.1 +0.2 +0.3 +0.4 +t +ϕ +FIG. 1: The background dynamics with the specific parameters (13). The upper channel shows the +evolution of the Hubble parameter and background energy density, while the lower channel shows +the dynamics of the scalar field φ. The bouncing phase happens at around t = 0 where H quickly +transfers from negative to positive. +it will remain sub-horizon in the whole expansion phase. This is in contrast with our general +belief that, the primordial perturbation should leave the horizon in the expansion phase (like +inflation) and freeze in, and re-enter the horizon in a later stage to set the initial condition +for structure formation. +However, the parity-violation signal is highly dependent on the subsequent expansion +phase after the bounce. In this paper, we want to compare the induction of parity-violation +between the contraction phase and the bouncing phase, so we wish to get a result independent +of the subsequent expansion phase. Unfortunately, we do not have a precise way to define +when the bouncing phase ends, so it is hard to directly get the parity-violation status at +the end of the bouncing phase. The advantage of our expansion phase (11) is that the +wave mode of interests will always be in the sub-horizon region. Thus, their dynamics can +be approximately described by the harmonic oscillator equation u′′ +k + c2 +Tk2uk = 0, whose +general solution is simply +uk ≃ uk,+eikτ + uk,−e−ikτ . +(12) +7 + +The information of parity-violation status when bounce ends is encoded in the function +uk,±, and we see that the expansion phase only changes their relative phase. +Thus, we +may alternatively get the physics of parity-violation at the end of the bouncing phase, by +tracing the statistical property of tensor perturbation during the expansion phase. We shall +elaborate more about this point in section III B. +We depict the background dynamics in figure 1, where we’ve adopted the following pa- +rameters +g = 1.5 , β = 2 , V0 = 10−7 , ω1 = 10 , ωV = +√ +10 . +(13) +C. +The effective action on high energy scale +As shown in figure 1, the background energy density at the bouncing phase is much higher +than the other phases. Hence, it is natural to introduce some actions effective only at a high +energy scale to eliminate the instability problem. In the context of effective field theory +(EFT) of non-singular cosmology [78, 79], certain EFT operators such as R(3)δg00 can help +to evade the instabilities without altering the background dynamics. +However, when come to the realization of such EFT operators, the dynamics of tensor +perturbation are generally influenced by such high-energy correction. For example, in [81, +85–87] where the EFT operator R(3)δg00 is written in a covariant form, there appears a +non-minimal coupling and the propagating speed of GWs is changed accordingly [109]. +There are also other approaches for LHE to eliminate the instabilities [82, 93, 95, 110–113], +while generically changes either the background dynamics or the propagation of gravitational +waves. It would be hard to combine all these approaches in a unified description. +In this paper, we will start with the simplest case where LHE has no influence on both +the background dynamic and propagation of gravitational waves (this is the case of the EFT +approach in [78]). The point is, we will use this case as fiducial to examine if the bouncing +phase contributes more to the parity-violation than the contraction phase. If this is true, we +would possibly have the opportunity to distinguish the above approaches through the GW +signals. +8 + +III. +TENSOR PERTURBATION +A. +Formalism +Now we come to the tensor mode. We’ve assumed that LHE doesn’t contribute to the +tensor mode, so the quadratic action for tensor perturbation is +S(2) +T += M 2 +p +8 +� +dτd3x +� +a2 � +γ′2 +ij − (∂γij)2� +− g′ +3 +M 2 +p +ϵijk +� +(∂iγjl)′(γl +k)′ − ∂i∂lγjq∂lγq +k +�� +, +(14) +where we’ve defined the conformal time τ ≡ +� +dt/a and a prime denotes differentiation with +respect to τ. Before proceeding, we see that the gCS term is suppressed by the factor M 2 +p, +so this term should be important at a high energy scale. Moreover, g′ +3 = a ˙φg3,φ, and from +figure 1 that ˙φ is non-trivial only during the bouncing phase. Thus, we can intuitively guess +that the gCS term should be important during the bouncing phase. +We work in the Fourier space where +γij(τ, ⃗x) = +� +s=L,R +� +d3k +(2π3)γ(s) +k (τ)p(s) +ij (⃗k)ei⃗k·⃗x , +(15) +with the polarization tensor satisfying +p(R) +ij pij(R) = p(L) +ij pij(L) = 0 , p(R) +ij pij(L) = 2 , iklϵqljp(s) +ij = kλ(s)pq(s) +i +. +(16) +The polarization mode is decided by the parameter λ, such that +λ(L) = −1 , λ(R) = 1 , λ(N) = 0 , +(17) +and here for convenience, we’ve defined a new N mode to represent the non-parity-violation +case. +Finally, the parity-violation is evaluated by the chiral parameter +∆χ ≡ P (L) +T +− P (R) +T +P (L) +T ++ P (R) +T +, +(18) +where P (s) +T +are the power spectrum of the corresponding polarization modes. +Although +the difference P (L) +T +− P (R) +T +is of observational interest, the absolute value of P (s) +T +is highly +dependent on the detailed bouncing models (for example, the tensor spectra index in our +model (8) is dependent on the model parameter ωV [90]). Thus for our purpose to compare +the parity-violation effect from a different phase, we shall concern with the parameter ∆χ. +9 + +L +R +-10000 +0 +10000 +20000 +30000 +40000 +-3. × 10-8 +-2. × 10-8 +-1. × 10-8 +0 +1. × 10-8 +2. × 10-8 +t +1 +a0 +2 +zT +'' +zT +L +N +R +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +t +1 +a0 +2 +zT +'' +zT +N +-0.5 0 0.5 1 +0. +0.02 +0.04 +1 +a02 +zT +'' +zt +FIG. 2: The function z(s)′′ +T +/z(s) +T +as a function of time for different polarization modes, rescaled by +a factor a−2 +0 . The left figure shows the time evolution of the whole cosmological history, and the +right figure shows the dynamics near the bouncing point. +B. +Dynamics of tensor perturbation +The dynamical equation for the tensor mode γ(s) +k +is +u(s)′′ +k ++ +� +k2 − z(s)′′ +T +z(s) +T +� +u(s) +k += 0 , +(19) +where we define the Mukhanov-Sasaki variable +u(s) +k +≡ z(s) +T γ(s) +k +, z(s) +T +≡ a +2 +� +1 − λ(s)k +a +g3,φφ′ +aM 2 +p +, +(20) +and the sound speed is set to be unity for all polarization modes. Notice that we require +the terms in the square root to be non-negative, otherwise, there will be ghost modes [114]. +Initially, all the perturbation modes of observational interests are on sub-horizon scales, +where the k2 terms in (19) dominates. Thus, we can take the vacuum initial condition +u(s) +k += e−ikτ +√ +2k +, τ → −∞ . +(21) +We can combine the equations (19) and (21) to get the dynamics of γij. +Firstly, we evaluate the term z(s)′′ +T +/z(s) +T +numerically, with a specific gCS coupling f3(φ) = +φ. +Moreover, we notice that the result depends only on the physical wavenumber k/a0 +instead of k, as long as we rescale the term z(s)′′ +T +/z(s) +T +by a factor a−2 +0 . At this point, we set +a specific scale k/a0 = 10−2, the averaged magnitude of maximum value of ˙φ and H. +We depict the term z(s)′′ +T +/z(s) +T +as a function of cosmic time in figure 2, with a rescale factor +a−2 +0 . Outside the bouncing phase, the L and R modes are almost identical; while during the +10 + +R +L +0 +2000 +4000 +6000 +8000 10000 +0.999999 +0.999999 +1.000000 +1.000000 +1.000000 +t +uk(s) +uk(N) +k/a0=10-4 +R +L +0 +2000 +4000 +6000 +8000 10000 +0.999985 +0.999990 +0.999995 +1.000000 +1.000010 +1.000010 +1.000010 +t +uk(s) +uk(N) +k/a0=10-3 +R +L +0 +2000 +4000 +6000 +8000 +10000 +0.99990 +0.99995 +1.00000 +1.00005 +1.00010 +t +uk(s) +uk(N) +k/a0=10-2 +FIG. 3: The dynamics of |u(s) +k |/|u(N) +k +| as a function of cosmic time. We specify the dynamics with +three characterised scale, k/a0 = 10−4, 10−3 and 10−2, respectively. +bouncing phase, the two polarization modes differ significantly, and the amplitude of L/R +mode is one order beyond the unpolarized mod N. +Now we come to the mode function u(s) +k . As we explained at the ending part of section +II B, the modes initially on the sub-horizon scale at t = t+ will stay in the sub-horizon region +during the expansion phase. Their evolution can then be approximated as +u(s) +k +≃ u(s) +k,+eikτ + u(s) +k,−e−ikτ , +(22) +so the amplitude of the mode function will oscillate during this phase. +We depict the dynamics of |u(s) +k |/|u(N) +k +| for different k/a0 value in figure 3. As we can +see, for large scale such as k/a0 = 10−4, the mode quickly becomes super-horizon during the +bouncing phase, and |u(s) +k |/|u(N) +k +| approaches constant. For intemediate scale like k = a0 = +10−3, the mode is sub-horizon but z(s)′′ +T +/z(s) +T +is still comparable to k2/a2 +0, so the dynamics is +oscillatory but not strictly identical. For small scale like k/a0 = 10−2, the oscillatory feature +is strong. +Now we conclude that, for sufficiently large wave mode, physical quantities such as the +mode function (and hence the tensor perturbation γ and parameter ∆χ) at the end of the +bouncing phase, can be represented by their statistics property at the expansion phase, since +the expansion phase only add an oscillating feature to them. +One additional advantage of our treatment is that the horizon-cross condition should +in principle determined by the behavior of z(s)′′ +T +/z(s) +T . While in the bouncing phase, this +term is highly non-trivial, it simplifies to a′′/a in the expansion phase and we have a simple +expression. +11 + +-10000 +-5000 +0 +5000 +10000 +0 +1. × 10-6 +2. × 10-6 +3. × 10-6 +4. × 10-6 +5. × 10-6 +6. × 10-6 +7. × 10-6 +t +zT +(R) - zT +(L) +zT +(N) +-0.5 +0 +0.5 +0 +0.003 +0.006 +-10000 +-5000 +0 +5000 +10000 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t +zT +(L) +-10000 +-5000 +0 +5000 +10000 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t +zT +(R) +FIG. 4: The dynamics of z(s) +T +in the whole cosmological history. The scale is chosen to be k/a0 = +10−2. We see from the left figure that z(s) +T +are almost identical, and for convenience, we also plot +the z(s) +T +for L and R mode respectively. +0 +2000 +4000 +6000 +8000 +10000 +-0.00002 +-0.00001 +0.00000 +0.00001 +0.00002 +t +Δχ +k/a0=2*10-3 +0 +200 +400 +600 +800 +1000 +-0.00005 +0.00000 +0.00005 +0.00010 +0.00015 +0.00020 +0.00025 +t +Δχ +k/a0=1*10-2 +0 +50 +100 +150 +200 +-0.005 +0.000 +0.005 +0.010 +0.015 +t +Δχ +k/a0=1*10-1 +FIG. 5: The parity-violation status as a function of t. The oscillatory feature is expected due to +the behavior of |u(s) +k |. Note that we adopted a smaller range of t for large k/a0, otherwise the +whole picture would be totally filled. +C. +Parity violation signal +With the dynamics of the mode function, we can evaluate the corresponding tensor +power spectrum. Notice that the tensor spectrum depends also on z(s) +T , which carries the +information on different polarization, so we should first evaluate γ(s) +k +≡ u(s) +k /z(s) +T . +However, in our case, the function z(s) +T +differs only slightly. As shown in figure 4, z(s) +T +for +L and R mode would have a maximum difference of order 10−2. Hence, we may simply take +P (L) +T +P (R) +T += |γ(L) +k |2 +|γ(R) +k +|2 = |u(L) +k |2 +|u(R) +k |2 +|z(R) +T |2 +|z(L) +T |2 ≃ |u(L) +k |2 +|u(R) +k |2 → ∆χ ≃ |u(L) +k |2 − |u(R) +k |2 +|u(L) +k |2 + |u(R) +k |2 , +(23) +with a loss of precision no more than O(10−2). +Now we can work out ∆χ. Since |u(s) +k | is oscillating, we expect ∆χ to be also oscillating, +as shown in figure 5. As stated in the last part in section III B, we will represent the parity- +violation state at the end of the bouncing phase, by the statistic property of uk (and hence +∆χ) during the expansion phase. Our strategy is, for each fixed k/a0, we take the value of +12 + +0.005 +0.010 +0.050 +0.100 +10-4 +0.001 +0.010 +0.100 +k/a0 +Δχ +FIG. 6: The parameter ∆χ as a function of physical wavenumber k/a0. Notice that for smaller +k/a0, the behavior of u(s) +k +would differ more from (22), so ∆χ would also receive more influence +from the expansion phase. Thus we shall treat the data from smaller k/a0 with less confidentiality. +∆χ’s amplitude A∆χ with a factor 1/ +√ +2, i.e. A∆χ/ +√ +2, to represent the corresponding ∆χ +at the end of the bouncing phase, ∆χb. Then, we can depict the dependence of ∆χb on the +physical wavenumber k/a0, in figure 6. +We see from figure 6 that, the parity-violation can be induced at the bouncing phase, +and for large k/a0, there are chances that the parameter ∆χ be large enough (i.e. of order +10−2) to generate detectable parity-violation signals. +D. +Comment on the resulting signal +Before proceeding, we shall comment on the result from section III C, and clarify some +potentially confusing points. +Firstly, we stress again that the signal obtained in the last section is in fact the repre- +sentation of the signal at the end of the bouncing phase. In order to confront the result of +observations, we need to design a more realistic expansion phase. Then, it is possible that +a large parity-violation signal at t = t+ is suppressed by the subsequent expansion phase. +Thus at the current stage, what we can conclude is that parity-violation feature can be pro- +duced at the bouncing phase where the energy scale is the highest in bouncing cosmology, +and it could potentially be detectable. +Besides, we see in figure 6 that ∆χ is proportional to k/a0, which seems to be in contrast +with the result from [46], where parity-violation signals are also generated by some NEC- +13 + +violation phase, but ∆χ is non-trivial only in selected wavelengths (see also [10, 11]), while +our result seems to be valid for a wide range of wavelengths. This is because the scenario +considered in [46] is in an inflation background. The NEC violation happens between two +inflation phases, and thus the NEC violation phase is in correspondence to a specific range +of wavenumber a−H− < k < a+H+, where the ± sign stands for the beginning and end +of the NEC violation phase, so k± = a±H± stands for the wave mode that exactly crosses +the Hubble horizon at t = t±. However, in our case, the bouncing point is characterized by +H(0) = 0, where all modes are inside the “Hubble horizon” 1/H → ∞. Thus, we expect all +modes “feel” the parity-violation physics during the bouncing phase. +Actually, the result displayed in figure 6 is consistent with that obtained in the bounce- +inflation scenario [10], where the effect of parity-violation measured by ∆χ is proportional +to k/H∗ for the GW modes which exit horizon during the contracting and bouncing phases +(i.e., before the inflationary phase), though the bouncing phase is assumed to be negligibly +short in [10]. +Finally, one may naturally ask, if ∆χ is proportional to k/a0 as that in figure 6, then +shouldn’t ∆χ be of higher order like O(1), and resulting in an unreasonably large parity- +violation signal? The problem is, we have to cut off at some k for at least two reasons. +Firstly, to avoid the appearance of ghosts, we require z(s) +T +to be real, so +����� +k +a +˙φ +M 2 +p +����� < 1 → +k +a0 +< max +� ˙φ +M 2 +p +� +, +(24) +and we have to cut off smaller scales. Besides, the effective description of our universe as a +homogeneous and isotropic ideal fluid breaks down for a sufficiently small scale, i.e. large +enough k. This means that the value of a0 cannot be arbitrary. Instead, it should have a +proper value such that the parity-violation happens at the correct scale, and the value of +k/a0 always satisfies the condition (24) for reasonable k. +We shall further mention that, in our toy model, the wave mode displayed in figure 6 is +in the sub-horizon region. However, in a realistic model, the tensor mode will experience +a decaying when evolving toward the horizon during the expansion phase. Smaller scales +would exit the horizon at a later time, and they would experience more decaying. Thus, +although ∆χ is approximately proportional to k/a0 at the end of the bouncing phase, it is +possible that smaller scales receive more suppression in the following expansion phase, and +the parity-violation effect is important only in some intermediate scales. +14 + +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-10 +-5 +0 +5 +10 +t +ϕ +... +L +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.04 +-0.02 +0.00 +0.02 +0.04 +t +zT +'' +zT +FIG. 7: We compare the dynamics of +... +φ and z(L)′′ +T +/z(L) +T +during the bouncing phase and find the +same feature from both of them. +E. +Semi-analytic investigation +Although we numerically verified the existence of parity-violation signals from the bounc- +ing phase, we wish to briefly explain the result analytically. Fortunately, the duration of the +bouncing phase is small from figure 1, so we may adopt the parametrization (10). Moreover, +in cosmic time, we have +z(s)′′ +T +z(s) +T += a2 +� +¨z(s) +T +z(s) +T ++ H ˙z(s) +T +z(s) +T +� +, +z(s) +T += a +2 +� +1 − λ(s)k +a +˙φ +M 2 +p +≃ a +2 − λ(s)k +4 +˙φ +M 2 +p +, +(25) +and we may write the expression in the following +z(s)′′ +T +z(s) +T +≃ a2 +� +¨a + H ˙a +2z(s) +T +− λ(s)k +4 +¨φH + +... +φ +M 2 +pz(s) +T +� +. +(26) +The term ¨a + H ˙a is suppressed by a factor t2, while the H ¨φ term suppressed by a factor t, +we concern on the term +... +φ. Now ˙φ is a δ-like function, so we expect ¨φ to have a positive +peak at t < 0 and a negative peak at t > 0. Subsequently, +... +φ should first have a positive +peak at t < 0, then a negative peak at t > 0, and finally followed by a second positive peak. +We illustrate this point by depicting both ¨φ and z(L)′′ +T +/z(L) +T +in figure 7 and see that they have +exactly the same feature. +We conclude that, the feature of z(s)′′ +T +/z(s) +T +comes from that of +... +φ, which is further decided +by the δ-function-like behavior of ˙φ. The mode function receives a non-trivial enhancement, +To intuitively understand how the peaks of z(s)′′ +T +/z(s) +T +affect the tensor mode, we may +approximately take each peak as a δ-like function. For simplicity, we take the realization of +15 + +Ai(x) +Bi(x) +-6 +-4 +-2 +0 +2 +-0.5 +0.0 +0.5 +1.0 +1.5 +FIG. 8: Airy function +these peaks to be a linear function +� +z(s)′′ +T +/z(s) +T +� +peak ≃ b|t − tc| , tp− < tc < tp+ , b > 0 , t ∈ (tp−, tp+) , +(27) +so for each region, the dynamical equation for the mode function becomes (for convenience +let’s temporarily take tc = 0, and also t ≃ τ during the bouncing phase since a is almost a +constant) +u(s)′′ +k ++ +� +k2 ± bt +� +u(s) +k += 0 , +(28) +whose general solution is the Airy function +u(s) +k += c1Ai +� +−k2 ∓ bt +|b| +2 +3 +� ++ c2Bi +� +−k2 ∓ bt +|b| +2 +3 +� +. +(29) +In figure 8 we depict the behavior of the Airy function. When the argument is negative, +both Airy functions oscillate. When the argument is positive, one branch increases while +the other shrinks. Thus, the amplitude of u(s) +k +will be enhanced in the positive-argument +region. +Note that the parametrization in (27) is rough, so at the current stage, we cannot go +further without the detailed expression of the peaks. Thus we can only conclude that the +peaks in z(s)′′ +T +/z(s) +T +enhance the amplitude of tensor perturbation, and different polarization +modes receive different enhancement due to the microscopic physics in the bouncing phase, +which causes the parity-violation. +Nextly, we shall intuitively explain why ∆χ has a linear dependence on k/a0. For this +purpose, we plot in figure 9. For large k/a0, the three peak value of z(s)′′ +T +/z(s) +T is approximately +linearly dependent on k/a0, so we expect the enhancement of u(s) +k +also depends on k/a0 +linearly. For smaller k/a0 when the peak value of L and R modes are comparable to that of +16 + +L +N +R +-0.4 -0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.06 +-0.04 +-0.02 +0.00 +0.02 +0.04 +0.06 +t +1 +a0 +2 +zT +'' +zT +k/a0=10-3 +L +N +R +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +t +1 +a0 +2 +zT +'' +zT +k/a0=10-2 +L +N +R +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-6 +-4 +-2 +0 +2 +4 +6 +t +1 +a0 +2 +zT +'' +zT +k/a0=10-1 +FIG. 9: The evolution of z(s)′′ +T +/z(s) +T +for different physical wavenumber k/a0. +N mode, the second peak destroys the linear relationship, so we expect the linear dependence +of ∆χ on k/a0 is ruined. Thus, the fitted function ∆χ in figure 6 is a little convex instead +of perfectly straight. +Finally, we emphasize how generic our result should be. +The suppression of parity- +violation signals during the contraction phase comes from the smallness of ˙φ. Although the +dynamics of ˙φ relies on the details of the contraction phase, for mainstream bouncing models +like matter bounce (contraction phase dominated by a stiff matter and small ˙φ like that in +[59, 115]) and Ekpyrotic bounce (the case described by our model where ˙φ = −2/ωV t), | ˙φ| +is always small for large |t|. Alternatively, a large ˙φ would correspond to a higher energy +scale, so the parity-violation effect is suppressed in the contraction phase because of the +low energy scale. Notice that the contraction phase will always have a lower energy scale +compared to the bouncing phase as long as we consider a classical bounce model where the +contraction phase happens with an initially classic configuration. Hence, the smallness of the +parity-violation signal in the contraction phase should be valid at least for many bouncing +models. +We may understand the smallness of ˙φ in the contraction phase by alternative arguments. +One common mechanism for NEC violation is ghost condensation [116], where the kinetic +Lagrangian L(X) has a non-trivial stationary point at X ̸= 0 with a negative vacuum +expectation (VEV). The contraction phase corresponds to the configuration of the false +vacuum X = 0, while the bouncing phase corresponds to the true vacuum. Thus, a small +˙φ is expected in this mechanism. Moreover, if the bouncing phase has a short duration, we +also expect ˙φ to have a sharp peak, whose magnitude is related to the VEV of the kinetic +Lagrangian. +In view of the above argument, we see that a short duration of the bouncing phase can +17 + +lead to both the sharp peak of ˙φ, and the vanishing of terms other than +... +φ in (26). This is +generically required by the ghost-free condition for the scalar mode, i.e. the coefficient of ¨φ +in (7) does not cross 0. One popular way to evade the scalar ghost is to let the bouncing +phase be short enough, such that bouncing ends before the coefficient approaches 0 [56]. In +this case, the duration is severely constrained. +In conclusion, we find that certain characteristics of our toy model, i.e. +˙φ small in the +contraction phase, one single sharp peak for ˙φ in the bouncing phase, and short duration of +bouncing phase, are generic in many bouncing models. We then expect our conclusion to +be also valid for these bouncing models. +IV. +CONCLUSION AND OUTLOOK +We investigate the possible parity-violation signals in bouncing cosmology, by a coupling +between the gCS term and the scalar field which triggers the bounce. Through numerical +studies of a toy bouncing model, we find that the parity-violation signals are enhanced +during the bouncing phase. Moreover, we study the numerical result in a semi-analytical +way and find that our result obtained in the toy model can be generalized in a wild range +of the bouncing models. +The significance of our result is twofold. On the one hand, we provide a possible mecha- +nism for the generation of parity-violation signals in the framework of bouncing cosmology, +enabling us to explain parity-violation physics in the GW background. On the other hand, +since the parity-violation signals come from the bouncing phase, where the energy scale +is the highest and new physics is believed to exist, our result provides a possible way to +explore the new physics through parity-violation signals. To our best knowledge, our result +is distinctive from many other phenomenological approaches, where the imprint from the +bouncing phase is minimized. +The current work is a preliminary check on parity-violation physics in bouncing cosmol- +ogy. There are a lot of following-up works to be finished in the future. +Firstly, since the tensor spectrum is dependent on the physics of the contraction phase and +expansion phase, it is important to construct a realistic bouncing model to predict the parity- +violation signals in the real world and confront them with observations. For example, for +the contraction phase, we may take either an Ekpyrotic contraction or a matter contraction; +18 + +for the expansion phase, we may take either an inflating as those bounce-inflation models +or an expansion dominated by radiation, such that the standard cosmology begins exactly +when the bouncing phase ends. +Furthermore, the physical scale at which the effect of +parity-violation appears depends on the scale of the bounce and a complete construction of +the evolution of the universe, which should also be addressed in future studies in order to +confront the observations. +Secondly, we shall study physics with the high energy correction LHE specified. In this +paper, we study the specific case where LHE has negligible impact on both the background +dynamics and the propagation of gravitational waves. To probe the physics of LHE, we shall +choose a specific form of LHE, study their effect at both background and perturbative levels, +and get their possible unique imprints. +Finally, there are issues beyond our current framework. For example, we are working +with a classical bounce. What would happen if we have a quantum bounce? Besides, there +could also be parity-violation signals from the coupling between the E mode and B mode. +It’s interesting to ask if our results hold in this scenario. Last but not least, it’s interesting +to consider alternative parity-violation mechanism [21, 24, 25, 36]. +These questions are +interesting to address and are open for the following study. +V. +ACKNOWLEDGEMENT +We thank Shingo Akama, Yi-Fu Cai, Chao Chen, Alexander Ganz, Chunshan Lin, As- +tuhisa Ota, Yun-Song Piao, Yi Wang and Yunlong Zheng for their helpful discussions and +comments. M. Z. is supported by grant No. UMO 2018/30/Q/ST9/00795 from the National +Science Centre, Poland. Y. 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Mukohyama, “Ghost condensation and +a consistent infrared modification of gravity,” JHEP 05 (2004) 074, +arXiv:hep-th/0312099. +28 + diff --git a/RtFRT4oBgHgl3EQfLTfA/content/tmp_files/load_file.txt b/RtFRT4oBgHgl3EQfLTfA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdfc1993a97c292a057f4eb8cb7c53492e0cf4e1 --- /dev/null +++ b/RtFRT4oBgHgl3EQfLTfA/content/tmp_files/load_file.txt @@ -0,0 +1,1416 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf,len=1415 +page_content='Parity-violation in bouncing cosmology Mian Zhu1,2∗ and Yong Cai1† 1 School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China and 2 Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krakow, Poland Abstract We investigate the possibility of the enhancement of parity-violation signal in bouncing cosmol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Specifically, we are interested in deciding which phase should generate the most significant parity-violation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We find that the dominant contribution comes from the bouncing phase, while the contraction phase has a smaller contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Therefore, bouncing cosmology can en- hance the parity-violation signals during the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, since the bouncing phase has the highest energy scale in bouncing cosmology, we can also probe new physics at this scale by studying the parity-violation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' PACS numbers: ∗ Corresponding author: mzhuan@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='hk † Corresponding author: yongcai phy@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='13502v1 [gr-qc] 31 Jan 2023 Contents I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Model 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Action 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Bouncing background 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The effective action on high energy scale 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Tensor perturbation 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Formalism 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Dynamics of tensor perturbation 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Parity violation signal 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Comment on the resulting signal 13 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Semi-analytic investigation 15 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Conclusion and Outlook 18 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Acknowledgement 19 References 19 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' INTRODUCTION The primordial gravitational waves (GWs) might encode rich information about the very early universe, which may help distinguish between different scenarios of the primordial universe, including inflation and its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Chirality is a distinct characteristic of GWs, which could be manifested in parity-violating theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Recently, it is found that the polarization data of Planck and WMAP [1–3] may be a hint of parity-violating physics in the cosmic microwave background, though further confirmations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The explorations of parity-violating primordial GWs have aroused a lot of interest, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' [4–36], see also [37–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In single-field slow-roll inflation models where inflaton is non-minimally coupled to a parity-violating term, such as the gravitational Chern-Simons (gCS) term [44, 45], the effect 2 of parity-violation should be suppressed by the slow-roll condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, in the modifi- cations or alternatives to the single field slow-roll inflation, the slow-roll condition could be violated at least for some moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' As a result, the effect of parity-violation could be en- hanced due to the dynamical coupling of the scalar field to the parity-violating term, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' [46] for the enhanced parity-violating GWs caused by violation of the null energy condition (NEC) [47] during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Therefore, observations of a parity-violating GW background might provide us with a new way to identify physics beyond single-field slow-roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Bouncing cosmology as a possible solution to the initial cosmological singularity problem of inflation and the Big Bang cosmology has attracted a lot of interest [48–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In the bouncing scenario, the universe originates from a contracting phase and enters an expanding phase after going through a non-singular bounce, where the NEC is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' One important issue is the ghost and gradient instabilities in the bouncing phase, which is a generic feature in a large class of theories [75–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' To acquire a healthy bouncing model, new physics effective at bouncing phase is introduced [78–95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In principle we may explore the new physics by studying their phenomenological predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Unfortunately, the signals from the new physics, which is generically effective only in the bouncing phase, are suppressed by the short duration of the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For example, in [90], it is found that the new physics at the bouncing phase has negligible contribution to the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Consequently, in many studies of non-singular cosmology [96–103], the signals from the bouncing phase are small, the main phenomenological contribution comes from the contraction phase, so it’s difficult to probe the new physics in bouncing phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Specifically, in previous literature addressing the parity-violation effect in bouncing cosmol- ogy [10], the bouncing phase is directly replaced by a simple junction condition so there is only a contraction and an expansion in their scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' It is then interesting to study if the parity-violation effect could be generated in bouncing cosmology, especially during the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Intuitively, the derivative of the scalar field is non-trivial around the bouncing phase, which may be able to amplify the effect of parity-violation, as long as the scalar field is non-minimally coupled to a parity-violating term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Additionally, the effective sound horizon of the primordial GW mode could also 1 Some counterexamples comes from the quantum bounce models [104–106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, this is beyond our scope since we consider purely classical bouncing cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 3 be nontrivial during the bouncing phase2, especially when chirality of the GW mode is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Therefore, we expect the non-trivial parity-violation signals to come from the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In this paper, we investigate the parity-violation effect in a toy bouncing model, where the source term is taken to be a gCS action coupled to the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We are especially interested in the following question: which phase, the contraction or the bouncing phase, contributes to the enhancement of the parity-violation effect dominantly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' As we will see in section III C, the bouncing phase can generate non-trivial parity-violation signals, while the contraction phase has negligible effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, the enhancement is sensitive to the detailed physics during the bouncing phase, so in principle, we can probe the new physics during bouncing through parity-violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Therefore, our result is twofold: we can not only explain the possibly observed parity-violation signal in the framework of bouncing cosmology, but also provide a possible way to probe new physics at the bouncing phase by studying their imprint on parity-violation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In section II we briefly introduce our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' After the basic formalism for tensor perturbation in III A, we numerically evaluate the dynamics of tensor perturbation in section III B and the parity-violation signal in section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We comment on some conceptual issues about our result in section III D and explain our numer- ical result in a semi-analytical way in section III E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' From the semi-analytical argument, we find that our numerical result should be qualitatively valid for a large variety of bouncing models, although the numerics are taken in a toy bouncing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We finally conclude in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Throughout this paper, we take the sign of the metric to be (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We will take ℏ = 1, c = 1, M 2 p = (8πG)−1 = 1, so that all quantities are in Planck units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The canonical kinetic term is defined as X ≡ −∇µφ∇µφ/2, such that X = ˙φ2/2 at the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 2 The bouncing phase is defined by dH/dt ≥ 0, where H is the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 4 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Action We take the action to be S = � d4x√−g �M 2 p 2 R + LH + LG + LHE � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (1) The term LH is responsible for setting the background evolution, where we set LH = M 2 pf1(φ)X + f2(φ)X2 − M 4 pV (φ) , (2) which is eligible for the background dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In the next section II B, we will use specific coupling functions f1 and f2 to construct a cosmological bouncing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The LG term is the gravitational CS term, with LG = f3(φ) 8 R ∧ R = f3(φ) 8 ϵαβρσRαβµνR µν ρσ , (3) and ϵαβρσ to be four-dimensional Levi-Civita symbol with ϵ0123 = −1/√−g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Finally, the term LHE represents the action effective at some high energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Since there will be ghost or gradient instability problems in the generic bouncing models [75, 76, 99], such terms are obligated to eliminate such instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We will discuss in details of this term in section II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We mention that in (1) we scale the scalar field φ to be dimensionless so that the coupling functions fi are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Bouncing background It is well-known that the gCS term will not contribute to the background dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We shall assume that the correction term LHE also satisfies this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Therefore, Fried- mann’s equations are totally determined by the Einstein-Hilbert action and the LH term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In a flat FLRW background ds2 = −dt2 + a2(t)d⃗x2 , (4) we have 3M 2 pH2 = M 2 p 2 f1 ˙φ2 + 3 4f2 ˙φ4 + M 4 pV (φ) , (5) 5 − 2M 2 p ˙H = M 2 pf1 ˙φ2 + f2 ˙φ4 , (6) or in terms of the scalar field φ: � M 2 pf1 + 3β ˙φ2� ¨φ + 3H ˙φ � M 2 pf1 + β ˙φ2� + M 4 p dV dφ + M 2 p 2 df1 dφ ˙φ2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (7) Now we choose a similar ansatz as that from [85, 90]: f1(φ) = 1 − g cosh ω1φ , f2 = β ≡ const , V (φ) = − V0 cosh ωV φ , (8) where the background dynamics are well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In the initial state of the universe where ˙φ → 0 and φ → −∞, the universe undergoes an Ekpyrotic contraction [107] φ ≃ − 1 ωV ln ω4 V V0t2 ω2 V − 6 , a(t) = a− � t − tc t− − tc � 2 ω2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (9) The Ekpyrotic phase makes us free from conceptual issues of bouncing cosmology [108], at the cost of requiring ω2 V > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Note that we set t = 0 to be the bouncing point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' the stage where the scale factor is minimal, so need an integration constant tc to correctly describe a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We also use the minus sign to denote the end of the Ekpyrotic phase, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' a− is the scale factor at the end of the Ekpyrotic contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' When |φ| → 1, the hyperbolic function approaches 1, and if we take g > 1, the f1X term inverses sign and NEC can be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The non-singular bounce phase starts when the NEC is violated, and the universe transit from contraction to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The dynamics during the bouncing phase are generically complicated, but for a short bounce, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' bouncing phase with short enough time, the following parameterization can be valid H = γM 2 pt , γ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' > 0 → a = a0e 1 2 γM2 pt2 , (10) where we have set a0 = a(0), which is the scale factor at the bouncing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' After the bouncing phase, the universe comes to an expansion phase, where the scale factor behaves as a(t) = a+ � t − te t+ − te � 1 3 , H(t) = 1 3(t − te) , (11) where we similarly use the “+” sign to denote the end of the bouncing phase, and te is another integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We shall comment more on the expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Notice that, the factor aH from (11) is proportional to (t − te)− 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Hence, for any wave mode that is initially sub-horizon at t = t+, 6 10000 5000 0 5000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0003 t H 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='003 t H 10000 5000 0 5000 10000 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-7 t ρ 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='51 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00003 t ρ 10000 5000 0 5000 10000 2 1 0 1 2 3 4 t ϕ 10000 5000 0 5000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0007 t ϕ\uf110 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 t ϕ\uf110 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 1: The background dynamics with the specific parameters (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The upper channel shows the evolution of the Hubble parameter and background energy density, while the lower channel shows the dynamics of the scalar field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The bouncing phase happens at around t = 0 where H quickly transfers from negative to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' it will remain sub-horizon in the whole expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' This is in contrast with our general belief that, the primordial perturbation should leave the horizon in the expansion phase (like inflation) and freeze in, and re-enter the horizon in a later stage to set the initial condition for structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, the parity-violation signal is highly dependent on the subsequent expansion phase after the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In this paper, we want to compare the induction of parity-violation between the contraction phase and the bouncing phase, so we wish to get a result independent of the subsequent expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Unfortunately, we do not have a precise way to define when the bouncing phase ends, so it is hard to directly get the parity-violation status at the end of the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The advantage of our expansion phase (11) is that the wave mode of interests will always be in the sub-horizon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, their dynamics can be approximately described by the harmonic oscillator equation u′′ k + c2 Tk2uk = 0, whose general solution is simply uk ≃ uk,+eikτ + uk,−e−ikτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (12) 7 The information of parity-violation status when bounce ends is encoded in the function uk,±, and we see that the expansion phase only changes their relative phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, we may alternatively get the physics of parity-violation at the end of the bouncing phase, by tracing the statistical property of tensor perturbation during the expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We shall elaborate more about this point in section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We depict the background dynamics in figure 1, where we’ve adopted the following pa- rameters g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 , β = 2 , V0 = 10−7 , ω1 = 10 , ωV = √ 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (13) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The effective action on high energy scale As shown in figure 1, the background energy density at the bouncing phase is much higher than the other phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Hence, it is natural to introduce some actions effective only at a high energy scale to eliminate the instability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In the context of effective field theory (EFT) of non-singular cosmology [78, 79], certain EFT operators such as R(3)δg00 can help to evade the instabilities without altering the background dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, when come to the realization of such EFT operators, the dynamics of tensor perturbation are generally influenced by such high-energy correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For example, in [81, 85–87] where the EFT operator R(3)δg00 is written in a covariant form, there appears a non-minimal coupling and the propagating speed of GWs is changed accordingly [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' There are also other approaches for LHE to eliminate the instabilities [82, 93, 95, 110–113], while generically changes either the background dynamics or the propagation of gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' It would be hard to combine all these approaches in a unified description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In this paper, we will start with the simplest case where LHE has no influence on both the background dynamic and propagation of gravitational waves (this is the case of the EFT approach in [78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The point is, we will use this case as fiducial to examine if the bouncing phase contributes more to the parity-violation than the contraction phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' If this is true, we would possibly have the opportunity to distinguish the above approaches through the GW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' TENSOR PERTURBATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Formalism Now we come to the tensor mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We’ve assumed that LHE doesn’t contribute to the tensor mode, so the quadratic action for tensor perturbation is S(2) T = M 2 p 8 � dτd3x � a2 � γ′2 ij − (∂γij)2� − g′ 3 M 2 p ϵijk � (∂iγjl)′(γl k)′ − ∂i∂lγjq∂lγq k �� , (14) where we’ve defined the conformal time τ ≡ � dt/a and a prime denotes differentiation with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Before proceeding, we see that the gCS term is suppressed by the factor M 2 p, so this term should be important at a high energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, g′ 3 = a ˙φg3,φ, and from figure 1 that ˙φ is non-trivial only during the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, we can intuitively guess that the gCS term should be important during the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We work in the Fourier space where γij(τ, ⃗x) = � s=L,R � d3k (2π3)γ(s) k (τ)p(s) ij (⃗k)ei⃗k·⃗x , (15) with the polarization tensor satisfying p(R) ij pij(R) = p(L) ij pij(L) = 0 , p(R) ij pij(L) = 2 , iklϵqljp(s) ij = kλ(s)pq(s) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (16) The polarization mode is decided by the parameter λ, such that λ(L) = −1 , λ(R) = 1 , λ(N) = 0 , (17) and here for convenience, we’ve defined a new N mode to represent the non-parity-violation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Finally, the parity-violation is evaluated by the chiral parameter ∆χ ≡ P (L) T − P (R) T P (L) T + P (R) T , (18) where P (s) T are the power spectrum of the corresponding polarization modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Although the difference P (L) T − P (R) T is of observational interest, the absolute value of P (s) T is highly dependent on the detailed bouncing models (for example, the tensor spectra index in our model (8) is dependent on the model parameter ωV [90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus for our purpose to compare the parity-violation effect from a different phase, we shall concern with the parameter ∆χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 9 L R 10000 0 10000 20000 30000 40000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=" × 10-8 t 1 a0 2 zT '' zT L N R 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content="6 t 1 a0 2 zT '' zT N 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content="04 1 a02 zT '' zt FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 2: The function z(s)′′ T /z(s) T as a function of time for different polarization modes, rescaled by a factor a−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The left figure shows the time evolution of the whole cosmological history, and the right figure shows the dynamics near the bouncing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Dynamics of tensor perturbation The dynamical equation for the tensor mode γ(s) k is u(s)′′ k + � k2 − z(s)′′ T z(s) T � u(s) k = 0 , (19) where we define the Mukhanov-Sasaki variable u(s) k ≡ z(s) T γ(s) k , z(s) T ≡ a 2 � 1 − λ(s)k a g3,φφ′ aM 2 p , (20) and the sound speed is set to be unity for all polarization modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Notice that we require the terms in the square root to be non-negative, otherwise, there will be ghost modes [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Initially, all the perturbation modes of observational interests are on sub-horizon scales, where the k2 terms in (19) dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, we can take the vacuum initial condition u(s) k = e−ikτ √ 2k , τ → −∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (21) We can combine the equations (19) and (21) to get the dynamics of γij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Firstly, we evaluate the term z(s)′′ T /z(s) T numerically, with a specific gCS coupling f3(φ) = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, we notice that the result depends only on the physical wavenumber k/a0 instead of k, as long as we rescale the term z(s)′′ T /z(s) T by a factor a−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' At this point, we set a specific scale k/a0 = 10−2, the averaged magnitude of maximum value of ˙φ and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We depict the term z(s)′′ T /z(s) T as a function of cosmic time in figure 2, with a rescale factor a−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Outside the bouncing phase, the L and R modes are almost identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' while during the 10 R L 0 2000 4000 6000 8000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='999999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='999999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000000 t uk(s) uk(N) k/a0=10-4 R L 0 2000 4000 6000 8000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='999985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='999990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='999995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000010 t uk(s) uk(N) k/a0=10-3 R L 0 2000 4000 6000 8000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='99990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='99995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00010 t uk(s) uk(N) k/a0=10-2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 3: The dynamics of |u(s) k |/|u(N) k | as a function of cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We specify the dynamics with three characterised scale, k/a0 = 10−4, 10−3 and 10−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' bouncing phase, the two polarization modes differ significantly, and the amplitude of L/R mode is one order beyond the unpolarized mod N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Now we come to the mode function u(s) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' As we explained at the ending part of section II B, the modes initially on the sub-horizon scale at t = t+ will stay in the sub-horizon region during the expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Their evolution can then be approximated as u(s) k ≃ u(s) k,+eikτ + u(s) k,−e−ikτ , (22) so the amplitude of the mode function will oscillate during this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We depict the dynamics of |u(s) k |/|u(N) k | for different k/a0 value in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' As we can see, for large scale such as k/a0 = 10−4, the mode quickly becomes super-horizon during the bouncing phase, and |u(s) k |/|u(N) k | approaches constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For intemediate scale like k = a0 = 10−3, the mode is sub-horizon but z(s)′′ T /z(s) T is still comparable to k2/a2 0, so the dynamics is oscillatory but not strictly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For small scale like k/a0 = 10−2, the oscillatory feature is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Now we conclude that, for sufficiently large wave mode, physical quantities such as the mode function (and hence the tensor perturbation γ and parameter ∆χ) at the end of the bouncing phase, can be represented by their statistics property at the expansion phase, since the expansion phase only add an oscillating feature to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' One additional advantage of our treatment is that the horizon-cross condition should in principle determined by the behavior of z(s)′′ T /z(s) T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' While in the bouncing phase, this term is highly non-trivial, it simplifies to a′′/a in the expansion phase and we have a simple expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 11 10000 5000 0 5000 10000 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' × 10-6 t zT (R) - zT (L) zT (N) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='006 10000 5000 0 5000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 t zT (L) 10000 5000 0 5000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 t zT (R) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 4: The dynamics of z(s) T in the whole cosmological history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The scale is chosen to be k/a0 = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We see from the left figure that z(s) T are almost identical, and for convenience, we also plot the z(s) T for L and R mode respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 0 2000 4000 6000 8000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00002 t Δχ k/a0=2*10-3 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00025 t Δχ k/a0=1*10-2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='015 t Δχ k/a0=1*10-1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 5: The parity-violation status as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The oscillatory feature is expected due to the behavior of |u(s) k |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Note that we adopted a smaller range of t for large k/a0, otherwise the whole picture would be totally filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Parity violation signal With the dynamics of the mode function, we can evaluate the corresponding tensor power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Notice that the tensor spectrum depends also on z(s) T , which carries the information on different polarization, so we should first evaluate γ(s) k ≡ u(s) k /z(s) T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, in our case, the function z(s) T differs only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' As shown in figure 4, z(s) T for L and R mode would have a maximum difference of order 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Hence, we may simply take P (L) T P (R) T = |γ(L) k |2 |γ(R) k |2 = |u(L) k |2 |u(R) k |2 |z(R) T |2 |z(L) T |2 ≃ |u(L) k |2 |u(R) k |2 → ∆χ ≃ |u(L) k |2 − |u(R) k |2 |u(L) k |2 + |u(R) k |2 , (23) with a loss of precision no more than O(10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Now we can work out ∆χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Since |u(s) k | is oscillating, we expect ∆χ to be also oscillating, as shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' As stated in the last part in section III B, we will represent the parity- violation state at the end of the bouncing phase, by the statistic property of uk (and hence ∆χ) during the expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Our strategy is, for each fixed k/a0, we take the value of 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='100 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='100 k/a0 Δχ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 6: The parameter ∆χ as a function of physical wavenumber k/a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Notice that for smaller k/a0, the behavior of u(s) k would differ more from (22), so ∆χ would also receive more influence from the expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus we shall treat the data from smaller k/a0 with less confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' ∆χ’s amplitude A∆χ with a factor 1/ √ 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' A∆χ/ √ 2, to represent the corresponding ∆χ at the end of the bouncing phase, ∆χb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Then, we can depict the dependence of ∆χb on the physical wavenumber k/a0, in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We see from figure 6 that, the parity-violation can be induced at the bouncing phase, and for large k/a0, there are chances that the parameter ∆χ be large enough (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' of order 10−2) to generate detectable parity-violation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Comment on the resulting signal Before proceeding, we shall comment on the result from section III C, and clarify some potentially confusing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Firstly, we stress again that the signal obtained in the last section is in fact the repre- sentation of the signal at the end of the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In order to confront the result of observations, we need to design a more realistic expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Then, it is possible that a large parity-violation signal at t = t+ is suppressed by the subsequent expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus at the current stage, what we can conclude is that parity-violation feature can be pro- duced at the bouncing phase where the energy scale is the highest in bouncing cosmology, and it could potentially be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Besides, we see in figure 6 that ∆χ is proportional to k/a0, which seems to be in contrast with the result from [46], where parity-violation signals are also generated by some NEC- 13 violation phase, but ∆χ is non-trivial only in selected wavelengths (see also [10, 11]), while our result seems to be valid for a wide range of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' This is because the scenario considered in [46] is in an inflation background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The NEC violation happens between two inflation phases, and thus the NEC violation phase is in correspondence to a specific range of wavenumber a−H− < k < a+H+, where the ± sign stands for the beginning and end of the NEC violation phase, so k± = a±H± stands for the wave mode that exactly crosses the Hubble horizon at t = t±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, in our case, the bouncing point is characterized by H(0) = 0, where all modes are inside the “Hubble horizon” 1/H → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, we expect all modes “feel” the parity-violation physics during the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Actually, the result displayed in figure 6 is consistent with that obtained in the bounce- inflation scenario [10], where the effect of parity-violation measured by ∆χ is proportional to k/H∗ for the GW modes which exit horizon during the contracting and bouncing phases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=', before the inflationary phase), though the bouncing phase is assumed to be negligibly short in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Finally, one may naturally ask, if ∆χ is proportional to k/a0 as that in figure 6, then shouldn’t ∆χ be of higher order like O(1), and resulting in an unreasonably large parity- violation signal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The problem is, we have to cut off at some k for at least two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Firstly, to avoid the appearance of ghosts, we require z(s) T to be real, so ����� k a ˙φ M 2 p ����� < 1 → k a0 < max � ˙φ M 2 p � , (24) and we have to cut off smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Besides, the effective description of our universe as a homogeneous and isotropic ideal fluid breaks down for a sufficiently small scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' large enough k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' This means that the value of a0 cannot be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Instead, it should have a proper value such that the parity-violation happens at the correct scale, and the value of k/a0 always satisfies the condition (24) for reasonable k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We shall further mention that, in our toy model, the wave mode displayed in figure 6 is in the sub-horizon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' However, in a realistic model, the tensor mode will experience a decaying when evolving toward the horizon during the expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Smaller scales would exit the horizon at a later time, and they would experience more decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, although ∆χ is approximately proportional to k/a0 at the end of the bouncing phase, it is possible that smaller scales receive more suppression in the following expansion phase, and the parity-violation effect is important only in some intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 10 5 0 5 10 t ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content="04 t zT '' zT FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 7: We compare the dynamics of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' φ and z(L)′′ T /z(L) T during the bouncing phase and find the same feature from both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Semi-analytic investigation Although we numerically verified the existence of parity-violation signals from the bounc- ing phase, we wish to briefly explain the result analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Fortunately, the duration of the bouncing phase is small from figure 1, so we may adopt the parametrization (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, in cosmic time, we have z(s)′′ T z(s) T = a2 � ¨z(s) T z(s) T + H ˙z(s) T z(s) T � , z(s) T = a 2 � 1 − λ(s)k a ˙φ M 2 p ≃ a 2 − λ(s)k 4 ˙φ M 2 p , (25) and we may write the expression in the following z(s)′′ T z(s) T ≃ a2 � ¨a + H ˙a 2z(s) T − λ(s)k 4 ¨φH + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' φ M 2 pz(s) T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (26) The term ¨a + H ˙a is suppressed by a factor t2, while the H ¨φ term suppressed by a factor t, we concern on the term .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Now ˙φ is a δ-like function, so we expect ¨φ to have a positive peak at t < 0 and a negative peak at t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Subsequently, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' φ should first have a positive peak at t < 0, then a negative peak at t > 0, and finally followed by a second positive peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We illustrate this point by depicting both ¨φ and z(L)′′ T /z(L) T in figure 7 and see that they have exactly the same feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We conclude that, the feature of z(s)′′ T /z(s) T comes from that of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' φ, which is further decided by the δ-function-like behavior of ˙φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The mode function receives a non-trivial enhancement, To intuitively understand how the peaks of z(s)′′ T /z(s) T affect the tensor mode, we may approximately take each peak as a δ-like function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For simplicity, we take the realization of 15 Ai(x) Bi(x) 6 4 2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 8: Airy function these peaks to be a linear function � z(s)′′ T /z(s) T � peak ≃ b|t − tc| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' tp− < tc < tp+ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' b > 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' t ∈ (tp−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' tp+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (27) so for each region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' the dynamical equation for the mode function becomes (for convenience let’s temporarily take tc = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' and also t ≃ τ during the bouncing phase since a is almost a constant) u(s)′′ k + � k2 ± bt � u(s) k = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (28) whose general solution is the Airy function u(s) k = c1Ai � −k2 ∓ bt |b| 2 3 � + c2Bi � −k2 ∓ bt |b| 2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' (29) In figure 8 we depict the behavior of the Airy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' When the argument is negative, both Airy functions oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' When the argument is positive, one branch increases while the other shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, the amplitude of u(s) k will be enhanced in the positive-argument region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Note that the parametrization in (27) is rough, so at the current stage, we cannot go further without the detailed expression of the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus we can only conclude that the peaks in z(s)′′ T /z(s) T enhance the amplitude of tensor perturbation, and different polarization modes receive different enhancement due to the microscopic physics in the bouncing phase, which causes the parity-violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Nextly, we shall intuitively explain why ∆χ has a linear dependence on k/a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For this purpose, we plot in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For large k/a0, the three peak value of z(s)′′ T /z(s) T is approximately linearly dependent on k/a0, so we expect the enhancement of u(s) k also depends on k/a0 linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For smaller k/a0 when the peak value of L and R modes are comparable to that of 16 L N R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content="06 t 1 a0 2 zT '' zT k/a0=10-3 L N R 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content="6 t 1 a0 2 zT '' zT k/a0=10-2 L N R 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content="0 6 4 2 0 2 4 6 t 1 a0 2 zT '' zT k/a0=10-1 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 9: The evolution of z(s)′′ T /z(s) T for different physical wavenumber k/a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' N mode, the second peak destroys the linear relationship, so we expect the linear dependence of ∆χ on k/a0 is ruined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, the fitted function ∆χ in figure 6 is a little convex instead of perfectly straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Finally, we emphasize how generic our result should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The suppression of parity- violation signals during the contraction phase comes from the smallness of ˙φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Although the dynamics of ˙φ relies on the details of the contraction phase, for mainstream bouncing models like matter bounce (contraction phase dominated by a stiff matter and small ˙φ like that in [59, 115]) and Ekpyrotic bounce (the case described by our model where ˙φ = −2/ωV t), | ˙φ| is always small for large |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Alternatively, a large ˙φ would correspond to a higher energy scale, so the parity-violation effect is suppressed in the contraction phase because of the low energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Notice that the contraction phase will always have a lower energy scale compared to the bouncing phase as long as we consider a classical bounce model where the contraction phase happens with an initially classic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Hence, the smallness of the parity-violation signal in the contraction phase should be valid at least for many bouncing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We may understand the smallness of ˙φ in the contraction phase by alternative arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' One common mechanism for NEC violation is ghost condensation [116], where the kinetic Lagrangian L(X) has a non-trivial stationary point at X ̸= 0 with a negative vacuum expectation (VEV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The contraction phase corresponds to the configuration of the false vacuum X = 0, while the bouncing phase corresponds to the true vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Thus, a small ˙φ is expected in this mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, if the bouncing phase has a short duration, we also expect ˙φ to have a sharp peak, whose magnitude is related to the VEV of the kinetic Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In view of the above argument, we see that a short duration of the bouncing phase can 17 lead to both the sharp peak of ˙φ, and the vanishing of terms other than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' φ in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' This is generically required by the ghost-free condition for the scalar mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' the coefficient of ¨φ in (7) does not cross 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' One popular way to evade the scalar ghost is to let the bouncing phase be short enough, such that bouncing ends before the coefficient approaches 0 [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In this case, the duration is severely constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In conclusion, we find that certain characteristics of our toy model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' ˙φ small in the contraction phase, one single sharp peak for ˙φ in the bouncing phase, and short duration of bouncing phase, are generic in many bouncing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' We then expect our conclusion to be also valid for these bouncing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK We investigate the possible parity-violation signals in bouncing cosmology, by a coupling between the gCS term and the scalar field which triggers the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Through numerical studies of a toy bouncing model, we find that the parity-violation signals are enhanced during the bouncing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Moreover, we study the numerical result in a semi-analytical way and find that our result obtained in the toy model can be generalized in a wild range of the bouncing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The significance of our result is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' On the one hand, we provide a possible mecha- nism for the generation of parity-violation signals in the framework of bouncing cosmology, enabling us to explain parity-violation physics in the GW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' On the other hand, since the parity-violation signals come from the bouncing phase, where the energy scale is the highest and new physics is believed to exist, our result provides a possible way to explore the new physics through parity-violation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' To our best knowledge, our result is distinctive from many other phenomenological approaches, where the imprint from the bouncing phase is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' The current work is a preliminary check on parity-violation physics in bouncing cosmol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' There are a lot of following-up works to be finished in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Firstly, since the tensor spectrum is dependent on the physics of the contraction phase and expansion phase, it is important to construct a realistic bouncing model to predict the parity- violation signals in the real world and confront them with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For example, for the contraction phase, we may take either an Ekpyrotic contraction or a matter contraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 18 for the expansion phase, we may take either an inflating as those bounce-inflation models or an expansion dominated by radiation, such that the standard cosmology begins exactly when the bouncing phase ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Furthermore, the physical scale at which the effect of parity-violation appears depends on the scale of the bounce and a complete construction of the evolution of the universe, which should also be addressed in future studies in order to confront the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Secondly, we shall study physics with the high energy correction LHE specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' In this paper, we study the specific case where LHE has negligible impact on both the background dynamics and the propagation of gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' To probe the physics of LHE, we shall choose a specific form of LHE, study their effect at both background and perturbative levels, and get their possible unique imprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Finally, there are issues beyond our current framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' For example, we are working with a classical bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' What would happen if we have a quantum bounce?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Besides, there could also be parity-violation signals from the coupling between the E mode and B mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' It’s interesting to ask if our results hold in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Last but not least, it’s interesting to consider alternative parity-violation mechanism [21, 24, 25, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' These questions are interesting to address and are open for the following study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' ACKNOWLEDGEMENT We thank Shingo Akama, Yi-Fu Cai, Chao Chen, Alexander Ganz, Chunshan Lin, As- tuhisa Ota, Yun-Song Piao, Yi Wang and Yunlong Zheng for their helpful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' is supported by grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' UMO 2018/30/Q/ST9/00795 from the National Science Centre, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' is supported in part by the National Natural Science Foun- dation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 11905224), the China Postdoctoral Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 2021M692942), and Zhengzhou University (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 32340282).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 19 [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Diego-Palazuelos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=', “Cosmic Birefringence from the Planck Data Release 4,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Lett.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' Mukohyama, “Ghost condensation and a consistent infrared modification of gravity,” JHEP 05 (2004) 074, arXiv:hep-th/0312099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtFRT4oBgHgl3EQfLTfA/content/2301.13502v1.pdf'} diff --git a/U9E3T4oBgHgl3EQfawpi/vector_store/index.pkl b/U9E3T4oBgHgl3EQfawpi/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d40e97fbb180f81de9ae65af5ad65d502540b13c --- /dev/null +++ b/U9E3T4oBgHgl3EQfawpi/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:658c0d2c1dc3a8e59a5febe99d713c463ff80d207d68af2fff1d39b26c2fd9fd +size 88630 diff --git a/VtFKT4oBgHgl3EQfmy5j/content/tmp_files/2301.11859v1.pdf.txt b/VtFKT4oBgHgl3EQfmy5j/content/tmp_files/2301.11859v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f856540e3fbbf345052f7528c30196ef37d5582 --- /dev/null +++ b/VtFKT4oBgHgl3EQfmy5j/content/tmp_files/2301.11859v1.pdf.txt @@ -0,0 +1,2766 @@ +Synthetic Difference In Differences Estimation +Damian Clarke +Department of Economics +University of Chile +dclarke@fen.uchile.cl +Daniel Paila˜nir +Department of Economics +University of Chile +dpailanir@fen.uchile.cl +Susan Athey +Graduate School of Business +Stanford University +athey@stanford.edu +Guido Imbens +Graduate School of Business +Stanford University +imbens@stanford.edu +Abstract. +In this paper, we describe a computational implementation of the +Synthetic difference-in-differences (SDID) estimator of Arkhangelsky et al. (2021) +for Stata. Synthetic difference-in-differences can be used in a wide class of cir- +cumstances where treatment effects on some particular policy or event are desired, +and repeated observations on treated and untreated units are available over time. +We lay out the theory underlying SDID, both when there is a single treatment +adoption date and when adoption is staggered over time, and discuss estimation +and inference in each of these cases. We introduce the sdid command which imple- +ments these methods in Stata, and provide a number of examples of use, discussing +estimation, inference, and visualization of results. +Keywords: synthetic difference-in-differences, synthetic control, difference-in-differences, +estimation, inference, visualization. +1 +Introduction +There has been an explosion in recent advances in econometric methods for policy anal- +ysis. A particularly active area is that applied to estimating the impact of exposure to +some particular event or policy, when observations are available in a panel or repeated +cross section of groups and time (see for example recent surveys by de Chaisemartin and +D’Haultfœuille (2022); Roth et al. (2022) for reviews of these methods). A modelling +challenge in this setting is in determining what would have happened to exposed units +had they been left unexposed. Should such a counterfactual be estimable from under- +lying data, causal inference can be conducted by comparing outcomes in treated units +to those in theoretical counterfactual untreated states, under the potential outcome +framework (see for example the discussion in Holland (1986); Rubin (2005). +A substantial number of empirical studies in economics and the social sciences more +generally seek to estimate effects in this setting using difference-in-difference (hereafter +DID) style designs. Here impacts are inferred by comparing treated to control units, +where time-invariant level differences between units are permitted as well as general +common trends. However, the drawing of causal inferences requires a parallel trend +assumption, which states that in the absence of treatment, treated units would have +followed parallel paths to untreated units. Whether this assumption is reasonable in +arXiv:2301.11859v1 [econ.EM] 27 Jan 2023 + +2 +Synthetic Difference In Differences +a particular context is an empirical issue. Recently, a number of methodologies have +sought to loosen this assumption. +This includes procedures in which counterfactual +trends can be assumed to deviate from parallel, leading to partial identification (Manski +and Pepper 2018; Rambachan and Roth 2019), flexible procedures to adequately control +for any prevailing differences between treated and control units (Bilinski and Hatfield +2018), often based on pre-treatment periods only (Bhuller et al. 2013; Goodman-Bacon +2021) and IV-style methods which explicitly consider dynamics in pre-treatment periods +(Freyaldenhoven et al. 2019). +In many cases, parallel trends may be a questionable modelling assumption. One +particular solution to the challenge has been the application of synthetic control meth- +ods. Early work in synthetic control explores the setting of comparative case studies, +where a single treated unit is observed, and one wishes to construct a matched synthetic +control from a larger number of potential donor units (Abadie and Gardeazabal 2003; +Abadie et al. 2010, 2015). These methods seek to generate a single synthetic control +from a unique convex weighting of underlying control units, such that this synthetic +control is as closely matched as possible to the treated unit in pre-treatment outcomes, +and potentially other covariates. This weights are optimally generated and fixed over +time, potentially assigning zero weight to certain control units, and larger weights to +others. This has attracted considerable attention in both empirical applications and the- +oretical extensions, with recent advances including debiasing procedures (Ben-Michael +et al. 2021) which can additionally house multiple treatment units (Abadie and L’Hour +2021), more flexible weighting schemes, or constant fixed differences between treated +and synthetic control units (Doudchenko and Imbens 2016; Ferman and Pinto 2021). +A recent particularly flexible modelling option which can be applied in panel data +settings seeks to bridge the DID and synthetic control (SC) procedures. Arkhangelsky +et al. (2021) propose the Synthetic Difference-in-Differences estimator (SDID), which +brings in strengths from both the DID and SC methods. Like DID models, SDID allows +for treated and control units to be trending on entirely different levels prior to a reform of +interest. And like SC methods, SDID seeks to optimally generate a matched control unit +which considerably loosens the need for parallel trend assumptions. Correspondingly, +SDID avoids common pitfalls in standard DID and SC methods – namely an inability +to estimate causal relationships if parallel trends are not met in aggregate data in the +case of DID, and a requirement that the treated unit be housed within a “convex hull” +of control units in the case of SC. Arkhangelsky et al. (2021) propose estimation and +inference procedures, formally proving consistency and asymptotic normality of the +proposed estimator. What’s more, the authors briefly discuss a number of important +applied points such as how their estimator can incorporate covariates, and how their +estimator can be applied to both multiple treatment units, and even multiple treatment +units which adopt treatment in different time periods. +In this paper we describe the sdid command (available for download as Paila˜nir and +Clarke (2022)) which implements the SDID estimator in Stata. This command allows +for the simple implementation of the SDID estimator provided that a panel or repeated +cross section of data is available covering groups and time periods, and which is strongly +balanced. The command, written principally in Mata, seamlessly incorporates cases + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +3 +where there is a single treated unit, multiple treatment units, and multiple treatment +periods. It reports treatment effects laid out in Arkhangelsky et al. (2021), additionally +implementing their proposed bootstrap, jackknife and placebo inference procedures. A +number of graphical output options are provided to examine the generation of the SDID +estimator and the underlying optimal weight matrices. While principally written to +conduct SDID estimation, the sdid command (and the SDID method) nests as possible +estimation procedures SC and DID, which can be easily generated to allow comparison +of estimation procedures and estimates.1 +In introducing the command, we first provide a primer on the core methodological +points of SDID (as well as comparisons to DID and SC), and then describe how these +procedures extend to a setting where treatment adoption occurs over multiple time +periods. We then lay out the command syntax of sdid, as well as the elements which +are returned to the user. We provide a number of examples to illustrate the use of the +SDID method in Stata, both based upon a well-known example of California’s passage +of Proposition 99, an anti-smoking measure previously presented in Abadie et al. (2010); +Arkhangelsky et al. (2021) in which a single state adopts a treatment at a given time, +as well as an example where exposure to a policy occurs at mutiple periods: the case of +parliamentary gender quotas studied by Bhalotra et al. (2022). We conclude by making +a number of practical points on the computational implementation of this estimator. +2 +Methods +2.1 +The Canonical Synthetic Difference-in-Differences Procedure +The synthetic DID procedure, hereafter SDID, is developed in Arkhangelsky et al. +(2021), and we lay out principal details here. +As input, SDID requires a balanced +panel of N units or groups, observed over T time periods. An outcome, denoted Yit, is +observed for each unit i in each period t. Some, but not all, of these observations are +treated with a specific binary variable of interest, denoted Wit. This treatment variable +Wit = 1 if observation i is treated by time t, otherwise, Wit = 0 indicates that unit i is +untreated at time t. Here, we assume that there is a single adoption period for treated +units, which Arkhangelsky et al. (2021) refer to as a ‘block treatment assignment’. In +section 2.3, we extend this to a ‘staggered adoption design’ (Athey and Imbens 2022), +where treated units adopt treatment at varying points. A key element of both of these +designs is that once treated, units are assumed to remain exposed to treatment forever +thereafter. In the particular setting of SDID, no always treated units can be included +in estimation. For estimation to proceed, we require at least two pre-treatment periods +off of which to determine control units. +The goal of SDID is to consistently estimate the causal effect of receipt of policy +1Code from the original paper was provided in R (Hirshberg undated), which can do many of the +procedures which sdid implements, and indeed, abstracting from differences in pseudo-random number +generation, give identical results in cases where similar procedures are possible. A number of useful +extensions are available in sdid, such as the implementation of estimates in cases where treatment +occurs in multiple periods, and alternative manners to include covariates. + +4 +Synthetic Difference In Differences +or treatment Wit, (an average treatment effect on the treated, or ATT) even if we do +not believe in the parallel trends assumption between all treatment and control units +on average. Estimation of the ATT proceeds as follows: +� +�τ sdid, �µ, �α, �β +� += arg min +τ,µ,α,β +� N +� +i=1 +T +� +t=1 +(Yit − µ − αi − βt − Witτ)2�ωsdid +i +�λsdid +t +� +(1) +where the estimand is the ATT, generated from a two-way fixed effect regression, with +optimally chosen weights �ωsdid +i +and �λsdid +t +discussed below. Note that here, this procedure +flexibly allows for shared temporal aggregate factors given the estimation of time fixed +effects βt and time invariant unit-specific factors given the estimation of unit fixed +effects αi. +As is standard in fully saturated fixed-effect models, one αi and one βt +fixed effect are normalized to zero to avoid multi-colinearity. The presence of unit-fixed +effects implies that SDID will simply seek to match treated and control units on pre- +treatment trends, and not necessarily on both pre-treatment trends and levels, allowing +for a constant difference between treatment and control units. +In this setting, it is illustrative to consider how the SDID procedure compares to the +traditional synthetic control method of Abadie et al. (2010), as well as the baseline DID +procedure. The standard DID procedure consists of precisely the same two-way fixed +effect OLS procedure, simply assigning equal weights to all time periods and groups: +� +�τ did, �µ, �α, �β +� += arg min +τ,µ,α,β +� N +� +i=1 +T +� +t=1 +(Yit − µ − αi − βt − Witτ)2 +� +. +(2) +The synthetic control, on the other hand, maintains optimally chosen unit-specific +weights ω (as laid out below), however does not seek to optimally consider time periods +via time weights, and omits unit fixed effects αi implying that the synthetic control and +treated units should maintain approximately equivalent pre-treatment levels, as well as +trends. +� +�τ sc, �µ, �β +� += arg min +τ,µ,β +� N +� +i=1 +T +� +t=1 +(Yit − µ − βt − Witτ)2�ωsc +i +� +(3) +From (2)-(3) it is clear that the SDID procedure offers greater flexibility than both the +DID and SC procedures; in the case of DID by permitting a violation of parallel trends +in aggregate data, and in the case of SC, by both additionally seeking to optimally +weight time periods when considering counterfactual outcomes, and allowing for level +differences between treatment and control groups. +The selection of unit weights, ω, as inputs to (1) (and (3)) seeks to ensure that com- +parison is made between treated units and controls which were approximately following +parallel trends prior to the adoption of treatments. The selection of time weights, λ in +the case of SDID seeks to draw more weight from pre-treatment periods which are more +similar to post-treatment periods, in the sense of finding a constant difference between +each control unit’s post treatment average, and pre-treatment weighted averages across +all selected controls. Specifically, as laid out in Arkhangelsky et al. (2021), unit-specific + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +5 +weights are found by resolving: +� +�ω0, �ωsdid� += +arg min +ω0∈R,ω∈Ω +Tpre +� +t=1 +� +ω0 + +Nco +� +i=1 +ωiYit − +1 +Ntr +N +� +i=Nco+1 +Yit +�2 ++ ζ2Tpre||ω||2 +2(4) +where Ω = +� +ω ∈ RN ++, with +Nco +� +i=1 +ωi = 1 and ωi = +1 +Ntr +for all i = Nco + 1, . . . , N +� +, +||ω||2 refers to the Euclidean norm and ζ is a regularization parameter laid out in +Arkhangelsky et al. (2021, pp. 4091-4092).2 This leads to a vector of Nco non-negative +weights plus an intercept ω0. The weights ωi for all i ∈ {1, . . . , Nco} imply that absolute +difference between control and treatment trends units should be minimized over all pre- +treatment periods, while ω0 initially allows for a constant difference between treatment +and controls over time. Together, these imply that units will follow parallel pre-trends, +though provided ω0 ̸= 0, not identical pre-trends. +In the case of time weights, a similar procedure is followed, finding weights which +minimize the following objective function: +� +�λ0, �λsdid� += +arg min +λ0∈R,λ∈Λ +Nco +� +i=1 +� +�λ0 + +Tpre +� +t=1 +λtYit − +1 +Tpost +T +� +t=Tpre+1 +Yit +� +� +2 ++ ζ2Nco||λ||2(5) +where Λ = +� +� +�λ ∈ RT ++, with +Tpre +� +t=1 +λt = 1 and λt = +1 +Tpost +for all t = Tpre + 1, . . . , T +� +� +� , +where the final term in (5) is a very small regularization term to ensure uniqueness of +time weights, where ζ = 1 × 10−6�σ, and �σ is defined as in footnote 2. +This estimation procedure is summarized in Arkhangelsky et al. (2021, algorithm +1), reproduced in Appendix 1 here for ease of access. Arkhangelsky et al. (2021) also +prove that the estimator is asymptotically normal, suggesting that confidence intervals +on τ can be constructed as: +�τ sdid ± zα/2 +� +�Vτ, +where zα/2 refers to the inverse normal density function at percentile α/2 should one +wish to compute 1-α confidence intervals. These confidence intervals thus simply require +an estimate of the variance of τ, �Vτ. Arkhangelsky et al. (2021) propose three specific +procedures to estimate this variance: a block bootstrap, a jackknife, or a permutation- +based approach. +2For the sake of completion, this regularization parameter is calculated as ζ = (Ntr × Tpost)1/4�σ, +where: +�σ2 = +1 +Nco(Tpre − 1) +Nco +� +i=1 +Tpre−1 +� +t=1 +(∆it− ¯∆)2, ∆it = Yi,(t+1)−Yit, and ¯∆ = +1 +Nco(Tpre − 1) +Nco +� +i=1 +Tpre−1 +� +t=1 +∆it. + +6 +Synthetic Difference In Differences +The block (also known as clustered) bootstrap approach, consists of taking some +large number, B, of bootstrap resamples over units, where units i are the resampled +blocks in the block bootstrap procedure. Provided that a given resample does not con- +sist entirely of treated, or entirely of control units, the quantity �τ sdid is re-estimated, +and denoted as �τ sdid +(b) +for each bootstrap resample. The bootstrap variance �V (b) +τ +is then +calculated as the variance of resampled estimates �τ sdid +(b) +across all B resamples. The boot- +strap algorithm is defined in Arkhangelsky et al. (2021, algorithm 2), reproduced here +in appendix 1. This bootstrap procedure is observed in simulation to have particularly +good properties, but has two particular drawbacks, justifying alternative inference pro- +cedures. The first is that it may be computationally costly, given that in each bootstrap +resample the entire synthetic DID procedure is re-estimated, including the estimation +of optimal weights. This is especially computationally expensive in cases where working +with large samples, or where covariates are included, as discussed at more length below. +The second, is that formal proofs of asymptotic normality rely on the number of treated +units being large, and as such, estimated variance, and confidence intervals, may be +unreliable when the number of treated units is small. +An alternative estimator which significantly reduces the computational burden in- +herent in the bootstrap is estimating a jackknife variance for �τ sdid. This procedure +consists of iterating over all units in the data, in each iteration removing the given +unit, and recalculating �τ sdid, denoted �τ sdid +(−i), maintaining fixed the optimal weights for ω +and λ calculated in the original SDID estimate. The jackknife variance, �V (jack) +τ +is then +calculated based on the variance of all �τ sdid +(−i) estimates, following Arkhangelsky et al. +(2021, Algorithm 3) (refer to Appendix 1 here). In this case, each iteration saves on re- +calculating optimal weights, and as documented by Arkhangelsky et al. (2021), provide +a variance leading to conservative confidence intervals, without the computational bur- +den imposed by the bootstrap. Once again, asymptotic normality relies on there being +a large number of treated units, and in particular if only 1 treated unit is observed – +as is often the case in comparative case studies – the jackknife will not even be defined +given that a �τ sdid +(−i) term will be undefined when removing the single treated unit. +Given limits to these inference options when the number of treated units is small, +an alternative placebo, or permutation-based, inference procedure is proposed. This +consists of, first, conserving just the control units, and then randomly assigning the +same treatment structure to these control units, as a placebo treatment. +Based on +this placebo treatment, we then re-estimate �τ sdid, denoted �τ sdid +(p) . +This procedure is +repeated many times, giving rise to a vector of estimates �τ sdid +(p) , and the placebo variance, +�V (placebo) +τ +, can be estimated as the variance of this vector. This is formally defined in +Arkhangelsky et al. (2021, algorithm 4), and appendix 1 here. It is important to note +that in the case of this placebo-based variance, homoskedasticity across units is required, +given that the variance is based off placebo assignments of treatment made only within +the control group. + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +7 +2.2 +Conditioning on Covariates +So far, we have limited exposition to cases where one wishes to study outcomes Yit, and +their evolution in treated and synthetic control units. However, in certain settings, it +may be of relevance to condition on exogenous time-varying covariates Xit. Arkhangel- +sky et al. (2021) note that in this case, we can proceed by applying the SDID algorithm +to the residuals calculated as: +Y res +it += Yit − Xit �β, +(6) +where �β comes from regression of Yit on Xit. This procedure, in which the synthetic +DID process will be applied to the residuals Y res +it +, is different to the logic of synthetic +controls following Abadie et al. (2010). In Abadie et al.’s conception, when covariates +are included the synthetic control is chosen to ensure that these covariates are as closely +matched as possible between treated and synthetic control units. However in the SDID +conception, covariate adjustment is viewed as a pre-processing task, which removes the +impact of changes in covariates from the outcome Yit prior to calculating the synthetic +control. Along with their paper, Arkhangelsky et al. (2021) provide an implementation +of their algorithm in R (Hirshberg undated), and in practice they condition out these +variables Xit by finding �β within an optimization procedure which additionally allows for +the efficient calculation of optimal weights ω and λ. In the sdid code described below, +we follow Hirshberg (undated) in implementing this efficient optimization procedure (the +Frank and Wolfe (1956) solver), however there are a number of potential complications +which can arise in this manner of dealing with covariates, and as such, alternative +procedures are also available. +A first potential issue is purely numerical. In the Frank-Wolfe solver discussed above, +a minimum point is assumed to be found when successive iterations of the solver lead to +arbitrarily small changes in all parameters estimated.3 Where these parameters include +coefficients on covariates, in extreme cases the solution found for (1) can be sensitive +to the scaling of covariates. This occurs in particular when covariates have very large +magnitudes and variances. In such cases, the inclusion of covariates in (1) can cause +optimization routines to suggest solutions which are not actually globally optimal, given +that successive movements in �β can be very small. In extreme cases, this can imply that +when multiplying all variables Xit by a large constant values, the estimated treatment +effect can vary. While this issue can be addressed by using smaller tolerances for defining +stopping rules in the optimization procedure, it can be addressed in a more simple way +if all covariates are first re-standardized as Z-scores, implying that no very-high-variance +variables are included, while capturing the same underlying variation in covariates. +A second, potentially more complicated point is described by Kranz (2022). +He +notes that in certain settings, specifically where the relationship between covariates +and the outcome vary over time differentially in treatment and control groups, the +procedure described above may fail to capture the true relationship between covariates +and the outcome of interest, and may subsequently lead to bias in estimated treatment +3In the case of convex functions such as that in (1), the Frank-Wolfe solver finds a global minima, +see for example Lawphongpanich (2009). + +8 +Synthetic Difference In Differences +effects. He proposes a slightly altered methodology of controlling for covariates. Namely, +his suggestion is to first estimate a two-way fixed effect regression of the dependent +variable on covariates (plus time and unit fixed effects), using only observations which +are not exposed, or not exposed yet, to treatment, i.e. sub-setting to observations for +which Wit = 0. +Based on the regression Yit = Xitβ + µi + λt + εit for units with +Wit = 0, coefficients �β estimated by OLS can be used to follow the procedure in (6), +and SDID can then be conducted. An additional benefit of this method is that unlike +the optimized method described previously, �β is calculated in a single step via OLS +rather than an interative optimization procedure, which can lead to substantial speed +ups in computation time. +We document these methods based on a number of empirical example in the fol- +lowing section. +It is worth noting that regardless of which procedure one uses, the +implementation of SDID follows the suggestion laid out in Arkhangelsky et al. (2021, +footnote 4) and (6) above. What varies between the former and the latter procedures +discussed here is the manner with which one estimates coefficients on covariates, �β. +2.3 +The Staggered Adoption Design +The design discussed up to this point assumes block assignment, or that all units are +either controls, or are treated in a single unit of time. However, in Arkhangelsky et al. +(2021, Appendix A), they note that this procedure can be extended to a staggered +adoption design, where treated units adopt treatment at varying moments of time. Here +we lay out the formal details related to the staggered adoption design, focusing first on +estimation of an aggregate treatment effect, and then on extending the three inference +procedures laid out previously into a staggered adoption setting. This proposal is one +potential way to deal with staggered adoption settings, though there are other possible +manners to proceed – see for example Ben-Michael et al. (2021), or Arkhangelsky et al. +(2021, Appendix A). In cases where the number of pure control units is small, this +proposal may not necessarily be very effective given challenges in finding appropriate +counterfactuals for each adoption-specific period. +Estimation Unlike the block assignment case where a single pre- versus post-treatment +date can be used to conduct estimation, in the staggered adoption design, multiple +adoption dates are observed. Consider for example the treatment matrix below, con- +sisting of 8 units, 2 of which (1 and 2) are untreated, while the other 6 are treated, + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +9 +however at varying points. +W = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +t1 +t2 +t3 +t4 +t5 +t6 +t7 +t8 +1 +0 +0 +0 +0 +0 +0 +0 +0 +2 +0 +0 +0 +0 +0 +0 +0 +0 +3 +0 +0 +0 +0 +0 +0 +1 +1 +4 +0 +0 +0 +0 +0 +0 +1 +1 +5 +0 +0 +0 +1 +1 +1 +1 +1 +6 +0 +0 +0 +1 +1 +1 +1 +1 +7 +0 +0 +1 +1 +1 +1 +1 +1 +8 +0 +0 +1 +1 +1 +1 +1 +1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +This single staggered treatment matrix W can be broken down into adoption date +specific matrices, W1, W2 and W3, or generically, W1, . . . , WA, where A indicates +the number of distinct adoption dates. Additionally, a row vector A consisting of A +elements contains these distinct adoption periods. In this specific setting where units +first adopt treatment in period t3 (units 7 and 8), t4 (units 5 and 6), and t7 (units 3 +and 4), the adoption date vector consists simple of periods 3, 4 and 7. +A = +�3 +4 +7� +Finally, adoption-specific matrices W1-W3 simply consist of pure treated units, and +units which adopt in this specific period, as below: +W1 = +� +� +� +� +� +� +t1 +t2 +t3 +t4 +t5 +t6 +t7 +t8 +1 +0 +0 +0 +0 +0 +0 +0 +0 +2 +0 +0 +0 +0 +0 +0 +0 +0 +3 +0 +0 +0 +0 +0 +0 +1 +1 +4 +0 +0 +0 +0 +0 +0 +1 +1 +� +� +� +� +� +� +, +W2 = +� +� +� +� +� +� +t1 +t2 +t3 +t4 +t5 +t6 +t7 +t8 +1 +0 +0 +0 +0 +0 +0 +0 +0 +2 +0 +0 +0 +0 +0 +0 +0 +0 +5 +0 +0 +0 +1 +1 +1 +1 +1 +6 +0 +0 +0 +1 +1 +1 +1 +1 +� +� +� +� +� +� +, +W3 = +� +� +� +� +� +� +t1 +t2 +t3 +t4 +t5 +t6 +t7 +t8 +1 +0 +0 +0 +0 +0 +0 +0 +0 +2 +0 +0 +0 +0 +0 +0 +0 +0 +7 +0 +0 +1 +1 +1 +1 +1 +1 +8 +0 +0 +1 +1 +1 +1 +1 +1 +� +� +� +� +� +� +. +As laid out in Arkhangelsky et al. (2021, Appendix A), the average treatment effect +on the treated can then be calculated by applying the synthetic DID estimator to each +of these 3 adoption-specific samples, and calculating a weighted average of the three +estimators, where weights are assigned based on the relative number of treated units + +10 +Synthetic Difference In Differences +and time periods in each adoption group. Generically, this ATT is calculated based on +adoption-specific SDID estimates as: +� +ATT = +� +for a∈A +T a +post +Tpost +× ˆτ sdid +a +(7) +where Tpost refers to the total number of post-treatment periods observed in treated +units. This estimation procedures is laid out formally in Algorithm 1 below. +Note that in this case, while the parameter interest is likely the treatment effect +ATT or adoption specific τ sdid +a +parameters, each adoption period is associated with an +optimal unit and time weight vector ωsdid +a +and λsdid +a +, which can be returned following +estimation. +Algorithm 1: Estimation of the ATT with staggered adoption +Data: Y, W, A. +Result: Point estimate � +ATT and adoption-specific values ˆτ sdid +a +, ˆωsdid +a +and ˆλsdid +a +for all a ∈ A. +for a ∈ A do +1. Subset Y and W to units who are pure controls, and who first adopt +treatment in period t = a. ; +2. Compute regularization parameter ζ ; +3. Compute unit weights ˆωsdid +a +; +4. Compute time weights ˆλsdid +a +; +5. Compute the SDID estimator via the weighted DID regression ; +� +ˆτ sdid +a +, ˆµa, ˆαa, ˆβa +� += arg min +τ,µ,α,β +� N +� +i=1 +T +� +t=1 +(Yit − µ − αi − βt − Witτ)2�ωsdid +a,i ˆλsdid +a,t +� +end +6. Compute ATT across adoption-specific SDID estimates +� +ATT = +� +for a∈A +T a +post +Tpost +× ˆτ sdid +a +Inference In the staggered adoption design, estimated treatment effects are simply a +multi-period extension of the underlying SDID algorithm, in each case working with +the relevant pure control and treated sub-sample. Thus, inference can be conducted in +the staggered adoption design under similar resample or placebo procedures. Here we +discuss inference following each of the bootstrap, jackknife, or placebo procedures laid +out in Arkhangelsky et al. (2021), applied to a staggered adoption setting. We note that + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +11 +in this design, it is likely the case that one wishes to conduct inference on the treatment +effect, ATT from (7). Thus in the below, we propose inference details for this estimand, +additionally noting that standard errors and confidence intervals on adoption-specific +SDID parameters τ sdid +a +come built-in as part of these procedures. +Consider first the case of bootstrap inference. Suppose that one wishes to estimate +standard errors or generate confidence intervals on the global treatment effect ATT. A +bootstrap procedure can be conducted based on many clustered bootstrap resamples +over the entire initial dataset, where in each case, a resampled ATT estimate � +ATT +b +is generated, following Algorithm 1. +Based on many such resampled estimates, the +bootstrap variance can be calculated as the variance of these resamples. We lay out the +bootstrap variance estimate below in Algorithm 2. Note that as in the block treatment +Algorithm 2: Bootstrap inference in the staggered adoption setting +Data: Y, W, A, B. +Result: Variance estimator �V cb +AT T . Additionally, variance for each adoption +specific estimate �V cb +τa for all a ∈ A. +for b ← 1 to B do +1. Construct a bootstrap dataset (Y(b), W(b), A(b)) by sampling N rows of +(Y, W) with replacement, and generating A as the corresponding adoption +vector ; +2. if the bootstrap sample has no treated units or no control units then +Discard resample and go to 1 ; +end +3. Compute SDID estimate ATT (b) following Algorithm 1 based on +(Y(b), W(b), A(b)). Similarly, generate a vector of adoption-date specific +resampled SDID estimates τ (b) +a +for all a ∈ A(b) ; +end +4. Define �V cb +AT T = 1 +B +�B +b=1 +� � +ATT (b) − 1 +B +�B +b=1 � +ATT (b) +�2 +. Similarly, estimate +adoption-date specific variances for each τ sdid +a +estimate as the variance over +each τ (b) +a ; +design, this bootstrap procedure requires the number of treated units to grow with N +within each adoption period. As such, if a very small number of treated units exist +for certain adoption periods, placebo inference is likely preferable. Similarly, as laid +out in the block treatment design, the bootstrap procedure re-estimates optimal weight +matrices in each resample, and can be computationally expensive in cases where samples +are large. +An alternative inference procedure, which is less computationally intensive but sim- +ilarly based on asymptotic arguments with a large number of states, and many treated +units, is based on the jackknife. +Here, optimal weight matrices calculated for each + +12 +Synthetic Difference In Differences +adoption-specific estimate τ sdid +a +in Algorithm 1 are treated as fixed, and provided as +inputs to a jackknife procedure, described below in Algorithm 3. Below, these matrices, +which consist of weights for each adoption period a ∈ A, are denoted as ω, λ.4 Note +that in Algorithm 3, notation (−i) refers to a standard jackknife estimator, removing +a single state (i) in each iteration. In cases where i refers to a treated unit, the ATT +will be calculated removing this particular treated unit. For this reason, the jackknife +estimator will not be defined in cases where any single adoption period has only one +treated unit, as in this case, �τ (−i) +a +will not be defined. +Algorithm 3: Jackknife inference in the staggered adoption setting +Inputs: Y, W, A, �ω, �λ, � +ATT. +Result: Variance estimator �VAT T +for i ← 1 to N do +1. Compute SDID estimate ATT (−i) following Algorithm 1 based on +(Y(−i), W(−i), A). Similarly, generate a vector of adoption-date specific +resampled SDID estimates τ (−i) +a +for all a ∈ A; +end +2. Compute �V jack +AT T = (N − 1)N −1 �N +i=1 +� +� +ATT +(−i) − � +ATT +�2 +; +Finally, in cases where the number of treated units is small, and concerns related to +the validity of the previous variance estimators exists, the placebo inference procedure +defined in algorithm 4 can be used. Here, this is defined for the staggered adoption case, +generalising Algorithm 4 of Arkhangelsky et al. (2021). +To conduct this procedure, +placebo treatments are randomly assigned based on the actual treatment structure, +however only to the control units. Based on these placebo assignments, placebo values +for ATT are generated, which can be used to calculate the variance as laid out in +Algorithm 4. It is important to note that such a procedure will only be feasible in cases +where the number of control units is strictly larger than the number of treated units +(or hence placebo assignments will not be feasible), and, as laid out in Arkhangelsky +et al. (2021); Conley and Taber (2011), such a procedure relies on homoskedasticity +across units, as otherwise the variance of the treatment effect on the treated could not +be inferred from variation in assignment of placebo treatments to control units. +4In the case of ω, this is an A × Nco matrix, where for each row a, the weight assigned to each +control unit in this particular adoption period is recorded. In the case of λ, this is a matrix containing +A columns, where each column consists of as many rows as the number of pre-treatment periods to +this specific adoption date. In each cell, the weight assigned to a particular pre-treatment year for each +adoption period is recorded. + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +13 +Algorithm 4: Placebo inference in the staggered adoption +Inputs: Yco, Ntr, B. +Result: Variance estimator �V placebo +AT T +for b ← 1 to B do +1. Sample Ntr out of the Nco control units without replacment to ‘receive +the placebo’ ; +2. Construct a placebo treatment matrix W(b) +co , for the controls ; +3. Compute SDID estimate ATT (b) following algorithm 1 based on +(Yco, W(b) +co , A(b)) ; +end +4. Define �V placebo +AT T += 1 +B +�B +b=1 +� � +ATT (b) − 1 +B +�B +b=1 � +ATT (b) +�2 +; +3 +The sdid command +Synthetic Difference-in-Differences can be implemented in Stata using the following +command syntax: +sdid depvar groupvar timevar treatment +� +if +� � +in +� +, vce(type) +� +covariates(varlist, type) seed(#) reps(#) method(type) graph g1on +g1 opt(string) g2 opt(string) graph export(string, type) +msize(markersizestyle) unstandardized mattitles +� +where depvar describes the dependent variable in a balanced panel of units (group- +var) and time periods (timevar). The variable which indicates units which are treated +at each time period, which accumulates over time, is indicated as treatment. Note that +here, it is not necessary for users to specify whether the design is a block or staggered +adoption design, as this will be inferred based off the data structure. Optionally, if and +in can be specified, provided that this does not result in imbalance in the panel. Re- +quired and permitted options are discussed below, followed by a description of objects +returned by sdid. +Options +vce(type) is a required option. This must be one of bootstrap, jackknife, placebo or +noinference, where in each case inference proceeds following the specified method. In +the case of bootstrap, this is only permitted if greater than one unit is treated. In +the case of jackknife, this is only permitted if greater than one unit is treated in each +treatment period (if multiple treatment periods are considered). In the case of placebo, +this requires at least one more control than treated unit to allow for permutations + +14 +Synthetic Difference In Differences +to be constructed. In each case, inference follows the specific algorithm laid out in +Arkhangelsky et al. (2021). We allow the no inference option (noinference) should one +wish to simply generate the point estimator. This is useful if you wish to plot outcome +trends without the added computational time associated with inference procedures. +covariates(varlist, type) Covariates should be included as a varlist, and if specified, +treatment and control units will be adjusted based on covariates in the synthetic +difference-in-differences procedure. Optionally, type may be specified, which indicates +how covariate adjustment will occur. +If the type is indicated as “optimized” (the +default) this will follow the method described in Arkhangelsky et al. (2021), footnote +4, where SDID is applied to the residuals of all units after regression adjustment. +However, this has been observed to be problematic at times (refer to Kranz (2022)), +and is also sensitive to optimization if covariates have high dispersion. +Thus, an +alternative type is implemented (“projected”), which consists of conducting regression +adjustment based on parameters estimated only in untreated units. This type follows +the procedure proposed by Kranz (2022) (xsynth in R), and is observed to be more +stable in some implementations (and at times, considerably faster). sdid will run +simple checks on the covariates indicated and return an error if covariates are constant, +to avoid multicolineality. +However, prior to running sdid, you are encouraged to +ensure that covariates are not perfectly multicolinear with other covariates and state +and year fixed effects, in a simple two-way fixed effect regression. If perfectly multi- +colinear covariates are included sdid will execute without errors, however where type is +“optimized”, the procedure may be sensitive to the inclusion of redundant covariates. +seed(#) Define the seed for pseudo-random numbers. +reps(#) Set the number of repetitions used in the calculation of bootstrap and placebo +standard errors. Default is 50 repetitions. Larger values should be preferred where +possible. +method(type) this option allows you to change the estimation method. The type must +be one of sdid, did or sc, where sdid refers to synthetic difference-in-differences, sc +refers to synthetic control, and did refers to difference-in-differences. By default, sdid +is enabled. +graph if this option is specified, graphs will be displayed showing unit and time weights +as well as outcome trends as per Figure 1 from Arkhangelsky et al. (2021). +g1on this option activates the unit-specific weight graph. By default g1 is off. +g1 opt(string) option to modify the appearance of the unit-specific weight graph. These +options adjust the underlying scatter plot, so should be consistent with twoway scatter +plots. +g2 opt(string) option to modify the appearance of the outcome trend graphs. These +options adjust the underlying line plot, so should be consistent with twoway line plots. +graph export(string, type) Graphs will be saved as weightsYYYY and trendsYYYY +for each of the unit-specific weights and outcome trends respectively, where YYYY +refers to each treatment adoption period. +Two graphs will be generated for each + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +15 +treatment adoption period provided that g1on is specified, otherwise a single graph +will be generated for each adoption period. If this option is specified, type must be +specified, which refers to a valid Stata graph type (eg “.eps”, “.pdf”, and so forth). +Optionally, a stub can be specified, in which case this will be prepended to exported +graph names. +msize(markersizestyle) allows you to modify the size of the marker for graph 1. +unstandardized if controls are included and the “optimized” method is specified, con- +trols will be standardized as Z-scores prior to finding optimal weights. This avoids +problems with optimization when control variables have very high dispersion. If un- +standardized is specified, controls will simply be entered in their original units. This +option should be used with care. +mattitles requests labels to be added to the returned e(omega) weight matrix provid- +ing names (in string) for the unit variables which generate the synthetic control group +in each case. If mattitles is not indicated, the returned weight matrix (e(omega)) will +store these weights with a final column providing the numerical ID of units, where +this numerical ID is either taken from the unit variable (if this variable is a numerical +format), or arranged in alphabetical order based on the unit variable, if this variable +is in string format. +Returned Objects +sdid stores the following in e(): +Scalars: +e(ATT) +Average Treatment Effect on the Treated +e(se) +Standard error for the ATT +e(reps) +Number of bootstrap/placebo replications +e(N clust) +Number of clusters +Macros: +e(cmd) +sdid +e(cmdline) +command as typed +e(depvar) +name of dependent variable +e(vce) +vcetype specified in vce() +e(clustvar) +name of cluster variable +Matrices: + +16 +Synthetic Difference In Differences +e(tau) +tau estimator for each adoption time-period +e(lambda) +lambda weights (time-specific weights) +e(omega) +omega weights (unit-specific weights) +e(adoption) +adoption times +e(beta) +beta vector corresponding to covariates +e(series) +control and treatment series of the graphs (only returned +when the graph option is indicated) +e(difference) +difference between treatment and control series (only returned +when the graph option is indicated) +4 +Examples based on an Empirical Application +In the sections below we provide a number of illustrations of the usage of, and per- +formance of, the sdid command, which operationalizes the Synthetic Difference-in- +Differences estimator in Stata. We consider both a block treatment design (with a +single adopting state), and a staggered adoption design, noting a number of points +covering estimation, inference, and visualization. +4.1 +A Block Design +In the first case, we consider the well-known example, also presented in Arkhangelsky +et al. (2021), of California’s “Proposition 99” tobacco control measure. This exam- +ple, based on the context described in Abadie et al. (2010) and data of Orzechowski +and Walker (2005), is frequently used to illustrate synthetic control style methods. +Proposition 99, which was passed by California in 1989, increased the taxes paid on a +packet of cigarettes by 25 cents. The impact of this reform is sought to be estimated +by comparing the evolution of sales of cigarettes in packs per capita in California (the +treated state) with those in 38 untreated states, which did not significantly increase +cigarette taxes during the study period. +The data used in analysis cover each of these 39 states over the period of 1970– +2000, with a single observation for each state and year. Adoption occurs in California +in 1989, implying Tpre = 19 pre-treatment periods and Tpost = 12 post-treatment +periods. There are Nco = 38 control and a single treated state, hence Ntr = 1. Using +the sdid command, we replicate the results from Arkhangelsky et al. (2021). In the +below code example, we first download the data, and then conduct the Synthetic +Difference-in-Differences implementation using a placebo inference procedure with (a +default) 50 placebo iterations. +. webuse set www.damianclarke.net/stata/ +. webuse prop99_example.dta, clear +. sdid packspercapita state year treated, vce(placebo) seed(1213) +Placebo replications (50). This may take some time. +----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 +.................................................. +50 + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +17 +Synthetic Difference-in-Differences Estimator +packsperca~a +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +treated +-15.60383 +9.53183 +-1.64 +0.102 +-34.28588 +3.07822 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +The third line of this code excerpt quite simply implements the synthetic difference- +in-differences estimator, returning identical point estimates to those documented in +Table 1 of Arkhangelsky et al. (2021). Standard errors are slightly different, as these +are based on pseudo-random placebo reshuffling, though can be replicated as presented +here provided that the same seed is set in the seed option. Note that in this case, +given that a small number (1) of treated units is present, placebo inference is the only +appropriate procedure, as indicated in the vce() option. +-110 +-90 +-70 +-50 +-30 +-10 +10 +30 +50 +Difference +Alabama +Arkansas +Colorado +Connecticut +Delaware +Georgia +Idaho +Illinois +Indiana +Iowa +Kansas +Kentucky +Louisiana +Maine +Minnesota +Mississippi +Missouri +Montana +Nebraska +Nevada +New Hampshire +New Mexico +North Carolina +North Dakota +Ohio +Oklahoma +Pennsylvania +Rhode Island +South Carolina +South Dakota +Tennessee +Texas +Utah +Vermont +Virginia +West Virginia +Wisconsin +Wyoming +(a) Unit-Specific Weights +0 +25 +50 +75 +100 +125 +150 +Packs per capita +1970 +1980 +1990 +2000 +Year +Control +Treated +(b) Outcome Trends and Time-Specific Weights +Figure 1: Proposition 99, example from Abadie et al. (2010); Arkhangelsky et al. (2021) +Should we wish to generate the same graphs as in Arkhangelsky et al. (2021), summa- +rizing both (a) unit specific weights, and (b) treatment and synthetic control outcome +trends along with time specific weights, this can be requested with the addition of the +graph option. This is displayed below, where we additionally modify plot aesthetics +via the g1 opt() and g2 opt() options for weight graphs (Figure 1(a)), and trend +graphs (Figure 1(b)) respectively. Finally, generated graphs can be saved to disk us- +ing the graph export() option, with a graph type (.eps below), and optionally a +pre-pended plot name. Output corresponding to the below command is provided in +Figure 1. +. sdid packspercapita state year treated, vce(placebo) seed(1213) graph g1on +> +g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) +> +g1_opt(xtitle("") scheme(sj)) g1on graph_export(sdid_, .eps) + +18 +Synthetic Difference In Differences +It is illustrative to compare the output of SDID estimation procedures with those of +standard synthetic control methods of Abadie et al. (2010), and unweighted difference- +in-difference estimates. By using the method() option one can request a standard +difference-in-differences output, requested with method(did), or synthetic control out- +put, requested with method(sc). In the interests of completeness, method(sdid) is +also accepted, although this is the default behaviour when method is not included in +command syntax. In each case, resulting graphs document matched treated and con- +trol/synthetic control trends, as well as weights received by each unit and time period. +These are displayed in Figure 2, with plots corresponding to each of the three calls +to sdid displayed below. In the left-hand panel, identical SDID plots are provided as +those noted above. In the middle plot, corresponding to method(did), a difference- +in-difference setting is displayed. Here, in the top panel, outcomes for California are +displayed as a solid line, while mean outcomes for all control states are documented as +a dashed line, where a clear divergence is observed in the pre-treatment period. The +bottom panel shows that in this case, each control unit receives an identical weight, +while time weights indicated at the base of the top plot note that each period is +weighted identically. Finally, in the case of synthetic control, output from the third +call to sdid is provided in the right-hand panel. In this case, treated and synthetic +control units are observed to overlap nearly exactly, with weights in figure (f) noted +to be more sparse, and placing relatively more weight on fewer control states. We +note that in each case, the vce(noinference) option is used, as here we are simply +interested in observing exported graphs, not the entire command output displaying +aggregate estimates, standard errors and confidence intervals. +sdid packspercapita state year treated, method(sdid) vce(noinference) graph +> +g1_opt(ylabel(-110(20)50) xtitle("") scheme(sj)) g1on +> +g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) +> +graph_export(sdid_, .eps) +sdid packspercapita state year treated, method(did) vce(noinference) graph msize(small) +> +g1_opt(ylabel(-110(20)50) xtitle("") scheme(sj)) g1on +> +g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) +> +graph_export(did_, .eps) +sdid packspercapita state year treated, method(sc) vce(noinference) graph +> +g1_opt(ylabel(-110(20)50) xtitle("") scheme(sj)) g1on +> +g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) +> +graph_export(sc_, .eps) +The sdid command returns multiple matrices containing treatment and control out- +come trends, weights, and other elements. These elements can be accessed simply +for use in post-estimation procedures or graphing. As a simple example, the follow- +ing code excerpt accesses treatment and synthetic control outcome trends (stored in +e(series), and time weights (stored in e(lambda)) and uses these elements to repli- +cate the plot presented in Figure 1b. Thus, if one wishes to have further control over +the precise nature of plotting, beyond that provided in the graphing options available +in sdid’s command syntax, one can simply work with elements returned in the ereturn +list. In Appendix 2, we show that with slightly more effort, returned elements can be + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +19 +0 +25 +50 +75 +100 +125 +150 +Packs per capita +1970 +1980 +1990 +2000 +Year +Control +Treated +(a) SDID: Outcome Trends +0 +25 +50 +75 +100 +125 +150 +Packs per capita +1970 +1980 +1990 +2000 +Year +Control +Treated +(b) DID: Outcome Trends +0 +25 +50 +75 +100 +125 +150 +Packs per capita +1970 +1980 +1990 +2000 +Year +Control +Treated +(c) SC: Outcome Trendss +-110 +-90 +-70 +-50 +-30 +-10 +10 +30 +50 +Difference +Alabama +Arkansas +Colorado +Connecticut +Delaware +Georgia +Idaho +Illinois +Indiana +Iowa +Kansas +Kentucky +Louisiana +Maine +Minnesota +Mississippi +Missouri +Montana +Nebraska +Nevada +New Hampshire +New Mexico +North Carolina +North Dakota +Ohio +Oklahoma +Pennsylvania +Rhode Island +South Carolina +South Dakota +Tennessee +Texas +Utah +Vermont +Virginia +West Virginia +Wisconsin +Wyoming +(d) SDID: Unit-Specific Weights +-110 +-90 +-70 +-50 +-30 +-10 +10 +30 +50 +Difference +Alabama +Arkansas +Colorado +Connecticut +Delaware +Georgia +Idaho +Illinois +Indiana +Iowa +Kansas +Kentucky +Louisiana +Maine +Minnesota +Mississippi +Missouri +Montana +Nebraska +Nevada +New Hampshire +New Mexico +North Carolina +North Dakota +Ohio +Oklahoma +Pennsylvania +Rhode Island +South Carolina +South Dakota +Tennessee +Texas +Utah +Vermont +Virginia +West Virginia +Wisconsin +Wyoming +(e) DID: Unit-Specific Weights +-110 +-90 +-70 +-50 +-30 +-10 +10 +30 +50 +Difference +Alabama +Arkansas +Colorado +Connecticut +Delaware +Georgia +Idaho +Illinois +Indiana +Iowa +Kansas +Kentucky +Louisiana +Maine +Minnesota +Mississippi +Missouri +Montana +Nebraska +Nevada +New Hampshire +New Mexico +North Carolina +North Dakota +Ohio +Oklahoma +Pennsylvania +Rhode Island +South Carolina +South Dakota +Tennessee +Texas +Utah +Vermont +Virginia +West Virginia +Wisconsin +Wyoming +(f) SC: Unit-Specific Weights +Figure 2: Comparison of estimators +used to construct the unit-specific weight plot from Figure 1a. +. preserve +. clear +. matrix series=e(series) +. matrix lambda=e(lambda) +. qui svmat series, names(col) +. qui svmat lambda +. tw line Yco1989 year, yaxis(1) || +> +line Ytr1989 year, yaxis(1) || +> +bar lambda1 year if year<=1988, yaxis(2) ylabel(0(1)5, axis(2)) yscale(off axis(2)) +> +xline(1989, lc(red)) legend(order(1 "Control" 2 "Treated") pos(12) col(2)) scheme(sj) +. graph export sdid replicate.eps, replace +. restore +4.2 +A Staggered Adoption Design +We present an example of a staggered adoption design, based on data and the context +studied in Bhalotra et al. (2022). In this case, the impact of parliamentary gender +quotas which reserve seats for women in parliament are estimated, first on rates of +women in parliament, and secondly on rates of maternal mortality. This is conducted +on a country by year panel, where for each of 1990-2015, 115 countries are observed, 9 + +20 +Synthetic Difference In Differences +50 +100 +150 +1970 +1980 +1990 +2000 +year +Control +Treated +Figure 3: Outcome Trends and Time-Specific Weights +of which implement a parliamentary gender quota.5 For each of these countries, data +on the rates of women in parliament and the maternal mortality ratio are collected, +as well as a number of covariates. +This example presents a staggered adoption configuration, given that in the period +under study, quota adoption occurred in seven different yearly periods between 2000 +and 2013. sdid handles a staggered adoption configuration seamlessly without any +particular changes to the syntax. +In the code below, we implement the synthetic +difference-in-differences estimator using the bootstrap procedure to calculate standard +errors. +The output by default reports the weighted ATT which is defined in (7) +above. +However, as laid out in (7), this is based on each adoption-period specific +synthetic difference-in-differences estimate. These adoption-period specific estimates +are returned in the matrix e(tau), which is tabulated below the standard command +output. +. webuse quota_example.dta, clear +. sdid womparl country year quota, vce(bootstrap) seed(1213) +Bootstrap replications (50). This may take some time. +----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 +.................................................. +50 +5This is a subset of the full sample studied in Bhalotra et al. (2022). +Here we only work with +countries for which observations of women in parliament and maternal mortality exist for the full time +period, without missing observations. + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +21 +Synthetic Difference-in-Differences Estimator +womparl +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +quota +8.03410 +3.74040 +2.15 +0.032 +0.70305 +15.36516 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +. matlist e(tau) +Tau +Time +r1 +8.388868 +2000 +r2 +6.967746 +2002 +r3 +13.95226 +2003 +r4 +-3.450543 +2005 +r5 +2.749036 +2010 +r6 +21.76272 +2012 +r7 +-.8203235 +2013 +All other elements are identical to those documented in the case of a single adoption +period, however generalised to multiple adoptions. For example, if requesting graphi- +cal output, a single treatment versus synthetic control trend graph and corresponding +unit-level weight graph is provided for each adoption date. Similarly, ereturned matri- +ces such as e(lambda), e(omega) and e(series) provide columns for each particular +adoption period. +Adding Covariates As laid out in Section 2.2, covariates can be handled in synthetic +difference-in-differences in a number of ways. Below we document the inclusion of a +single covariate (the natural logarithm of GDP per capita). As sdid is based on a +balanced panel of observations, we must first ensure that there are no missing observa- +tions for all covariates, in this case dropping a small number of (control) countries for +which this measure is not available. We then include covariates via the covariates() +option. In the first case, this is conducted exactly following the procedure discussed +by Arkhangelsky et al. (2021), in which parameters on covariates are estimated within +the optimization routines in Mata. +This is analogous to indicating covariates(, +optimized). Estimates in this particular case suggest that the inclusion of this con- +trol does little to dampen effects. After estimation, the coefficients on the covariates +can be inspected as part of e(beta), where an adoption-specific value for each co- +variate is provided, given that the underlying SDID estimate is calculated for each +adoption period. +. drop if lngdp==. +. sdid womparl country year quota, vce(bootstrap) seed(1213) covariates(lngdp) +Bootstrap replications (50). This may take some time. +----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 +.................................................. +50 +Synthetic Difference-in-Differences Estimator + +22 +Synthetic Difference In Differences +womparl +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +quota +8.05150 +3.09252 +2.60 +0.009 +1.99027 +14.11272 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +The inclusion of covariates in the previous implementation adds considerably to the +computational time as it increases the complexity of the underlying optimization rou- +tine, and this is conducted in each adoption period and each bootstrap replicate. An +alternative manner to capture covariates described in section 2.2 above is that of Kranz +(2022), where the impact of covariates are projected out using a baseline regression of +the outcome on covariates and fixed effects only in units where the treatment status +is equal to zero. This is implemented as below, with covariates(, projected). +. sdid womparl country year quota, vce(bootstrap) seed(1213) covariates(lngdp, projected) +Bootstrap replications (50). This may take some time. +----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 +.................................................. +50 +Synthetic Difference-in-Differences Estimator +womparl +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +quota +8.05927 +3.11913 +2.58 +0.010 +1.94589 +14.17264 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +Here, results are slightly different, though quantitatively comparable to those when +using alternative procedures for conditioning out covariates. In this case, if examining +the e(beta) matrix, only a single coefficient will be provided, as the regression used to +estimate the coefficient vector is always based on the same sample. This additionally +offers a non-trivial speed up in the execution of the code. For example, on a particular +personal computer with Stata SE 15.1 and relatively standard specifications, using +the optimized method above requires 324 seconds of computational time while using +projected requires 61 seconds (compared with 58 seconds where covariates are not +included in sdid). +Post-Estimation Commands While sdid provides a standard tabular and graphical +output as displayed previously, the command can used to provide output in alternative +formats. For example, the sdid command interacts seamlessly with routines such as +estout (Jann 2004) for the exportation of results tables. To see this, the below block +of code estimates three specific versions of the model discussed above, storing each +model using an eststo: prefix, before finally exporting estimated ATTs and standard +errors to a LaTeX file, which can be included in tabular form as displayed in Table 1. +Similar such procedures could be conducted with routines such as outreg or outreg2, + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +23 +and tabular output could be further enriched using additional options within esttab +if desired. +. webuse set www.damianclarke.net/stata/ +. webuse quota_example.dta, clear +. lab var quota "Parliamentary Gender Quota" +. eststo sdid_1: sdid womparl country year quota, vce(bootstrap) seed(2022) +. drop if lngdp==. +. eststo sdid_2: sdid womparl country year quota, vce(bootstrap) seed(2022) +> +covariates(lngdp, optimized) +. eststo sdid_3: sdid womparl country year quota, vce(bootstrap) seed(2022) +> +covariates(lngdp, projected) +. esttab sdid_1 sdid_2 sdid_3 using "example1.tex", +> +nonotes nomtitles stats(N, labels("Observations") fmt(%9.0fc)) +> +addnotes("* p$<$0.10, ** p$<$0.05, *** p$<$0.01") +> +starlevel ("*" 0.10 "**" 0.05 "***" 0.01) lab +> +b(%-9.3f) se(%-9.3f) style(tex) replace +(1) +(2) +(3) +Parliamentary Gender Quota +8.034** +8.051*** +8.059*** +(3.940) +(3.047) +(3.099) +Observations +3,094 +2,990 +2,990 +* p<0.10, ** p<0.05, *** p<0.01 +Table 1: Tabular Output Following sdid +4.3 +Inference Options +In this section we provide examples of the implementation of alternative inference +options, as laid out in algorithms 2-4. For this illustration we will keep only treated +units which adopt gender quotas in 2002 and 2003, as otherwise adoption periods +will exist in which only a single unit is treated, and jackknife procedures will not be +feasible. +. webuse quota example.dta, clear +. drop if country=="Algeria" +| country=="Kenya" | country=="Samoa" | +> +country=="Swaziland" | country=="Tanzania" +In the following three code blocks we document bootstrap, placebo and jackknife +inference procedures. The difference in implementation in each case is very minor, +simply indicating either bootstrap, placebo or jaccknife in the vce() option. For +example, in the case of bootstrap inference, where block bootstraps over the variable +country are performed, the syntax is as follows: +. sdid womparl country year quota, vce(bootstrap) seed(1213) + +24 +Synthetic Difference In Differences +Bootstrap replications (50). This may take some time. +----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 +.................................................. +50 +Synthetic Difference-in-Differences Estimator +womparl +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +quota +10.33066 +4.72911 +2.18 +0.029 +1.06178 +19.59954 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +By default, only 50 bootstrap replicates are performed, though in practice, a sub- +stantially higher number should be used, and this can be indicated in the reps(#) +option. In the case of placebo, the syntax and output are virtually identical. The +suitability of each method depends on the underlying structure of the panel, and in +this particular case, given the relatively small number of treated units, it may be the +case that placebo procedures are preferred. +. sdid womparl country year quota, vce(placebo) seed(1213) +Placebo replications (50). This may take some time. +----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 +.................................................. +50 +Synthetic Difference-in-Differences Estimator +womparl +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +quota +10.33066 +5.14741 +2.01 +0.045 +0.24191 +20.41941 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +Finally, in the interests of completeness, the jackknife procedure, which is by far +the fastest of the three to execute6, is provided below. +Note that unlike the case +with placebo or bootstrap inference, it is not necessary (or relevant) to set a seed, +nor indicate the number of replications, as the jackknife procedure implies conducting +a leave-one-out procedure over each unit. In this particular case, jackknife inference +appears to be more conservative than bootstrap procedures, in line with what may be +expected based on Arkhangelsky et al. (2021)’s demonstration that jackknife inference +is in general conservative. +. sdid womparl country year quota, vce(jackknife) +Synthetic Difference-in-Differences Estimator +6As an example, with 50 replicates for bootstrap and placebo, and on a standard personal computer +running Stata SE, 15.1, the execution time for bootstrap was 18.2 seconds, for placebo permutations +was 10.09 seconds, and for jackknife was 0.7 seconds. This time scales approximately linearly with the +number of replicates in the case of bootstrap and placebo. With 500 replicates the time was 178.1 and +101.6 for bootstrap and placebo procedures respectively. + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +25 +womparl +ATT +Std. Err. +t +P>|t| +[95% Conf. Interval] +quota +10.33066 +6.00560 +1.72 +0.085 +-1.44009 +22.10141 +95% CIs and p-values are based on Large-Sample approximations. +Refer to Arkhangelsky et al., (2020) for theoretical derivations. +4.4 +Event Study Style Output +While sdid offers a simple implementation to conduct standard synthetic difference-in- +difference procedures and provide output, with some work results can also be visualized +in alternative ways. +For example, consider the standard ‘panel event-study’ style +setting (see e.g. Freyaldenhoven et al. (2019); Schmidheiny and Siegloch (2019); Clarke +and Tapia-Schythe (2021)), where one wishes to visualize how the dynamics of some +treatment effect evolve over time, as well as how differences between treated and +control units evolve prior to the adoption of treatment. Such graphs are frequently +used to efficiently provide information on both the credibility of parallel pre-trends in +an observational setting, as well as the emergence of any impact owing to treatment +once treatment is switched on. +-5 +0 +5 +10 +15 +20 +Women in Parliament +1990 +1995 +2000 +2005 +2010 +2015 +Year +Control +Treated +(a) Outcome Trends +-5 +0 +5 +10 +15 +Women in Parliament +1990 +1991 +1992 +1993 +1994 +1995 +1996 +1997 +1998 +1999 +2000 +2001 +2002 +2003 +2004 +2005 +2006 +2007 +2008 +2009 +2010 +2011 +2012 +2013 +2014 +2015 +Point Estimate +95% CI +(b) Event Study +Figure 4: Outcome trends and event study style estimate of the impact of quotas on % +women in parliament +What such an analysis seeks to document is the differential evolution of treated and +(synthetic) control units, abstracting away from any baseline difference between the +groups. As an example, refer to Figure 4(a), which is based on the adoption of gender +quotas laid out in section 4.2, and in particular quota adoption year 2002. This is +standard output from sdid, presenting trends in rates of women in parliament in +countries which adopted quotas in 2002 (solid line), and synthetic control countries +which did not adopt quotas (dashed line). We will refer to the values plotted in these + +26 +Synthetic Difference In Differences +trend lines as ¯Y T r +t +for treated units in year t, and ¯Y Co +t +for synthetic control units in +year t. While this standard output allows us to visualize trends in the two groups in +a simple way, it is not immediately clear how the differences in these outcomes evolve +over time compared to baseline differences, nor the confidence intervals on any such +changes over time. +For this to resemble the logic of an event study analysis, we wish to consider, for +each period t, whether differences between treated units and synthetic controls have +changed when compared to baseline differences. Namely, for each period t, we wish +to calculate: +� ¯Y T r +t +− ¯Y Co +t +� +− +� ¯Y T r +baseline − ¯Y Co +baseline +� +, +(8) +along with the confidence interval for this quantity. Here ¯Y T r +baseline and ¯Y Co +baseline refer +to baseline (pre-treatment) means for treated and synthetic control units respectively. +In standard panel event studies, some arbitrary baseline period is chosen off of which +to estimate pre-treatment differences. This is often one year prior to treatment. In +the case of SDID where pre-treatment weights are optimally chosen as �λsdid +t +(refer to +section 2), this suggests an alternative quantity for ¯Y T r +baseline and ¯Y Co +baseline, namely: +¯Y T r +baseline = +Tpre +� +t=1 +�λsdid +t +¯Y T r +t +¯Y Co +baseline = +Tpre +� +t=1 +�λsdid +t +¯Y Co +t +. +(9) +In words these baseline outcomes are simply pre-treatment aggregates, where weights +are determined by optimal pre-treatment weights (indicated by the shaded gray area +in Figure 4(a)). The event study then plots the quantities defined in (8), for each time +t. +An example of such an event study style plot is presented in Figure 4(b). Here, +blue points present the quantity indicated in (8) for each year. In this case, t ranges +from 1990 to 2015. While all these points are based off a simple implementation of +sdid comparing outcomes between treated and control units following (8), confidence +intervals documented in gray shaded areas of Figure 4(b) can be generated following +the resampling or permutation procedures discussed earlier in this paper. Specifically, +in the case of re-sampling, a block bootstrap can be conducted, and in each iteration +the quantity in (8) can be re-calculated for each t. The confidence interval associated +with each of these quantities can then be calculated based on its variance across many +(block)-bootstrap resamples. +Figure 4(b), and graphs following this principle more generally, can be generated +following the use of sdid. However, by default sdid simply provides output on trends +among the treated and synthetic control units (as displayed in Figure 4(a)). In the +code below, we lay out how one can move from these trends to the event study in +panel (b). As this procedure requires conducting the inference portion of the plot +manually (unlike most other procedures involving sdid where inference is conducted +automatically as part of the command) the code is somewhat more involved. +For +this reason, we discuss the code below in a number of blocks, terminating with the +generation of the plot displayed in Figure 4(b). + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +27 +In a first code block, we will open the parliamentary gender quota data which we +used in section 4.2, and keep the particular adoption period considered here (countries +which adopt quotas in 2002), as well as un-treated units: +. webuse set www.damianclarke.net/stata/ +. webuse quota_example.dta, clear +. egen m=min(year) if quota==1, by(country) +//indicator for the year of adoption +. egen mm=mean(m), by(country) +. keep if mm==2002 | mm==. +//keep only one time of adoption +. drop if lngdp==. +//keep if covariates observed +We can then implement the standard SDID procedure, additionally exporting the +trend graphs which is displayed in Figure 4(a). This is done in the first line below, +after which a number of vectors are stored. These vectors allow us to calculate the +quantity ( ¯Y T r +baseline − ¯Y Co +baseline) indicated in (8), which is generated from �λsdid, from +the returned matrix e(lambda), and pre-treatment values for ¯Y T r +t +and ¯Y Co +t +, from +the returned matrix e(series). This baseline quantity is referred to as meanpre o +below. Finally, the quantity of interest in (8) for each time period t is generated as +the variable d, which is plotted below as the blue points on the event study in Figure +4(b). +. qui sdid womparl country year quota, vce(noinference) graph g2_opt(ylab(-5(5)20) +> +ytitle("Women in Parliament") scheme(sj)) graph_export(groups, .pdf) +> +covariates(lngdp, projected) +. matrix lambda = e(lambda)[1..12,1] +//save lambda weight +. matrix yco = e(series)[1..12,2] +//control baseline +. matrix ytr = e(series)[1..12,3] +//treated baseline +. matrix aux = lambda´*(ytr - yco) +//calculate the pre-treatment mean +. scalar meanpre_o = aux[1,1] +. matrix difference = e(difference)[1..26,1..2] +// Store Ytr-Yco +. svmat difference +. ren (difference1 difference2) (time d) +. replace d = d - meanpre_o +// Calculate vector in (8) +Perhaps the most complicated portion of code is that which implements the boot- +strap procedure. This is provided below, where for ease of replication we consider a +relatively small number of bootstrap resamples, which are set as the local B = 100. +In each bootstrap resample, we first ensure that both treatment and control units are +present (using the locals r1 and r2), and then re-estimate the sdid procedure with the +new bootstrap sample generated using Stata’s bsample command. This is precisely +the same block bootstrap procedure laid out by Arkhangelsky et al. (2021), and which +sdid conducts internally, however here we are interested in collecting, for each boot- +strap resample, the same quantity estimated above with the main sample as d, which +captures the estimate defined in (8) for each t. To do so, we simply follow an identical +procedure as that conducted above, however now save the resulting resampled values +of the quantities from (8) as a series of matrices d‘b’ for later processing to generate +confidence intervals in the event study. +. local b = 1 + +28 +Synthetic Difference In Differences +. local B = 100 +. while `b´<=`B´ { +. +preserve +. +bsample, cluster(country) idcluster(c2) +. +qui count if quota == 0 +. +local r1 = r(N) +. +qui count if quota != 0 +. +local r2 = r(N) +. +if (`r1´!=0 & `r2´!=0) { +. +qui sdid womparl c2 year quota, vce(noinference) graph covariates(lngdp, projected) +. +matrix lambda_b = e(lambda)[1..12,1] +//save lambda weight +. +matrix yco_b = e(series)[1..12,2] +//control baseline +. +matrix ytr_b = e(series)[1..12,3] +//treated baseline +. +matrix aux_b = lambda_b´*(ytr_b - yco_b) +//calculate the pre-treatment mean +. +matrix meanpre_b = J(26,1,aux_b[1,1]) +. +matrix d`b´ = e(difference)[1..26,2] - meanpre_b +. +local ++b +. +} +. +restore +. } +The final step is to calculate the standard deviation of each estimate from (8) based +on the bootstrap resamples, and then to generate confidence intervals for each pa- +rameter based on the estimates generated above (d), as well as their standard errors. +This is conducted in the first lines of the code below. For each of the B = 100 re- +samples conducted above, we import the vector of resampled estimates from (8), and +then using rowsd() calculate the standard deviation of the estimates across each time +period t. This is the bootstrap standard error, which is used below to calculate the +upper and lower bounds of 95% confidence intervals as [LCI;UCI]. Finally, based on +these generated elements (d, as blue points on the event study, and LCI, UCI as the +end points of confidence intervals) we generate the output for Figure 4(b) in the final +lines of code. +. preserve +. keep time d +. keep if time!=. +. forval b=1/`B´ { +. +svmat d`b´ +// import each bootstrap replicate of difference between trends +. } +. egen rsd = rowsd(d11 - d`B´1) +//calculate standard deviation of this difference +. gen LCI = d + invnormal(0.025)*rsd //lower bounds on bootstrap CIs +. gen UCI = d + invnormal(0.975)*rsd //upper bounds on bootstrap CIs +. *generate plot +. tw rarea UCI LCI time, color(gray%40) || scatter d time, color(blue) m(d) +> +xtitle("") ytitle("Women in Parliament") xlab(1990(1)2015, angle(45)) +> +legend(order(2 "Point Estimate" 1 "95% CI") pos(12) col(2)) +> +xline(2002, lc(black) lp(solid)) yline(0, lc(red) lp(shortdash)) +> +scheme(sj) +. graph export "event_sdid.pdf", replace +. restore + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +29 +As noted above, the outcome of this graph is provided in Figure 4(b), where we +observe that, as expected, the synthetic difference-in-difference algorithm has resulted +in quite closely matched trends between the synthetic control and treatment group in +the pre-treatment period, given that all pre-treatment estimates lie close to zero. The +observed impact of quotas on women in parliament occurs from the treatment year +onward, where these differences are observed to be large and statistically significant. +This process of estimating an event study style plot is conducted here for a specific +adoption year. In the case of a block adoption design where there is only one adoption +period, this will be the only resulting event study to consider. However in a staggered +adoption design, a single event study could be generated for each adoption period. +Potentially, such event studies could be combined, but some way would be required to +deal with unbalanced lags and leads, and additionally some weighting function would +be required to group treatment lags and leads where multiple such lags and leads are +available. One such procedure has been proposed in Sun and Abraham (2021), and +could be a way forward here. +5 +Conclusions +In this paper we have laid out the details behind Arkhangelsky et al. (2021)’s SDID +method, and discussed its implementation in Stata using the sdid command. We have +briefly discussed the methods behind this command, as well as laid out extensions into +a staggered adoption setting. We provide two empirical examples to demonstrate the +usage of the command. +It is important to note that given the nature of the algorithm, a number of require- +ments must be met for this to be applied to data. We lay these out below, as key +considerations for empirical researchers wishing to conduct estimation and inference +using the SDID estimator. +• Firstly, and most importantly, a balanced panel of data is required that provides +outcomes and treatment measures for each unit in all periods under study. Should +missing values in such outcomes be present in the panel, these either must be +eliminated from the estimation sample, or data should be sought to fill in gaps. +• Secondly, no units can be considered if they were exposed to treatment from the +first period in which data is observed. If this occurs, there is no pre-treatment +period on which to generate synthetic control cohorts. If always treated units are +present in the data, these either need to be eliminated, or earlier data sought. +• Third, pure control units are required. At least some units must never be treated +in order to act as donor units. If all units are treated at some point in the panel, +no donor units exist, and synthetic controls cannot be generated. +• Fourth, in cases where covariates are included, these covariates must be present +in all observations. +If missing observations are present in covariates, this will +generate similar problems as when outcomes or treatment measures are missing. + +30 +Synthetic Difference In Differences +If missing observations are present, these treated units shold be removed from the +estimation sample, or data should be sought to complete the covariate coverage. +• Finally, in the case of inference, a number of additional requirements must be +met. In the case of bootstrap or jackknife procedures, the number of treated units +should be larger than 1 (and ideally considerably larger than this). Should only 1 +treated unit be present, placebo inference should be conducted. Additionally, in +the case of placebo inference, this can only be conducted if the number of control +units exceeds the number of treated units. +Should a balanced panel of data be available, the SDID method, and the sdid +command described here, offers a flexible, easy to implement and robust option for +the analysis of impacts of policies or treatments in certain groups at certain times. +These methods provide clear graphical results to describe outcomes, and an explicit +description of how counterfactual outcomes are inferred. These methods are likely +well suited to a large body of empirical work in social sciences, where treatment +assignment is not random, and offer benefits over both difference-in-differences and +synthetic control methods. + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +31 +6 +References +Abadie, A., A. Diamond, and J. Hainmueller. 2010. 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American Economic Review 111(12): 4088–4118. +https://www.aeaweb.org/articles?id=10.1257/aer.20190159. +Athey, +S., +and +G. +W. +Imbens. +2022. +Design-based +analysis +in +Difference- +In-Differences +settings +with +staggered +adoption. +Journal +of +Economet- +rics +226(1): +62–79. +Annals +Issue +in +Honor +of +Gary +Chamberlain. +https://www.sciencedirect.com/science/article/pii/S0304407621000488. +Ben-Michael, E., A. Feller, and J. Rothstein. 2021. The Augmented Synthetic Control +Method. Journal of the American Statistical Association 116(536): 1789–1803. +Bhalotra, S. R., D. Clarke, J. F. Gomes, and A. Venkataramani. 2022. Maternal Mortal- +ity and Women’s Political Power. Working Paper 30103, National Bureau of Economic +Research. http://www.nber.org/papers/w30103. +Bhuller, M., T. Havnes, E. Leuven, and M. Mogstad. 2013. Broadband Internet: An +Information Superhighway to Sex Crime? 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The Review of Economics and Statistics 93(1): +113–125. + +32 +Synthetic Difference In Differences +Doudchenko, +N., +and +G. +W. +Imbens. +2016. +Balancing, +Regression, +Difference-In-Differences +and +Synthetic +Control +Methods: +A +Synthesis. +https://arxiv.org/abs/1610.07748. +Ferman, +B., +and +C. +Pinto. +2021. +Synthetic +controls +with +imper- +fect +pretreatment +fit. +Quantitative +Economics +12(4): +1197–1221. +https://onlinelibrary.wiley.com/doi/abs/10.3982/QE1596. +Frank, +M., +and +P. +Wolfe. +1956. +An +algorithm +for +quadratic +pro- +gramming. +Naval +Research +Logistics +Quarterly +3(1-2): +95–110. +https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800030109. +Freyaldenhoven, S., C. Hansen, and J. M. Shapiro. 2019. +Pre-event Trends in +the Panel Event-Study Design. +American Economic Review 109(9): +3307–38. +http://www.aeaweb.org/articles?id=10.1257/aer.20180609. +Goodman-Bacon, +A. +2021. +The +Long-Run +Effects +of +Childhood +In- +surance +Coverage: +Medicaid +Implementation, +Adult +Health, +and +La- +bor +Market +Outcomes. +American +Economic +Review +111(8): +2550–93. +https://www.aeaweb.org/articles?id=10.1257/aer.20171671. +Hirshberg, D. A. Undated. synthdid: Synthetic Difference in Differences Estimation. +https://synth-inference.github.io/synthdid/. +Holland, P. W. 1986. Statistics and Causal Inference. Journal of the American Statistical +Association 81(396): 945–960. +Jann, +B. +2004. +ESTOUT: +Stata +module to +make +regression +tables. +Sta- +tistical +Software +Components, +Boston +College +Department +of +Economics. +https://ideas.repec.org/c/boc/bocode/s439301.html. +Kranz, +S. +2022. +Synthetic +Difference-in-Differences +with +Time- +Varying +Covariates. +Technical +report. +Available +online +at: +https://github.com/skranz/xsynthdid/blob/main/paper/synthdid with covariates.pdf. +Lawphongpanich, S. 2009. Encyclopedia of Optimization, chap. Frank–Wolfe Algorithm, +1094–1097. Boston, MA: Springer US. +Manski, C. F., and J. V. Pepper. 2018. How Do Right-to-Carry Laws Affect Crime +Rates? Coping with Ambiguity Using Bounded-Variation Assumptions. The Review +of Economics and Statistics 100(2): 232–244. https://doi.org/10.1162/REST a 00689. +Orzechowski, W., and R. C. Walker. 2005. The Tax Burden on Tobacco. Historical +Compilation Volume 40, Arlington, VA. +Paila˜nir, +D., +and +D. +Clarke. +2022. +SDID: +Stata +module +to +perform +syn- +thetic +difference-in-differences +estimation, +inference, +and +visualization. +Sta- +tistical +Software +Components, +Boston +College +Department +of +Economics. +https://ideas.repec.org/c/boc/bocode/s459058.html. + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +33 +Rambachan, A., and J. Roth. 2019. An Honest Approach to Parallel Trends. +Roth, J., P. H. C. Sant’Anna, A. Bilinski, and J. Poe. 2022. +What’s Trending +in Difference-in-Differences? +A Synthesis of the Recent Econometrics Literature. +https://arxiv.org/abs/2201.01194. +Rubin, D. B. 2005. Causal Inference Using Potential Outcomes. Journal of the American +Statistical Association 100(469): 322–331. +Schmidheiny, K., and S. Siegloch. 2019. +On Event Study Designs and Distributed- +Lag Models: Equivalence, Generalization and Practical Implications. IZA Discussion +Papers 12079, Institute of Labor Economics (IZA). +Sun, L., and S. Abraham. 2021. Estimating dynamic treatment effects in event stud- +ies with heterogeneous treatment effects. Journal of Econometrics 225(2): 175–199. +https://EconPapers.repec.org/RePEc:eee:econom:v:225:y:2021:i:2:p:175-199. +About the authors +Susan Athey is the Economics of Technology Professor at Stanford Graduate School of Busi- +ness. +Damian Clarke is an Associate Professor at The Department of Economics of The Universidad +de Chile, a Research Fellow at IZA and an Associate at the Millennium Institute for Market +Imperfections and Public Policy and CAGE, Warwick. +Guido Imbens is the Applied Econometrics Professor and Professor of Economics at Stanford +Graduate School of Business. +Daniel Paila˜nir is an MA student at The Department of Economics of The Universidad de +Chile, and a young researcher associated with the Millennium Nucleus for the Study of the +Life Course and Vulnerability. +Acknowledgments +We are grateful to Asjad Naqvi for comments relating to this code, and many users of the +sdid ado for sending feedback and suggestions related to certain features implemented here. + +34 +Synthetic Difference In Differences +Appendices +1 +Estimation Algorithms for the Block Design +In this appendix, we replicate the estimation algorithm and inference algorithms de- +fined in Arkhangelsky et al. (2021). These are referred to in the text, and follow the +same notation as in section 2 here. +Algorithm A1: Algorithm 1 from Arkhangelsky et al. (2021) +Data: Y, W. +Result: Point estimate �τ sdid. +1. Compute regularization parameter ζ; +2. Compute unit weights ˆωsdid; +3. Compute time weights ˆλsdid; +4. Compute the SDID estimator via the weighted DID regression; +� +ˆτ sdid, ˆµ, ˆα, ˆβ +� += arg min +τ,µ,α,β +� N +� +i=1 +T +� +t=1 +(Yit − µ − αi − βt − Witτ)2�ωsdid +i +ˆλsdid +t +� +Algorithm A2: Algorithm 2 from Arkhangelsky et al. (2021) +Data: Y, W, B. +Result: Variance estimator �V cb +τ +for b ← 1 to B do +1. Construct a bootstrap dataset (Y(b), W(b)) by sampling N rows of +(Y, W) with replacement; +2. if the bootstrap sample has no treated units or no control units then +Discard resample and go to 1; +end +3. Compute SDID estimate τ (b) following algorithm A1 based on +(Y(b), W(b)); +end +4. Define �V cb +τ += 1 +B +�B +b=1 +�� +τ (b) − 1 +B +�B +b=1 � +τ (b) +�2 +; + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +35 +Algorithm A3: Algorithm 3 from Arkhangelsky et al. (2021) +Data: �ω, �λ Y, W, �τ. +Result: Variance estimator �Vτ +for i ← 1 to N do +1. Compute �τ (−i) : arg minτ,{αj,βt}j̸=i,t +� +j̸=i,t(Yit − αj − βt − Witτ)2�ωjˆλt; +end +2. Compute �V jack +τ += (N − 1)N −1 �N +i=1 +� +�τ (−i) − �τ +�2; +Algorithm A4: Algorithm 4 from Arkhangelsky et al. (2021) +Data: Yco, Ntr, B. +Result: Variance estimator �V placebo +τ +for b ← 1 to B do +1. Sample Ntr out of the Nco control units without replacment to ‘receive +the placebo’; +2. Construct a placebo treatment matrix W(b) +co , for the controls; +3. Compute SDID estimate τ (b) based on (Yco, W(b) +co ); +end +4. Define �V placebo +τ += 1 +B +�B +b=1 +�� +τ (b) − 1 +B +�B +b=1 � +τ (b) +�2 +; + +36 +Synthetic Difference In Differences +2 +Replicating Weight Graphs +After implementing the sdid estimator, the unit specific weights can be used to re- +create the weight graph provided as output automatically with the graph option. +While this is somewhat involved, and likely would not be conducted by hand, it may be +illustrative to see how this is generated, combining both unit-specific weights, and unit- +specific difference-in-difference estimates. This code is displayed below, first saving +time weights which are used to calculate DID estimates, secondly saving unit weights +for graphing, thirdly combining all elements and calculating the DID estimates, and +finally, generating the graph, which is displayed after this code excerpt. +. preserve +. clear +. matrix lambda=e(lambda) +. svmat lambda +. ren (lambda1 lambda2) (lambda year) +. keep if year<=1988 +. tempfile dlambda +. save `dlambda´ +. restore +. preserve +. clear +. matrix omega=e(omega) +. svmat omega +. ren (omega1 omega2) (omega id) +. keep if id<=39 +. tempfile domega +. save `domega´ +. restore +. merge m:1 year using `dlambda´, nogen +. bys state: egen y1=mean(packspercapita) if year>=1989 +. bys state: egen y2=mean(packspercapita*lambda) if year<=1988 +. replace y2=y2*19 +. egen ypost=mean(y1), by(state) +. egen ypre=mean(y2), by(state) +. keep state ypre ypost +. duplicates drop +. gen delta=ypost-ypre +. qui sum delta if state=="California" +. gen sdelta=`r(mean)´ +. gen difference=sdelta-delta +. egen id=group(state) +. merge 1:1 id using `domega´, nogen +. drop if omega==. +. encode state, gen(state2) l(id) +. tw scatter difference state2 if omega!=0 [aw=omega], msize(tiny) || +> +scatter difference state2 if omega==0, m(X) +> +xlabel(1(1)38, angle(vertical) labs(vsmall) valuelabel) +> +yline(-15.60383, lc(red)) legend(off) xtitle("") +> +ytitle("Difference") scheme(sj) +. graph export sdid panela.eps, replace + +Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir +37 +-30 +-20 +-10 +0 +10 +Difference +Alabama +Arkansas +Colorado +Connecticut +Delaware +Georgia +Idaho +Illinois +Indiana +Iowa +Kansas +Kentucky +Louisiana +Maine +Minnesota +Mississippi +Missouri +Montana +Nebraska +Nevada +New Hampshire +New Mexico +North Carolina +North Dakota +Ohio +Oklahoma +Pennsylvania +Rhode Island +South Carolina +South Dakota +Tennessee +Texas +Utah +Vermont +Virginia +West Virginia +Wisconsin +Wyoming +Figure A1: Unit-Specific Weight Graph + diff --git a/VtFKT4oBgHgl3EQfmy5j/content/tmp_files/load_file.txt b/VtFKT4oBgHgl3EQfmy5j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7bde0e50630a6d42b59c6cb9ec3a4c460d6338ae --- /dev/null +++ b/VtFKT4oBgHgl3EQfmy5j/content/tmp_files/load_file.txt @@ -0,0 +1,1696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf,len=1695 +page_content='Synthetic Difference In Differences Estimation Damian Clarke Department of Economics University of Chile dclarke@fen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='cl Daniel Paila˜nir Department of Economics University of Chile dpailanir@fen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='cl Susan Athey Graduate School of Business Stanford University athey@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='edu Guido Imbens Graduate School of Business Stanford University imbens@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this paper, we describe a computational implementation of the Synthetic difference-in-differences (SDID) estimator of Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) for Stata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Synthetic difference-in-differences can be used in a wide class of cir- cumstances where treatment effects on some particular policy or event are desired, and repeated observations on treated and untreated units are available over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We lay out the theory underlying SDID, both when there is a single treatment adoption date and when adoption is staggered over time, and discuss estimation and inference in each of these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We introduce the sdid command which imple- ments these methods in Stata, and provide a number of examples of use, discussing estimation, inference, and visualization of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Keywords: synthetic difference-in-differences, synthetic control, difference-in-differences, estimation, inference, visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 1 Introduction There has been an explosion in recent advances in econometric methods for policy anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A particularly active area is that applied to estimating the impact of exposure to some particular event or policy, when observations are available in a panel or repeated cross section of groups and time (see for example recent surveys by de Chaisemartin and D’Haultfœuille (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2022) for reviews of these methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A modelling challenge in this setting is in determining what would have happened to exposed units had they been left unexposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Should such a counterfactual be estimable from under- lying data, causal inference can be conducted by comparing outcomes in treated units to those in theoretical counterfactual untreated states, under the potential outcome framework (see for example the discussion in Holland (1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Rubin (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A substantial number of empirical studies in economics and the social sciences more generally seek to estimate effects in this setting using difference-in-difference (hereafter DID) style designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here impacts are inferred by comparing treated to control units, where time-invariant level differences between units are permitted as well as general common trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, the drawing of causal inferences requires a parallel trend assumption, which states that in the absence of treatment, treated units would have followed parallel paths to untreated units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Whether this assumption is reasonable in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='11859v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='EM] 27 Jan 2023 2 Synthetic Difference In Differences a particular context is an empirical issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Recently, a number of methodologies have sought to loosen this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This includes procedures in which counterfactual trends can be assumed to deviate from parallel, leading to partial identification (Manski and Pepper 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Rambachan and Roth 2019), flexible procedures to adequately control for any prevailing differences between treated and control units (Bilinski and Hatfield 2018), often based on pre-treatment periods only (Bhuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Goodman-Bacon 2021) and IV-style methods which explicitly consider dynamics in pre-treatment periods (Freyaldenhoven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In many cases, parallel trends may be a questionable modelling assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' One particular solution to the challenge has been the application of synthetic control meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Early work in synthetic control explores the setting of comparative case studies, where a single treated unit is observed, and one wishes to construct a matched synthetic control from a larger number of potential donor units (Abadie and Gardeazabal 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2010, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These methods seek to generate a single synthetic control from a unique convex weighting of underlying control units, such that this synthetic control is as closely matched as possible to the treated unit in pre-treatment outcomes, and potentially other covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This weights are optimally generated and fixed over time, potentially assigning zero weight to certain control units, and larger weights to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This has attracted considerable attention in both empirical applications and the- oretical extensions, with recent advances including debiasing procedures (Ben-Michael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2021) which can additionally house multiple treatment units (Abadie and L’Hour 2021), more flexible weighting schemes, or constant fixed differences between treated and synthetic control units (Doudchenko and Imbens 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Ferman and Pinto 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A recent particularly flexible modelling option which can be applied in panel data settings seeks to bridge the DID and synthetic control (SC) procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) propose the Synthetic Difference-in-Differences estimator (SDID), which brings in strengths from both the DID and SC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Like DID models, SDID allows for treated and control units to be trending on entirely different levels prior to a reform of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' And like SC methods, SDID seeks to optimally generate a matched control unit which considerably loosens the need for parallel trend assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Correspondingly, SDID avoids common pitfalls in standard DID and SC methods – namely an inability to estimate causal relationships if parallel trends are not met in aggregate data in the case of DID, and a requirement that the treated unit be housed within a “convex hull” of control units in the case of SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) propose estimation and inference procedures, formally proving consistency and asymptotic normality of the proposed estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' What’s more, the authors briefly discuss a number of important applied points such as how their estimator can incorporate covariates, and how their estimator can be applied to both multiple treatment units, and even multiple treatment units which adopt treatment in different time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this paper we describe the sdid command (available for download as Paila˜nir and Clarke (2022)) which implements the SDID estimator in Stata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This command allows for the simple implementation of the SDID estimator provided that a panel or repeated cross section of data is available covering groups and time periods, and which is strongly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The command, written principally in Mata, seamlessly incorporates cases Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 3 where there is a single treated unit, multiple treatment units, and multiple treatment periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' It reports treatment effects laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), additionally implementing their proposed bootstrap, jackknife and placebo inference procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A number of graphical output options are provided to examine the generation of the SDID estimator and the underlying optimal weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' While principally written to conduct SDID estimation, the sdid command (and the SDID method) nests as possible estimation procedures SC and DID, which can be easily generated to allow comparison of estimation procedures and estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1 In introducing the command, we first provide a primer on the core methodological points of SDID (as well as comparisons to DID and SC), and then describe how these procedures extend to a setting where treatment adoption occurs over multiple time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We then lay out the command syntax of sdid, as well as the elements which are returned to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We provide a number of examples to illustrate the use of the SDID method in Stata, both based upon a well-known example of California’s passage of Proposition 99, an anti-smoking measure previously presented in Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) in which a single state adopts a treatment at a given time, as well as an example where exposure to a policy occurs at mutiple periods: the case of parliamentary gender quotas studied by Bhalotra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We conclude by making a number of practical points on the computational implementation of this estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2 Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1 The Canonical Synthetic Difference-in-Differences Procedure The synthetic DID procedure, hereafter SDID, is developed in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), and we lay out principal details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As input, SDID requires a balanced panel of N units or groups, observed over T time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' An outcome, denoted Yit, is observed for each unit i in each period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Some, but not all, of these observations are treated with a specific binary variable of interest, denoted Wit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This treatment variable Wit = 1 if observation i is treated by time t, otherwise, Wit = 0 indicates that unit i is untreated at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here, we assume that there is a single adoption period for treated units, which Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) refer to as a ‘block treatment assignment’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='3, we extend this to a ‘staggered adoption design’ (Athey and Imbens 2022), where treated units adopt treatment at varying points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A key element of both of these designs is that once treated, units are assumed to remain exposed to treatment forever thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the particular setting of SDID, no always treated units can be included in estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For estimation to proceed, we require at least two pre-treatment periods off of which to determine control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The goal of SDID is to consistently estimate the causal effect of receipt of policy 1Code from the original paper was provided in R (Hirshberg undated), which can do many of the procedures which sdid implements, and indeed, abstracting from differences in pseudo-random number generation, give identical results in cases where similar procedures are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A number of useful extensions are available in sdid, such as the implementation of estimates in cases where treatment occurs in multiple periods, and alternative manners to include covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4 Synthetic Difference In Differences or treatment Wit, (an average treatment effect on the treated, or ATT) even if we do not believe in the parallel trends assumption between all treatment and control units on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Estimation of the ATT proceeds as follows: � �τ sdid, �µ, �α, �β � = arg min τ,µ,α,β � N � i=1 T � t=1 (Yit − µ − αi − βt − Witτ)2�ωsdid i �λsdid t � (1) where the estimand is the ATT, generated from a two-way fixed effect regression, with optimally chosen weights �ωsdid i and �λsdid t discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Note that here, this procedure flexibly allows for shared temporal aggregate factors given the estimation of time fixed effects βt and time invariant unit-specific factors given the estimation of unit fixed effects αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As is standard in fully saturated fixed-effect models, one αi and one βt fixed effect are normalized to zero to avoid multi-colinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The presence of unit-fixed effects implies that SDID will simply seek to match treated and control units on pre- treatment trends, and not necessarily on both pre-treatment trends and levels, allowing for a constant difference between treatment and control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this setting, it is illustrative to consider how the SDID procedure compares to the traditional synthetic control method of Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2010), as well as the baseline DID procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The standard DID procedure consists of precisely the same two-way fixed effect OLS procedure, simply assigning equal weights to all time periods and groups: � �τ did, �µ, �α, �β � = arg min τ,µ,α,β � N � i=1 T � t=1 (Yit − µ − αi − βt − Witτ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2) The synthetic control, on the other hand, maintains optimally chosen unit-specific weights ω (as laid out below), however does not seek to optimally consider time periods via time weights, and omits unit fixed effects αi implying that the synthetic control and treated units should maintain approximately equivalent pre-treatment levels, as well as trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' � �τ sc, �µ, �β � = arg min τ,µ,β � N � i=1 T � t=1 (Yit − µ − βt − Witτ)2�ωsc i � (3) From (2)-(3) it is clear that the SDID procedure offers greater flexibility than both the DID and SC procedures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' in the case of DID by permitting a violation of parallel trends in aggregate data, and in the case of SC, by both additionally seeking to optimally weight time periods when considering counterfactual outcomes, and allowing for level differences between treatment and control groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The selection of unit weights, ω, as inputs to (1) (and (3)) seeks to ensure that com- parison is made between treated units and controls which were approximately following parallel trends prior to the adoption of treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The selection of time weights, λ in the case of SDID seeks to draw more weight from pre-treatment periods which are more similar to post-treatment periods, in the sense of finding a constant difference between each control unit’s post treatment average, and pre-treatment weighted averages across all selected controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Specifically, as laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), unit-specific Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 5 weights are found by resolving: � �ω0, �ωsdid� = arg min ω0∈R,ω∈Ω Tpre � t=1 � ω0 + Nco � i=1 ωiYit − 1 Ntr N � i=Nco+1 Yit �2 + ζ2Tpre||ω||2 2(4) where Ω = � ω ∈ RN +, with Nco � i=1 ωi = 1 and ωi = 1 Ntr for all i = Nco + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' , N � , ||ω||2 refers to the Euclidean norm and ζ is a regularization parameter laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4091-4092).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2 This leads to a vector of Nco non-negative weights plus an intercept ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The weights ωi for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' , Nco} imply that absolute difference between control and treatment trends units should be minimized over all pre- treatment periods, while ω0 initially allows for a constant difference between treatment and controls over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Together, these imply that units will follow parallel pre-trends, though provided ω0 ̸= 0, not identical pre-trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of time weights, a similar procedure is followed, finding weights which minimize the following objective function: � �λ0, �λsdid� = arg min λ0∈R,λ∈Λ Nco � i=1 � �λ0 + Tpre � t=1 λtYit − 1 Tpost T � t=Tpre+1 Yit � � 2 + ζ2Nco||λ||2(5) where Λ = � � �λ ∈ RT +, with Tpre � t=1 λt = 1 and λt = 1 Tpost for all t = Tpre + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' , T � � � , where the final term in (5) is a very small regularization term to ensure uniqueness of time weights, where ζ = 1 × 10−6�σ, and �σ is defined as in footnote 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This estimation procedure is summarized in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, algorithm 1), reproduced in Appendix 1 here for ease of access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) also prove that the estimator is asymptotically normal, suggesting that confidence intervals on τ can be constructed as: �τ sdid ± zα/2 � �Vτ, where zα/2 refers to the inverse normal density function at percentile α/2 should one wish to compute 1-α confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These confidence intervals thus simply require an estimate of the variance of τ, �Vτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) propose three specific procedures to estimate this variance: a block bootstrap, a jackknife, or a permutation- based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2For the sake of completion, this regularization parameter is calculated as ζ = (Ntr × Tpost)1/4�σ, where: �σ2 = 1 Nco(Tpre − 1) Nco � i=1 Tpre−1 � t=1 (∆it− ¯∆)2, ∆it = Yi,(t+1)−Yit, and ¯∆ = 1 Nco(Tpre − 1) Nco � i=1 Tpre−1 � t=1 ∆it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 6 Synthetic Difference In Differences The block (also known as clustered) bootstrap approach, consists of taking some large number, B, of bootstrap resamples over units, where units i are the resampled blocks in the block bootstrap procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Provided that a given resample does not con- sist entirely of treated, or entirely of control units, the quantity �τ sdid is re-estimated, and denoted as �τ sdid (b) for each bootstrap resample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The bootstrap variance �V (b) τ is then calculated as the variance of resampled estimates �τ sdid (b) across all B resamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The boot- strap algorithm is defined in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, algorithm 2), reproduced here in appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This bootstrap procedure is observed in simulation to have particularly good properties, but has two particular drawbacks, justifying alternative inference pro- cedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The first is that it may be computationally costly, given that in each bootstrap resample the entire synthetic DID procedure is re-estimated, including the estimation of optimal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is especially computationally expensive in cases where working with large samples, or where covariates are included, as discussed at more length below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The second, is that formal proofs of asymptotic normality rely on the number of treated units being large, and as such, estimated variance, and confidence intervals, may be unreliable when the number of treated units is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' An alternative estimator which significantly reduces the computational burden in- herent in the bootstrap is estimating a jackknife variance for �τ sdid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This procedure consists of iterating over all units in the data, in each iteration removing the given unit, and recalculating �τ sdid, denoted �τ sdid (−i), maintaining fixed the optimal weights for ω and λ calculated in the original SDID estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The jackknife variance, �V (jack) τ is then calculated based on the variance of all �τ sdid (−i) estimates, following Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, Algorithm 3) (refer to Appendix 1 here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this case, each iteration saves on re- calculating optimal weights, and as documented by Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), provide a variance leading to conservative confidence intervals, without the computational bur- den imposed by the bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Once again, asymptotic normality relies on there being a large number of treated units, and in particular if only 1 treated unit is observed – as is often the case in comparative case studies – the jackknife will not even be defined given that a �τ sdid (−i) term will be undefined when removing the single treated unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Given limits to these inference options when the number of treated units is small, an alternative placebo, or permutation-based, inference procedure is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This consists of, first, conserving just the control units, and then randomly assigning the same treatment structure to these control units, as a placebo treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Based on this placebo treatment, we then re-estimate �τ sdid, denoted �τ sdid (p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This procedure is repeated many times, giving rise to a vector of estimates �τ sdid (p) , and the placebo variance, �V (placebo) τ , can be estimated as the variance of this vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is formally defined in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, algorithm 4), and appendix 1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' It is important to note that in the case of this placebo-based variance, homoskedasticity across units is required, given that the variance is based off placebo assignments of treatment made only within the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2 Conditioning on Covariates So far, we have limited exposition to cases where one wishes to study outcomes Yit, and their evolution in treated and synthetic control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, in certain settings, it may be of relevance to condition on exogenous time-varying covariates Xit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangel- sky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) note that in this case, we can proceed by applying the SDID algorithm to the residuals calculated as: Y res it = Yit − Xit �β, (6) where �β comes from regression of Yit on Xit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This procedure, in which the synthetic DID process will be applied to the residuals Y res it , is different to the logic of synthetic controls following Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In Abadie et al.’s conception, when covariates are included the synthetic control is chosen to ensure that these covariates are as closely matched as possible between treated and synthetic control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However in the SDID conception, covariate adjustment is viewed as a pre-processing task, which removes the impact of changes in covariates from the outcome Yit prior to calculating the synthetic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Along with their paper, Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) provide an implementation of their algorithm in R (Hirshberg undated), and in practice they condition out these variables Xit by finding �β within an optimization procedure which additionally allows for the efficient calculation of optimal weights ω and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the sdid code described below, we follow Hirshberg (undated) in implementing this efficient optimization procedure (the Frank and Wolfe (1956) solver), however there are a number of potential complications which can arise in this manner of dealing with covariates, and as such, alternative procedures are also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A first potential issue is purely numerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the Frank-Wolfe solver discussed above, a minimum point is assumed to be found when successive iterations of the solver lead to arbitrarily small changes in all parameters estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='3 Where these parameters include coefficients on covariates, in extreme cases the solution found for (1) can be sensitive to the scaling of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This occurs in particular when covariates have very large magnitudes and variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In such cases, the inclusion of covariates in (1) can cause optimization routines to suggest solutions which are not actually globally optimal, given that successive movements in �β can be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In extreme cases, this can imply that when multiplying all variables Xit by a large constant values, the estimated treatment effect can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' While this issue can be addressed by using smaller tolerances for defining stopping rules in the optimization procedure, it can be addressed in a more simple way if all covariates are first re-standardized as Z-scores, implying that no very-high-variance variables are included, while capturing the same underlying variation in covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A second, potentially more complicated point is described by Kranz (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' He notes that in certain settings, specifically where the relationship between covariates and the outcome vary over time differentially in treatment and control groups, the procedure described above may fail to capture the true relationship between covariates and the outcome of interest, and may subsequently lead to bias in estimated treatment 3In the case of convex functions such as that in (1), the Frank-Wolfe solver finds a global minima, see for example Lawphongpanich (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 8 Synthetic Difference In Differences effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' He proposes a slightly altered methodology of controlling for covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Namely, his suggestion is to first estimate a two-way fixed effect regression of the dependent variable on covariates (plus time and unit fixed effects), using only observations which are not exposed, or not exposed yet, to treatment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sub-setting to observations for which Wit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Based on the regression Yit = Xitβ + µi + λt + εit for units with Wit = 0, coefficients �β estimated by OLS can be used to follow the procedure in (6), and SDID can then be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' An additional benefit of this method is that unlike the optimized method described previously, �β is calculated in a single step via OLS rather than an interative optimization procedure, which can lead to substantial speed ups in computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We document these methods based on a number of empirical example in the fol- lowing section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' It is worth noting that regardless of which procedure one uses, the implementation of SDID follows the suggestion laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, footnote 4) and (6) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' What varies between the former and the latter procedures discussed here is the manner with which one estimates coefficients on covariates, �β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='3 The Staggered Adoption Design The design discussed up to this point assumes block assignment, or that all units are either controls, or are treated in a single unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, Appendix A), they note that this procedure can be extended to a staggered adoption design, where treated units adopt treatment at varying moments of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here we lay out the formal details related to the staggered adoption design, focusing first on estimation of an aggregate treatment effect, and then on extending the three inference procedures laid out previously into a staggered adoption setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This proposal is one potential way to deal with staggered adoption settings, though there are other possible manners to proceed – see for example Ben-Michael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), or Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In cases where the number of pure control units is small, this proposal may not necessarily be very effective given challenges in finding appropriate counterfactuals for each adoption-specific period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Estimation Unlike the block assignment case where a single pre- versus post-treatment date can be used to conduct estimation, in the staggered adoption design, multiple adoption dates are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Consider for example the treatment matrix below, con- sisting of 8 units, 2 of which (1 and 2) are untreated, while the other 6 are treated, Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 9 however at varying points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' W = � � � � � � � � � � � � � � t1 t2 t3 t4 t5 t6 t7 t8 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 1 1 5 0 0 0 1 1 1 1 1 6 0 0 0 1 1 1 1 1 7 0 0 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 � � � � � � � � � � � � � � This single staggered treatment matrix W can be broken down into adoption date specific matrices, W1, W2 and W3, or generically, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' , WA, where A indicates the number of distinct adoption dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Additionally, a row vector A consisting of A elements contains these distinct adoption periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this specific setting where units first adopt treatment in period t3 (units 7 and 8), t4 (units 5 and 6), and t7 (units 3 and 4), the adoption date vector consists simple of periods 3, 4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A = �3 4 7� Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' adoption-specific matrices W1-W3 simply consist of pure treated units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' and units which adopt in this specific period,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' as below: W1 = � � � � � � t1 t2 t3 t4 t5 t6 t7 t8 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 1 1 � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' W2 = � � � � � � t1 t2 t3 t4 t5 t6 t7 t8 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 5 0 0 0 1 1 1 1 1 6 0 0 0 1 1 1 1 1 � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' W3 = � � � � � � t1 t2 t3 t4 t5 t6 t7 t8 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 7 0 0 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021, Appendix A), the average treatment effect on the treated can then be calculated by applying the synthetic DID estimator to each of these 3 adoption-specific samples, and calculating a weighted average of the three estimators, where weights are assigned based on the relative number of treated units 10 Synthetic Difference In Differences and time periods in each adoption group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Generically, this ATT is calculated based on adoption-specific SDID estimates as: � ATT = � for a∈A T a post Tpost × ˆτ sdid a (7) where Tpost refers to the total number of post-treatment periods observed in treated units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This estimation procedures is laid out formally in Algorithm 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Note that in this case, while the parameter interest is likely the treatment effect ATT or adoption specific τ sdid a parameters, each adoption period is associated with an optimal unit and time weight vector ωsdid a and λsdid a , which can be returned following estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Algorithm 1: Estimation of the ATT with staggered adoption Data: Y, W, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Point estimate � ATT and adoption-specific values ˆτ sdid a , ˆωsdid a and ˆλsdid a for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' for a ∈ A do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Subset Y and W to units who are pure controls, and who first adopt treatment in period t = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute regularization parameter ζ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute unit weights ˆωsdid a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute time weights ˆλsdid a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute the SDID estimator via the weighted DID regression ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' � ˆτ sdid a , ˆµa, ˆαa, ˆβa � = arg min τ,µ,α,β � N � i=1 T � t=1 (Yit − µ − αi − βt − Witτ)2�ωsdid a,i ˆλsdid a,t � end 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute ATT across adoption-specific SDID estimates � ATT = � for a∈A T a post Tpost × ˆτ sdid a Inference In the staggered adoption design, estimated treatment effects are simply a multi-period extension of the underlying SDID algorithm, in each case working with the relevant pure control and treated sub-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Thus, inference can be conducted in the staggered adoption design under similar resample or placebo procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here we discuss inference following each of the bootstrap, jackknife, or placebo procedures laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), applied to a staggered adoption setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We note that Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 11 in this design, it is likely the case that one wishes to conduct inference on the treatment effect, ATT from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Thus in the below, we propose inference details for this estimand, additionally noting that standard errors and confidence intervals on adoption-specific SDID parameters τ sdid a come built-in as part of these procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Consider first the case of bootstrap inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Suppose that one wishes to estimate standard errors or generate confidence intervals on the global treatment effect ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A bootstrap procedure can be conducted based on many clustered bootstrap resamples over the entire initial dataset, where in each case, a resampled ATT estimate � ATT b is generated, following Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Based on many such resampled estimates, the bootstrap variance can be calculated as the variance of these resamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We lay out the bootstrap variance estimate below in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Note that as in the block treatment Algorithm 2: Bootstrap inference in the staggered adoption setting Data: Y, W, A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Variance estimator �V cb AT T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Additionally, variance for each adoption specific estimate �V cb τa for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' for b ← 1 to B do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Construct a bootstrap dataset (Y(b), W(b), A(b)) by sampling N rows of (Y, W) with replacement, and generating A as the corresponding adoption vector ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' if the bootstrap sample has no treated units or no control units then Discard resample and go to 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute SDID estimate ATT (b) following Algorithm 1 based on (Y(b), W(b), A(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Similarly, generate a vector of adoption-date specific resampled SDID estimates τ (b) a for all a ∈ A(b) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Define �V cb AT T = 1 B �B b=1 � � ATT (b) − 1 B �B b=1 � ATT (b) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Similarly, estimate adoption-date specific variances for each τ sdid a estimate as the variance over each τ (b) a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' design, this bootstrap procedure requires the number of treated units to grow with N within each adoption period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As such, if a very small number of treated units exist for certain adoption periods, placebo inference is likely preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Similarly, as laid out in the block treatment design, the bootstrap procedure re-estimates optimal weight matrices in each resample, and can be computationally expensive in cases where samples are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' An alternative inference procedure, which is less computationally intensive but sim- ilarly based on asymptotic arguments with a large number of states, and many treated units, is based on the jackknife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here, optimal weight matrices calculated for each 12 Synthetic Difference In Differences adoption-specific estimate τ sdid a in Algorithm 1 are treated as fixed, and provided as inputs to a jackknife procedure, described below in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Below, these matrices, which consist of weights for each adoption period a ∈ A, are denoted as ω, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='4 Note that in Algorithm 3, notation (−i) refers to a standard jackknife estimator, removing a single state (i) in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In cases where i refers to a treated unit, the ATT will be calculated removing this particular treated unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For this reason, the jackknife estimator will not be defined in cases where any single adoption period has only one treated unit, as in this case, �τ (−i) a will not be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Algorithm 3: Jackknife inference in the staggered adoption setting Inputs: Y, W, A, �ω, �λ, � ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Variance estimator �VAT T for i ← 1 to N do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute SDID estimate ATT (−i) following Algorithm 1 based on (Y(−i), W(−i), A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Similarly, generate a vector of adoption-date specific resampled SDID estimates τ (−i) a for all a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute �V jack AT T = (N − 1)N −1 �N i=1 � � ATT (−i) − � ATT �2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, in cases where the number of treated units is small, and concerns related to the validity of the previous variance estimators exists, the placebo inference procedure defined in algorithm 4 can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here, this is defined for the staggered adoption case, generalising Algorithm 4 of Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' To conduct this procedure, placebo treatments are randomly assigned based on the actual treatment structure, however only to the control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Based on these placebo assignments, placebo values for ATT are generated, which can be used to calculate the variance as laid out in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' It is important to note that such a procedure will only be feasible in cases where the number of control units is strictly larger than the number of treated units (or hence placebo assignments will not be feasible), and, as laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Conley and Taber (2011), such a procedure relies on homoskedasticity across units, as otherwise the variance of the treatment effect on the treated could not be inferred from variation in assignment of placebo treatments to control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4In the case of ω, this is an A × Nco matrix, where for each row a, the weight assigned to each control unit in this particular adoption period is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of λ, this is a matrix containing A columns, where each column consists of as many rows as the number of pre-treatment periods to this specific adoption date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In each cell, the weight assigned to a particular pre-treatment year for each adoption period is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 13 Algorithm 4: Placebo inference in the staggered adoption Inputs: Yco, Ntr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Variance estimator �V placebo AT T for b ← 1 to B do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Sample Ntr out of the Nco control units without replacment to ‘receive the placebo’ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Construct a placebo treatment matrix W(b) co , for the controls ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute SDID estimate ATT (b) following algorithm 1 based on (Yco, W(b) co , A(b)) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Define �V placebo AT T = 1 B �B b=1 � � ATT (b) − 1 B �B b=1 � ATT (b) �2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 3 The sdid command Synthetic Difference-in-Differences can be implemented in Stata using the following command syntax: sdid depvar groupvar timevar treatment � if � � in � , vce(type) � covariates(varlist, type) seed(#) reps(#) method(type) graph g1on g1 opt(string) g2 opt(string) graph export(string, type) msize(markersizestyle) unstandardized mattitles � where depvar describes the dependent variable in a balanced panel of units (group- var) and time periods (timevar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The variable which indicates units which are treated at each time period, which accumulates over time, is indicated as treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Note that here, it is not necessary for users to specify whether the design is a block or staggered adoption design, as this will be inferred based off the data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Optionally, if and in can be specified, provided that this does not result in imbalance in the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Re- quired and permitted options are discussed below, followed by a description of objects returned by sdid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Options vce(type) is a required option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This must be one of bootstrap, jackknife, placebo or noinference, where in each case inference proceeds following the specified method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of bootstrap, this is only permitted if greater than one unit is treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of jackknife, this is only permitted if greater than one unit is treated in each treatment period (if multiple treatment periods are considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of placebo, this requires at least one more control than treated unit to allow for permutations 14 Synthetic Difference In Differences to be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In each case, inference follows the specific algorithm laid out in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We allow the no inference option (noinference) should one wish to simply generate the point estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is useful if you wish to plot outcome trends without the added computational time associated with inference procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' covariates(varlist, type) Covariates should be included as a varlist, and if specified, treatment and control units will be adjusted based on covariates in the synthetic difference-in-differences procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Optionally, type may be specified, which indicates how covariate adjustment will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If the type is indicated as “optimized” (the default) this will follow the method described in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), footnote 4, where SDID is applied to the residuals of all units after regression adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, this has been observed to be problematic at times (refer to Kranz (2022)), and is also sensitive to optimization if covariates have high dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Thus, an alternative type is implemented (“projected”), which consists of conducting regression adjustment based on parameters estimated only in untreated units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This type follows the procedure proposed by Kranz (2022) (xsynth in R), and is observed to be more stable in some implementations (and at times, considerably faster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid will run simple checks on the covariates indicated and return an error if covariates are constant, to avoid multicolineality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, prior to running sdid, you are encouraged to ensure that covariates are not perfectly multicolinear with other covariates and state and year fixed effects, in a simple two-way fixed effect regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If perfectly multi- colinear covariates are included sdid will execute without errors, however where type is “optimized”, the procedure may be sensitive to the inclusion of redundant covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' seed(#) Define the seed for pseudo-random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' reps(#) Set the number of repetitions used in the calculation of bootstrap and placebo standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Default is 50 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Larger values should be preferred where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' method(type) this option allows you to change the estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The type must be one of sdid, did or sc, where sdid refers to synthetic difference-in-differences, sc refers to synthetic control, and did refers to difference-in-differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' By default, sdid is enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' graph if this option is specified, graphs will be displayed showing unit and time weights as well as outcome trends as per Figure 1 from Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' g1on this option activates the unit-specific weight graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' By default g1 is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' g1 opt(string) option to modify the appearance of the unit-specific weight graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These options adjust the underlying scatter plot, so should be consistent with twoway scatter plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' g2 opt(string) option to modify the appearance of the outcome trend graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These options adjust the underlying line plot, so should be consistent with twoway line plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' graph export(string, type) Graphs will be saved as weightsYYYY and trendsYYYY for each of the unit-specific weights and outcome trends respectively, where YYYY refers to each treatment adoption period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Two graphs will be generated for each Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 15 treatment adoption period provided that g1on is specified, otherwise a single graph will be generated for each adoption period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If this option is specified, type must be specified, which refers to a valid Stata graph type (eg “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps”, “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='pdf”, and so forth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Optionally, a stub can be specified, in which case this will be prepended to exported graph names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' msize(markersizestyle) allows you to modify the size of the marker for graph 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' unstandardized if controls are included and the “optimized” method is specified, con- trols will be standardized as Z-scores prior to finding optimal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This avoids problems with optimization when control variables have very high dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If un- standardized is specified, controls will simply be entered in their original units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This option should be used with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' mattitles requests labels to be added to the returned e(omega) weight matrix provid- ing names (in string) for the unit variables which generate the synthetic control group in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If mattitles is not indicated, the returned weight matrix (e(omega)) will store these weights with a final column providing the numerical ID of units, where this numerical ID is either taken from the unit variable (if this variable is a numerical format), or arranged in alphabetical order based on the unit variable, if this variable is in string format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Returned Objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='sdid stores the following in e(): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Scalars: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(ATT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Average Treatment Effect on the Treated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(se) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Standard error for the ATT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(reps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Number of bootstrap/placebo replications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(N clust) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Number of clusters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Macros: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(cmd) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='sdid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(cmdline) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='command as typed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(depvar) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='name of dependent variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(vce) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='vcetype specified in vce() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(clustvar) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='name of cluster variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Matrices: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Synthetic Difference In Differences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(tau) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='tau estimator for each adoption time-period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(lambda) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='lambda weights (time-specific weights) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(omega) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='omega weights (unit-specific weights) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(adoption) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='adoption times ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(beta) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='beta vector corresponding to covariates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(series) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='control and treatment series of the graphs (only returned ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='when the graph option is indicated) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='e(difference) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='difference between treatment and control series (only returned ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='when the graph option is indicated) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Examples based on an Empirical Application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='In the sections below we provide a number of illustrations of the usage of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' and per- formance of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' the sdid command,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' which operationalizes the Synthetic Difference-in- Differences estimator in Stata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We consider both a block treatment design (with a single adopting state), and a staggered adoption design, noting a number of points covering estimation, inference, and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1 A Block Design In the first case, we consider the well-known example, also presented in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), of California’s “Proposition 99” tobacco control measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This exam- ple, based on the context described in Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2010) and data of Orzechowski and Walker (2005), is frequently used to illustrate synthetic control style methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Proposition 99, which was passed by California in 1989, increased the taxes paid on a packet of cigarettes by 25 cents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The impact of this reform is sought to be estimated by comparing the evolution of sales of cigarettes in packs per capita in California (the treated state) with those in 38 untreated states, which did not significantly increase cigarette taxes during the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The data used in analysis cover each of these 39 states over the period of 1970– 2000, with a single observation for each state and year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Adoption occurs in California in 1989, implying Tpre = 19 pre-treatment periods and Tpost = 12 post-treatment periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' There are Nco = 38 control and a single treated state, hence Ntr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Using the sdid command, we replicate the results from Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the below code example, we first download the data, and then conduct the Synthetic Difference-in-Differences implementation using a placebo inference procedure with (a default) 50 placebo iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse set www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='damianclarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='net/stata/ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse prop99_example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='dta, clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid packspercapita state year treated, vce(placebo) seed(1213) Placebo replications (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This may take some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='. 50 Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 17 Synthetic Difference-in-Differences Estimator packsperca~a ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] treated 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='60383 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='53183 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='102 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='28588 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='07822 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The third line of this code excerpt quite simply implements the synthetic difference- in-differences estimator, returning identical point estimates to those documented in Table 1 of Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Standard errors are slightly different, as these are based on pseudo-random placebo reshuffling, though can be replicated as presented here provided that the same seed is set in the seed option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Note that in this case, given that a small number (1) of treated units is present, placebo inference is the only appropriate procedure, as indicated in the vce() option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='30 ' metadata={'source': 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+page_content='South Dakota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Tennessee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Texas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Utah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Vermont ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Virginia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='West Virginia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Wisconsin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Wyoming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(a) Unit-Specific Weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Packs per capita ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1970 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Treated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(b) Outcome Trends and Time-Specific Weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Figure 1: Proposition 99,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' example from Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) Should we wish to generate the same graphs as in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), summa- rizing both (a) unit specific weights, and (b) treatment and synthetic control outcome trends along with time specific weights, this can be requested with the addition of the graph option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is displayed below, where we additionally modify plot aesthetics via the g1 opt() and g2 opt() options for weight graphs (Figure 1(a)), and trend graphs (Figure 1(b)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, generated graphs can be saved to disk us- ing the graph export() option, with a graph type (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps below), and optionally a pre-pended plot name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Output corresponding to the below command is provided in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid packspercapita state year treated, vce(placebo) seed(1213) graph g1on > g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) > g1_opt(xtitle("") scheme(sj)) g1on graph_export(sdid_, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps) 18 Synthetic Difference In Differences It is illustrative to compare the output of SDID estimation procedures with those of standard synthetic control methods of Abadie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2010), and unweighted difference- in-difference estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' By using the method() option one can request a standard difference-in-differences output, requested with method(did), or synthetic control out- put, requested with method(sc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the interests of completeness, method(sdid) is also accepted, although this is the default behaviour when method is not included in command syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In each case, resulting graphs document matched treated and con- trol/synthetic control trends, as well as weights received by each unit and time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These are displayed in Figure 2, with plots corresponding to each of the three calls to sdid displayed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the left-hand panel, identical SDID plots are provided as those noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the middle plot, corresponding to method(did), a difference- in-difference setting is displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here, in the top panel, outcomes for California are displayed as a solid line, while mean outcomes for all control states are documented as a dashed line, where a clear divergence is observed in the pre-treatment period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The bottom panel shows that in this case, each control unit receives an identical weight, while time weights indicated at the base of the top plot note that each period is weighted identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, in the case of synthetic control, output from the third call to sdid is provided in the right-hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this case, treated and synthetic control units are observed to overlap nearly exactly, with weights in figure (f) noted to be more sparse, and placing relatively more weight on fewer control states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We note that in each case, the vce(noinference) option is used, as here we are simply interested in observing exported graphs, not the entire command output displaying aggregate estimates, standard errors and confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid packspercapita state year treated, method(sdid) vce(noinference) graph > g1_opt(ylabel(-110(20)50) xtitle("") scheme(sj)) g1on > g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) > graph_export(sdid_, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps) sdid packspercapita state year treated, method(did) vce(noinference) graph msize(small) > g1_opt(ylabel(-110(20)50) xtitle("") scheme(sj)) g1on > g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) > graph_export(did_, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps) sdid packspercapita state year treated, method(sc) vce(noinference) graph > g1_opt(ylabel(-110(20)50) xtitle("") scheme(sj)) g1on > g2_opt(ylabel(0(25)150) ytitle("Packs per capita") scheme(sj)) > graph_export(sc_, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps) The sdid command returns multiple matrices containing treatment and control out- come trends, weights, and other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These elements can be accessed simply for use in post-estimation procedures or graphing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As a simple example, the follow- ing code excerpt accesses treatment and synthetic control outcome trends (stored in e(series), and time weights (stored in e(lambda)) and uses these elements to repli- cate the plot presented in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Thus, if one wishes to have further control over the precise nature of plotting, beyond that provided in the graphing options available in sdid’s command syntax, one can simply work with elements returned in the ereturn list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In Appendix 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' we show that with slightly more effort,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' returned elements can be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Packs per capita ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1970 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Treated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(a) SDID: Outcome Trends ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Packs per capita ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1970 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Treated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(b) DID: Outcome Trends ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Packs per capita ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1970 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Year ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Difference ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Difference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Alabama ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Arkansas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Colorado ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Connecticut ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Delaware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Georgia ' 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+page_content='Pennsylvania ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Rhode Island ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='South Carolina ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='South Dakota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Tennessee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Texas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Utah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Vermont ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Virginia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='West Virginia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Wisconsin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Wyoming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(f) SC: Unit-Specific Weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Figure 2: Comparison of estimators ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='used to construct the unit-specific weight plot from Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' preserve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix series=e(series) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix lambda=e(lambda) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui svmat series, names(col) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui svmat lambda .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' tw line Yco1989 year, yaxis(1) || > line Ytr1989 year, yaxis(1) || > bar lambda1 year if year<=1988, yaxis(2) ylabel(0(1)5, axis(2)) yscale(off axis(2)) > xline(1989, lc(red)) legend(order(1 "Control" 2 "Treated") pos(12) col(2)) scheme(sj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' graph export sdid replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='eps, replace .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' restore 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2 A Staggered Adoption Design We present an example of a staggered adoption design, based on data and the context studied in Bhalotra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this case, the impact of parliamentary gender quotas which reserve seats for women in parliament are estimated, first on rates of women in parliament, and secondly on rates of maternal mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is conducted on a country by year panel, where for each of 1990-2015, 115 countries are observed, 9 20 Synthetic Difference In Differences 50 100 150 1970 1980 1990 2000 year Control Treated Figure 3: Outcome Trends and Time-Specific Weights of which implement a parliamentary gender quota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='5 For each of these countries, data on the rates of women in parliament and the maternal mortality ratio are collected, as well as a number of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This example presents a staggered adoption configuration, given that in the period under study, quota adoption occurred in seven different yearly periods between 2000 and 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid handles a staggered adoption configuration seamlessly without any particular changes to the syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the code below, we implement the synthetic difference-in-differences estimator using the bootstrap procedure to calculate standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The output by default reports the weighted ATT which is defined in (7) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, as laid out in (7), this is based on each adoption-period specific synthetic difference-in-differences estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These adoption-period specific estimates are returned in the matrix e(tau), which is tabulated below the standard command output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse quota_example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='dta, clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid womparl country year quota, vce(bootstrap) seed(1213) Bootstrap replications (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This may take some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='. 50 5This is a subset of the full sample studied in Bhalotra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here we only work with countries for which observations of women in parliament and maternal mortality exist for the full time period, without missing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 21 Synthetic Difference-in-Differences Estimator womparl ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] quota 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='03410 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='74040 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='70305 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='36516 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matlist e(tau) Tau Time r1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='388868 2000 r2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='967746 2002 r3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='95226 2003 r4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='450543 2005 r5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='749036 2010 r6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='76272 2012 r7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='8203235 2013 All other elements are identical to those documented in the case of a single adoption period, however generalised to multiple adoptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For example, if requesting graphi- cal output, a single treatment versus synthetic control trend graph and corresponding unit-level weight graph is provided for each adoption date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Similarly, ereturned matri- ces such as e(lambda), e(omega) and e(series) provide columns for each particular adoption period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Adding Covariates As laid out in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2, covariates can be handled in synthetic difference-in-differences in a number of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Below we document the inclusion of a single covariate (the natural logarithm of GDP per capita).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As sdid is based on a balanced panel of observations, we must first ensure that there are no missing observa- tions for all covariates, in this case dropping a small number of (control) countries for which this measure is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We then include covariates via the covariates() option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the first case, this is conducted exactly following the procedure discussed by Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), in which parameters on covariates are estimated within the optimization routines in Mata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is analogous to indicating covariates(, optimized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Estimates in this particular case suggest that the inclusion of this con- trol does little to dampen effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' After estimation, the coefficients on the covariates can be inspected as part of e(beta), where an adoption-specific value for each co- variate is provided, given that the underlying SDID estimate is calculated for each adoption period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' drop if lngdp==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid womparl country year quota, vce(bootstrap) seed(1213) covariates(lngdp) Bootstrap replications (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This may take some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='. 50 Synthetic Difference-in-Differences Estimator 22 Synthetic Difference In Differences womparl ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] quota 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='05150 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='09252 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='99027 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='11272 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The inclusion of covariates in the previous implementation adds considerably to the computational time as it increases the complexity of the underlying optimization rou- tine, and this is conducted in each adoption period and each bootstrap replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' An alternative manner to capture covariates described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2 above is that of Kranz (2022), where the impact of covariates are projected out using a baseline regression of the outcome on covariates and fixed effects only in units where the treatment status is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is implemented as below, with covariates(, projected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid womparl country year quota, vce(bootstrap) seed(1213) covariates(lngdp, projected) Bootstrap replications (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This may take some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='. 50 Synthetic Difference-in-Differences Estimator womparl ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] quota 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='05927 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='11913 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='94589 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='17264 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here, results are slightly different, though quantitatively comparable to those when using alternative procedures for conditioning out covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this case, if examining the e(beta) matrix, only a single coefficient will be provided, as the regression used to estimate the coefficient vector is always based on the same sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This additionally offers a non-trivial speed up in the execution of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For example, on a particular personal computer with Stata SE 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1 and relatively standard specifications, using the optimized method above requires 324 seconds of computational time while using projected requires 61 seconds (compared with 58 seconds where covariates are not included in sdid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Post-Estimation Commands While sdid provides a standard tabular and graphical output as displayed previously, the command can used to provide output in alternative formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For example, the sdid command interacts seamlessly with routines such as estout (Jann 2004) for the exportation of results tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' To see this, the below block of code estimates three specific versions of the model discussed above, storing each model using an eststo: prefix, before finally exporting estimated ATTs and standard errors to a LaTeX file, which can be included in tabular form as displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Similar such procedures could be conducted with routines such as outreg or outreg2, Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 23 and tabular output could be further enriched using additional options within esttab if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse set www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='damianclarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='net/stata/ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse quota_example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='dta, clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' lab var quota "Parliamentary Gender Quota" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' eststo sdid_1: sdid womparl country year quota, vce(bootstrap) seed(2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' drop if lngdp==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' eststo sdid_2: sdid womparl country year quota, vce(bootstrap) seed(2022) > covariates(lngdp, optimized) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' eststo sdid_3: sdid womparl country year quota, vce(bootstrap) seed(2022) > covariates(lngdp, projected) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' esttab sdid_1 sdid_2 sdid_3 using "example1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='tex", > nonotes nomtitles stats(N, labels("Observations") fmt(%9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='0fc)) > addnotes("* p$<$0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='10, ** p$<$0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='05, *** p$<$0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='01") > starlevel ("*" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='10 "**" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='05 "***" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='01) lab > b(%-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='3f) se(%-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='3f) style(tex) replace (1) (2) (3) Parliamentary Gender Quota 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='034** 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='051*** 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='059*** (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='940) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='047) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='099) Observations 3,094 2,990 2,990 p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='10, ** p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='05, *** p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='01 Table 1: Tabular Output Following sdid 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='3 Inference Options In this section we provide examples of the implementation of alternative inference options, as laid out in algorithms 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For this illustration we will keep only treated units which adopt gender quotas in 2002 and 2003, as otherwise adoption periods will exist in which only a single unit is treated, and jackknife procedures will not be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse quota example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='dta, clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' drop if country=="Algeria" | country=="Kenya" | country=="Samoa" | > country=="Swaziland" | country=="Tanzania" In the following three code blocks we document bootstrap, placebo and jackknife inference procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The difference in implementation in each case is very minor, simply indicating either bootstrap, placebo or jaccknife in the vce() option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For example, in the case of bootstrap inference, where block bootstraps over the variable country are performed, the syntax is as follows: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid womparl country year quota, vce(bootstrap) seed(1213) 24 Synthetic Difference In Differences Bootstrap replications (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This may take some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='. 50 Synthetic Difference-in-Differences Estimator womparl ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] quota 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='33066 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='72911 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='029 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='06178 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='59954 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' By default, only 50 bootstrap replicates are performed, though in practice, a sub- stantially higher number should be used, and this can be indicated in the reps(#) option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of placebo, the syntax and output are virtually identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The suitability of each method depends on the underlying structure of the panel, and in this particular case, given the relatively small number of treated units, it may be the case that placebo procedures are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid womparl country year quota, vce(placebo) seed(1213) Placebo replications (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This may take some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='. 50 Synthetic Difference-in-Differences Estimator womparl ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] quota 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='33066 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='14741 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='24191 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='41941 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, in the interests of completeness, the jackknife procedure, which is by far the fastest of the three to execute6, is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Note that unlike the case with placebo or bootstrap inference, it is not necessary (or relevant) to set a seed, nor indicate the number of replications, as the jackknife procedure implies conducting a leave-one-out procedure over each unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this particular case, jackknife inference appears to be more conservative than bootstrap procedures, in line with what may be expected based on Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021)’s demonstration that jackknife inference is in general conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' sdid womparl country year quota, vce(jackknife) Synthetic Difference-in-Differences Estimator 6As an example, with 50 replicates for bootstrap and placebo, and on a standard personal computer running Stata SE, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1, the execution time for bootstrap was 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2 seconds, for placebo permutations was 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='09 seconds, and for jackknife was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='7 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This time scales approximately linearly with the number of replicates in the case of bootstrap and placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' With 500 replicates the time was 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='1 and 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='6 for bootstrap and placebo procedures respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 25 womparl ATT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' t P>|t| [95% Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Interval] quota 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='33066 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='00560 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='085 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='44009 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='10141 95% CIs and p-values are based on Large-Sample approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Refer to Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', (2020) for theoretical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='4 Event Study Style Output While sdid offers a simple implementation to conduct standard synthetic difference-in- difference procedures and provide output, with some work results can also be visualized in alternative ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For example, consider the standard ‘panel event-study’ style setting (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Freyaldenhoven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Schmidheiny and Siegloch (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Clarke and Tapia-Schythe (2021)), where one wishes to visualize how the dynamics of some treatment effect evolve over time, as well as how differences between treated and control units evolve prior to the adoption of treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Such graphs are frequently used to efficiently provide information on both the credibility of parallel pre-trends in an observational setting, as well as the emergence of any impact owing to treatment once treatment is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2012 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2013 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Point Estimate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='95% CI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(b) Event Study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='Figure 4: Outcome trends and event study style estimate of the impact of quotas on % ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='women in parliament ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='What such an analysis seeks to document is the differential evolution of treated and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='(synthetic) control units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' abstracting away from any baseline difference between the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As an example, refer to Figure 4(a), which is based on the adoption of gender quotas laid out in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2, and in particular quota adoption year 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is standard output from sdid, presenting trends in rates of women in parliament in countries which adopted quotas in 2002 (solid line), and synthetic control countries which did not adopt quotas (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We will refer to the values plotted in these 26 Synthetic Difference In Differences trend lines as ¯Y T r t for treated units in year t, and ¯Y Co t for synthetic control units in year t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' While this standard output allows us to visualize trends in the two groups in a simple way, it is not immediately clear how the differences in these outcomes evolve over time compared to baseline differences, nor the confidence intervals on any such changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For this to resemble the logic of an event study analysis, we wish to consider, for each period t, whether differences between treated units and synthetic controls have changed when compared to baseline differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Namely, for each period t, we wish to calculate: � ¯Y T r t − ¯Y Co t � − � ¯Y T r baseline − ¯Y Co baseline � , (8) along with the confidence interval for this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here ¯Y T r baseline and ¯Y Co baseline refer to baseline (pre-treatment) means for treated and synthetic control units respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In standard panel event studies, some arbitrary baseline period is chosen off of which to estimate pre-treatment differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is often one year prior to treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of SDID where pre-treatment weights are optimally chosen as �λsdid t (refer to section 2), this suggests an alternative quantity for ¯Y T r baseline and ¯Y Co baseline, namely: ¯Y T r baseline = Tpre � t=1 �λsdid t ¯Y T r t ¯Y Co baseline = Tpre � t=1 �λsdid t ¯Y Co t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (9) In words these baseline outcomes are simply pre-treatment aggregates, where weights are determined by optimal pre-treatment weights (indicated by the shaded gray area in Figure 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The event study then plots the quantities defined in (8), for each time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' An example of such an event study style plot is presented in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Here, blue points present the quantity indicated in (8) for each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In this case, t ranges from 1990 to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' While all these points are based off a simple implementation of sdid comparing outcomes between treated and control units following (8), confidence intervals documented in gray shaded areas of Figure 4(b) can be generated following the resampling or permutation procedures discussed earlier in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Specifically, in the case of re-sampling, a block bootstrap can be conducted, and in each iteration the quantity in (8) can be re-calculated for each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The confidence interval associated with each of these quantities can then be calculated based on its variance across many (block)-bootstrap resamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Figure 4(b), and graphs following this principle more generally, can be generated following the use of sdid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However, by default sdid simply provides output on trends among the treated and synthetic control units (as displayed in Figure 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the code below, we lay out how one can move from these trends to the event study in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' As this procedure requires conducting the inference portion of the plot manually (unlike most other procedures involving sdid where inference is conducted automatically as part of the command) the code is somewhat more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For this reason, we discuss the code below in a number of blocks, terminating with the generation of the plot displayed in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 27 In a first code block, we will open the parliamentary gender quota data which we used in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='2, and keep the particular adoption period considered here (countries which adopt quotas in 2002), as well as un-treated units: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse set www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='damianclarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='net/stata/ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' webuse quota_example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='dta, clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' egen m=min(year) if quota==1, by(country) //indicator for the year of adoption .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' egen mm=mean(m), by(country) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' keep if mm==2002 | mm==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' //keep only one time of adoption .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' drop if lngdp==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' //keep if covariates observed We can then implement the standard SDID procedure, additionally exporting the trend graphs which is displayed in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is done in the first line below, after which a number of vectors are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These vectors allow us to calculate the quantity ( ¯Y T r baseline − ¯Y Co baseline) indicated in (8), which is generated from �λsdid, from the returned matrix e(lambda), and pre-treatment values for ¯Y T r t and ¯Y Co t , from the returned matrix e(series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This baseline quantity is referred to as meanpre o below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, the quantity of interest in (8) for each time period t is generated as the variable d, which is plotted below as the blue points on the event study in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui sdid womparl country year quota, vce(noinference) graph g2_opt(ylab(-5(5)20) > ytitle("Women in Parliament") scheme(sj)) graph_export(groups, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='pdf) > covariates(lngdp, projected) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix lambda = e(lambda)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.12,1] //save lambda weight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix yco = e(series)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.12,2] //control baseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix ytr = e(series)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.12,3] //treated baseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix aux = lambda´*(ytr - yco) //calculate the pre-treatment mean .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' scalar meanpre_o = aux[1,1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix difference = e(difference)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.26,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.2] // Store Ytr-Yco .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' svmat difference .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ren (difference1 difference2) (time d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' replace d = d - meanpre_o // Calculate vector in (8) Perhaps the most complicated portion of code is that which implements the boot- strap procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is provided below, where for ease of replication we consider a relatively small number of bootstrap resamples, which are set as the local B = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In each bootstrap resample, we first ensure that both treatment and control units are present (using the locals r1 and r2), and then re-estimate the sdid procedure with the new bootstrap sample generated using Stata’s bsample command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is precisely the same block bootstrap procedure laid out by Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021), and which sdid conducts internally, however here we are interested in collecting, for each boot- strap resample, the same quantity estimated above with the main sample as d, which captures the estimate defined in (8) for each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' To do so, we simply follow an identical procedure as that conducted above, however now save the resulting resampled values of the quantities from (8) as a series of matrices d‘b’ for later processing to generate confidence intervals in the event study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' local b = 1 28 Synthetic Difference In Differences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' local B = 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' while `b´<=`B´ { .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' preserve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' bsample, cluster(country) idcluster(c2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui count if quota == 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' local r1 = r(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui count if quota !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' local r2 = r(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' if (`r1´!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='=0 & `r2´!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='=0) { .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui sdid womparl c2 year quota, vce(noinference) graph covariates(lngdp, projected) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix lambda_b = e(lambda)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.12,1] //save lambda weight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix yco_b = e(series)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.12,2] //control baseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix ytr_b = e(series)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.12,3] //treated baseline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix aux_b = lambda_b´*(ytr_b - yco_b) //calculate the pre-treatment mean .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix meanpre_b = J(26,1,aux_b[1,1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix d`b´ = e(difference)[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='.26,2] - meanpre_b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' local ++b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' restore .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' } The final step is to calculate the standard deviation of each estimate from (8) based on the bootstrap resamples, and then to generate confidence intervals for each pa- rameter based on the estimates generated above (d), as well as their standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is conducted in the first lines of the code below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' For each of the B = 100 re- samples conducted above, we import the vector of resampled estimates from (8), and then using rowsd() calculate the standard deviation of the estimates across each time period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This is the bootstrap standard error, which is used below to calculate the upper and lower bounds of 95% confidence intervals as [LCI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='UCI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, based on these generated elements (d, as blue points on the event study, and LCI, UCI as the end points of confidence intervals) we generate the output for Figure 4(b) in the final lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' preserve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' keep time d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' keep if time!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' forval b=1/`B´ { .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' svmat d`b´ // import each bootstrap replicate of difference between trends .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' egen rsd = rowsd(d11 - d`B´1) //calculate standard deviation of this difference .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' gen LCI = d + invnormal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='025)*rsd //lower bounds on bootstrap CIs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' gen UCI = d + invnormal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='975)*rsd //upper bounds on bootstrap CIs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' *generate plot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' tw rarea UCI LCI time, color(gray%40) || scatter d time, color(blue) m(d) > xtitle("") ytitle("Women in Parliament") xlab(1990(1)2015, angle(45)) > legend(order(2 "Point Estimate" 1 "95% CI") pos(12) col(2)) > xline(2002, lc(black) lp(solid)) yline(0, lc(red) lp(shortdash)) > scheme(sj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' graph export "event_sdid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='pdf", replace .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' restore Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 29 As noted above, the outcome of this graph is provided in Figure 4(b), where we observe that, as expected, the synthetic difference-in-difference algorithm has resulted in quite closely matched trends between the synthetic control and treatment group in the pre-treatment period, given that all pre-treatment estimates lie close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' The observed impact of quotas on women in parliament occurs from the treatment year onward, where these differences are observed to be large and statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This process of estimating an event study style plot is conducted here for a specific adoption year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of a block adoption design where there is only one adoption period, this will be the only resulting event study to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' However in a staggered adoption design, a single event study could be generated for each adoption period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Potentially, such event studies could be combined, but some way would be required to deal with unbalanced lags and leads, and additionally some weighting function would be required to group treatment lags and leads where multiple such lags and leads are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' One such procedure has been proposed in Sun and Abraham (2021), and could be a way forward here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 5 Conclusions In this paper we have laid out the details behind Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021)’s SDID method, and discussed its implementation in Stata using the sdid command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We have briefly discussed the methods behind this command, as well as laid out extensions into a staggered adoption setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We provide two empirical examples to demonstrate the usage of the command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' It is important to note that given the nature of the algorithm, a number of require- ments must be met for this to be applied to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' We lay these out below, as key considerations for empirical researchers wishing to conduct estimation and inference using the SDID estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Firstly, and most importantly, a balanced panel of data is required that provides outcomes and treatment measures for each unit in all periods under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Should missing values in such outcomes be present in the panel, these either must be eliminated from the estimation sample, or data should be sought to fill in gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Secondly, no units can be considered if they were exposed to treatment from the first period in which data is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If this occurs, there is no pre-treatment period on which to generate synthetic control cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If always treated units are present in the data, these either need to be eliminated, or earlier data sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Third, pure control units are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' At least some units must never be treated in order to act as donor units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If all units are treated at some point in the panel, no donor units exist, and synthetic controls cannot be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Fourth, in cases where covariates are included, these covariates must be present in all observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' If missing observations are present in covariates, this will generate similar problems as when outcomes or treatment measures are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 30 Synthetic Difference In Differences If missing observations are present, these treated units shold be removed from the estimation sample, or data should be sought to complete the covariate coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Finally, in the case of inference, a number of additional requirements must be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' In the case of bootstrap or jackknife procedures, the number of treated units should be larger than 1 (and ideally considerably larger than this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Should only 1 treated unit be present, placebo inference should be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Additionally, in the case of placebo inference, this can only be conducted if the number of control units exceeds the number of treated units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Should a balanced panel of data be available, the SDID method, and the sdid command described here, offers a flexible, easy to implement and robust option for the analysis of impacts of policies or treatments in certain groups at certain times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These methods provide clear graphical results to describe outcomes, and an explicit description of how counterfactual outcomes are inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These methods are likely well suited to a large body of empirical work in social sciences, where treatment assignment is not random, and offer benefits over both difference-in-differences and synthetic control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 31 6 References Abadie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Diamond, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Hainmueller.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Abadie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' L’Hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A Penalized Synthetic Control Estimator for Disag- gregated Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Journal of the American Statistical Association 116(536): 1817–1834.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' What’s Trending in Difference-in-Differences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' A Synthesis of the Recent Econometrics Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='01194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Causal Inference Using Potential Outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Journal of the American Statistical Association 100(469): 322–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Schmidheiny, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Siegloch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' On Event Study Designs and Distributed- Lag Models: Equivalence, Generalization and Practical Implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' IZA Discussion Papers 12079, Institute of Labor Economics (IZA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Sun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Abraham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Estimating dynamic treatment effects in event stud- ies with heterogeneous treatment effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Journal of Econometrics 225(2): 175–199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' https://EconPapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='repec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='org/RePEc:eee:econom:v:225:y:2021:i:2:p:175-199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' About the authors Susan Athey is the Economics of Technology Professor at Stanford Graduate School of Busi- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Damian Clarke is an Associate Professor at The Department of Economics of The Universidad de Chile, a Research Fellow at IZA and an Associate at the Millennium Institute for Market Imperfections and Public Policy and CAGE, Warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Guido Imbens is the Applied Econometrics Professor and Professor of Economics at Stanford Graduate School of Business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Daniel Paila˜nir is an MA student at The Department of Economics of The Universidad de Chile, and a young researcher associated with the Millennium Nucleus for the Study of the Life Course and Vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Acknowledgments We are grateful to Asjad Naqvi for comments relating to this code, and many users of the sdid ado for sending feedback and suggestions related to certain features implemented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 34 Synthetic Difference In Differences Appendices 1 Estimation Algorithms for the Block Design In this appendix, we replicate the estimation algorithm and inference algorithms de- fined in Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' These are referred to in the text, and follow the same notation as in section 2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Algorithm A1: Algorithm 1 from Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) Data: Y, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Point estimate �τ sdid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute regularization parameter ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute unit weights ˆωsdid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute time weights ˆλsdid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute the SDID estimator via the weighted DID regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' � ˆτ sdid, ˆµ, ˆα, ˆβ � = arg min τ,µ,α,β � N � i=1 T � t=1 (Yit − µ − αi − βt − Witτ)2�ωsdid i ˆλsdid t � Algorithm A2: Algorithm 2 from Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) Data: Y, W, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Variance estimator �V cb τ for b ← 1 to B do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Construct a bootstrap dataset (Y(b), W(b)) by sampling N rows of (Y, W) with replacement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' if the bootstrap sample has no treated units or no control units then Discard resample and go to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute SDID estimate τ (b) following algorithm A1 based on (Y(b), W(b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Define �V cb τ = 1 B �B b=1 �� τ (b) − 1 B �B b=1 � τ (b) �2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Susan Athey and Damian Clarke and Guido Imbens and Daniel Paila˜nir 35 Algorithm A3: Algorithm 3 from Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) Data: �ω, �λ Y, W, �τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Variance estimator �Vτ for i ← 1 to N do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute �τ (−i) : arg minτ,{αj,βt}j̸=i,t � j̸=i,t(Yit − αj − βt − Witτ)2�ωjˆλt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute �V jack τ = (N − 1)N −1 �N i=1 � �τ (−i) − �τ �2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Algorithm A4: Algorithm 4 from Arkhangelsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' (2021) Data: Yco, Ntr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Result: Variance estimator �V placebo τ for b ← 1 to B do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Sample Ntr out of the Nco control units without replacment to ‘receive the placebo’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Construct a placebo treatment matrix W(b) co , for the controls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Compute SDID estimate τ (b) based on (Yco, W(b) co );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' Define �V placebo τ = 1 B �B b=1 �� τ (b) − 1 B �B b=1 � τ (b) �2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' 36 Synthetic Difference In Differences 2 Replicating Weight Graphs After implementing the sdid estimator, the unit specific weights can be used to re- create the weight graph provided as output automatically with the graph option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' While this is somewhat involved, and likely would not be conducted by hand, it may be illustrative to see how this is generated, combining both unit-specific weights, and unit- specific difference-in-difference estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' This code is displayed below, first saving time weights which are used to calculate DID estimates, secondly saving unit weights for graphing, thirdly combining all elements and calculating the DID estimates, and finally, generating the graph, which is displayed after this code excerpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' preserve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix lambda=e(lambda) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' svmat lambda .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ren (lambda1 lambda2) (lambda year) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' keep if year<=1988 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' tempfile dlambda .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' save `dlambda´ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' restore .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' preserve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' matrix omega=e(omega) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' svmat omega .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' ren (omega1 omega2) (omega id) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' keep if id<=39 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' tempfile domega .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' save `domega´ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' restore .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' merge m:1 year using `dlambda´, nogen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' bys state: egen y1=mean(packspercapita) if year>=1989 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' bys state: egen y2=mean(packspercapita*lambda) if year<=1988 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' replace y2=y2*19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' egen ypost=mean(y1), by(state) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' egen ypre=mean(y2), by(state) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' keep state ypre ypost .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' duplicates drop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' gen delta=ypost-ypre .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' qui sum delta if state=="California" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' gen sdelta=`r(mean)´ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' gen difference=sdelta-delta .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' egen id=group(state) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' merge 1:1 id using `domega´, nogen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' drop if omega==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' encode state, gen(state2) l(id) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content=' tw scatter difference state2 if omega!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFKT4oBgHgl3EQfmy5j/content/2301.11859v1.pdf'} +page_content='=0 [aw=omega], msize(tiny) || > scatter difference state2 if omega==0, m(X) > 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Astrophys. Astr. (0000) 000: #### +DOI +Investigating the morphology and CO gas kinematics of Sh2-112 region +Kshitiz K. Mallick1, Saurabh Sharma1, Lokesh K. Dewangan2, Devendra K. Ojha3, Neelam +Panwar1, Tapas Baug4 +1Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital, 263002, India. +2Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India. +3Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mum- +bai 400005, India +4S.N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, West Bengal, +India +*Corresponding author. E-mail: kshitiz@aries.res.in +MS received 1 January 2015; accepted 1 January 2015 +Abstract. +We present a study of the molecular cloud in Sh2-112 massive star forming region using the 3–2 +transition of CO isotopologues - CO, 13CO, and C18O; supplemented in part by CGPS H I line emission and MSX +data. Sh2-112 is an optically visible region powered by an O8V type massive star BD +45 3216, and hosts two Red +MSX Survey sources – G083.7962+03.3058 and G083.7071+03.2817 – classified as H II region and young stellar +object, respectively. Reduced spectral data products from the James Clerk Maxwell Telescope archive, centered +on the two RMS objects with ∼7’×7’ field of view each, were utilised for the purpose. The 13CO(3-2) channel +map of the region shows the molecular cloud to have filamentary extensions directed away from the massive star, +which also seems to be at the edge of a cavity like structure. Multiple molecular cloud protrusions into this cavity +structure host local peaks of emission. The integrated emission map of the region constructed from only those +emission clumps detected above 5σ level in the position-position-velocity space affirms the same. MSX sources +were found distributed along the cavity boundary where the gas has the been compressed. Spectral extraction +at these positions yielded high Mach numbers and low ratios of thermal to non-thermal pressure, suggesting a +dominance of supersonic and non-thermal motion in the cloud. +Keywords. +Interstellar filaments—H II regions—Millimeter astronomy—Star formation—Massive stars +1. Introduction +Formation and evolution of massive stars is a significant +area of research (Zinnecker & Yorke, 2007), as such +stars can affect the evolution of the Galaxy via their im- +mense matter and radiation output. Though it has been +difficult to establish an evolutionary sequence for mas- +sive stars, due to their faster evolution and relatively +rarer occurrences, various high-mass precursors have +been sought to be established, such as massive cold +molecular cores, massive starless clumps, infrared dark +clouds (IRDCs) (Motte et al., 2018), massive young +stellar objects (MYSOs) (Hoare et al., 2005), and so on. +The Lyman continuum radiation from high-mass stars +ionizes the surrounding medium, and the subsequent +expanding H II regions have been subjects of investi- +gation for understanding the impact of radiation output +in triggering or quenching further star formation in the +natal cloud (Elmegreen, 1998, 2011; Ogura, 2010). +Sh2-112 is an optically visible H II region (Dickel +et al., 1969) (Figure 1) (l ∼ 83.758°, b ∼ +03.275°) at +a distance of 2 kpc (Panwar et al., 2020) in the north- +ern Galactic plane. It is associated with the Cygnus +superbubble, and is one of the many Sharpless (Sharp- +less, 1959) H II regions in Cygnus (Uyanıker et al., +2001). The region appears nearly circular at optical +wavelengths, and displays a blister morphology in radio +emission (Israel, 1978). Figure 1 displays a prominent +dark lane against the optical emission. The massive star +BD +45 3216 – which has been estimated to be of spec- +tral type O8V (Lahulla, 1985; Panwar et al., 2020) – lies +close to this lane, offcenter from the optical nebulos- +ity. The Red MSX Source (RMS) survey by Lumsden +et al. (2013) – which aims to catalog massive young +stellar population in the Galaxy – has identified two +sources associated with the Sh2-112 region, namely +G083.7962+03.3058 and G083.7071+03.2817, classi- +fied as H II region and YSO, respectively, in the cata- +log. Though the molecular cloud in this region has been +subject to investigations in literature in various molec- +© Indian Academy of Sciences +1 +arXiv:2301.02048v1 [astro-ph.GA] 5 Jan 2023 + +#### Page 2 of 1 +J. Astrophys. Astr. (0000) 000: #### +83.900 +83.800 +83.700 +3.400 +3.300 +3.200 +Galactic longitude +Galactic latitude +2 pc +Figure 1. DSS2-Red image of the Sh2-112 region. +Plus +symbol marks the massive star BD +45 3216. +Circle and +diamond denote the locations of the two RMS sources, +G083.7962+03.3058 +and +G083.7071+03.2817, +respec- +tively. Dashed boxes show the field of view of the two JCMT +fields. +ular transitions, such as HCN (J=1-0) (Burov et al., +1988), CO (Blitz et al., 1982), 13CO (J=1-0) (Dobashi +et al., 1994, 1996), and CO isotopologues (Urquhart et +al., 2008; Maud et al., 2015b; Panja et al., 2022), they +have either been at a low resolution, and/or as a part of +a large statistical study or a larger region encompass- +ing Sh2-112. As such, a detailed examination of the +molecular cloud associated with Sh2-112 H II region is +pending, a void we try to fill in this paper. +The organisation of this paper is as follows. In sec- +tion 2., we list the datasets used and any processing +steps. This is followed by section 3., where we present +the analysis results for the molecular gas kinematics. +Finally we discuss our results in section 4., followed by +a summary and conclusions section in section 5.. +2. Data Used +2.1 Archival Spectral Data Products +We obtained the archival reduced and calibrated JCMT +(James Clerk Maxwell Telescope) spectral cubes from +the Canadian Astronomy Data Centre1 (CADC), for +1https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ +CO(3-2) (rest frequency = 345.79599 GHz), 13CO(3- +2) (rest frequency = 330.587960 GHz),and C18O(3-2) +(rest frequency = 329.330545 GHz). The J=3–2 tran- +sition traces gas at a critical density of ∼ 104−5 cm−3 +(Buckle et al., 2010). In the direction of the Sh2-112 +region, two fields have been observed by the JCMT, +namely G083.7962+03.3058 and G083.7071+03.2817 +(see Figure 1), using the HARP/ACSIS (Heterodyne +Array Receiver Programme/Auto-Correlation Spectral +Imaging System; Buckle et al., 2009) spectral imag- +ing system. +Cubes for both of these fields were re- +trieved. The temperature scale used for the pixel bright- +ness units is T∗ +A (antenna temperature). A basic pro- +cessing using the starlink kappa (Currie et al., 2014) +package was carried out, wherein the downloaded re- +duced cubes for the two fields were mosaiced; the spec- +tral axis was converted to LSRK velocity scale; and +the coordinate system was transformed to Galactic from +FK5 for ease of analysis. Thereafter the cubes were re- +binned along spectral axis to 0.5 km s−1 channel width. +Table 1 provides the details of the JCMT data used. +While the rebinned cubes were used for the detection +of spatial structures (sections 3.1 and 3.2) as they have +lower RMS noise; for the calculation of physical pa- +rameters (section 3.3) the native channel width (13CO +and/or C18O) cubes were used as high velocity resolu- +tion is required for the same. +For the Sh2-112 region, we also procured archival +spectral cube for 21 cm H I line emission produced by +the Canadian Galactic Plane Survey (CGPS) Consor- +tium (Taylor et al., 2003). We use the cube on an as is +basis for examining the morphology of the region. The +21 cm cube has an angular resolution of ∼1’×1’ cosecδ, +channel width of ∼ 0.8 km s−1, and a velocity resolution +of ∼ 1.3 km s−1. +2.2 Archival Imaging Data Products +We obtained the stellar sources present in MSX (Mid- +course Space Experiment) Catalog 6 (Egan et al., 2003) +for our JCMT field of view, via the NASA/IPAC In- +frared Science Archive2. Though observations by the +MSX satellite were carried out in 6 bands ranging from +4-21 µm, the most sensitive observation has been in the +mid-infrared (MIR) A band (8.28 µm) (Lumsden et al., +2002). The other three MIR bands used by MSX are C +(12.13 µm), D (14.65 µm), and E (21.3 µm). MSX has +surveyed the Galactic plane at MIR bands with a spa- +tial resolution of ∼ 18.3” (Price et al., 2001), and has +helped in uncovering its MIR population, especially the +deeply embedded stellar objects (Lumsden et al., 2002). +2https://irsa.ipac.caltech.edu/ + +J. Astrophys. Astr. (0000)000: #### +Page 3 of 1 #### +Table 1. JCMT Data Used +Line/Wavelength +Spatial +Native Channel +Rebinned Channel +Program +Resolution +Width +Width (& Noise) +IDs Used +CO(3-2) +∼14” +∼0.42 km s−1 +0.5 km s−1 +M08AU19 +(∼0.37 K) +13CO(3-2) +∼14” +∼0.05 km s−1 +0.5 km s−1 +M08AU19, +(∼0.35 K) +M08BU18 +C18O(3-2) +∼14” +∼0.05 km s−1 +0.5 km s−1 +M08AU19, +(∼0.45 K) +M08BU18 +Besides the above, Hα image from the IPHAS3 +(Drew et al., 2005; Barentsen et al., 2014) survey +(Obs date:2003-08-14); the DSS2-Red image4; and the +NVSS (NRAO VLA Sky Survey; Condon et al., 1998) +1.4 GHz continuum image was downloaded for a visual +examination of the region. +3. Results +3.1 Channel Maps +Figure 2 shows the channel map for the Sh2-112 region +in 13CO(3-2) emission. The massive star BD +45 3216, +and the two RMS sources (G083.7962+03.3058 and +G083.7071+03.2817) have been marked on the image. +The filamentary nature of the molecular emission is +clearly brought out in the channel maps. While in the +eastern part, different filamentary structures seem to +join at one end at the RMS source G083.7962+03.3058 +(circle); in the western portion such structures seem to +converge on G083.7071+03.2817 (diamond symbol). +In the velocity range -4.0 to -2.0 km s−1, the two parts +seem to be connected by molecular filaments. +An- +other noticeable feature is that in the eastern part, the +molecular emission forms an arc like structure directed +away from the massive star BD +45 3216 (plus sym- +bol), best delineated in the [-2.0,-1.0] km s−1 panel. +While in the velocity range [-2.0,0.0] km s−1, this arc +appears to be directed towards north; at the blueshifted +velocities, for example [-7.0,-5.0] km s−1 range, it ap- +pears curved towards the south. In the middle veloc- +ity ranges, [-5.0,-2.0] km s−1, no proper pattern can be +made out. It is possible that two separate filaments, +seen at blueshifted and redshifted velocities, respec- +tively, are merging at this juncture. Towards the north- +west of the massive star BD +45 3216, there appears +to be a cavity where hardly any molecular emission is +seen. Furthermore, the western emission centered on +3https://www.iphas.org/images/ +4https://archive.eso.org/dss/dss +the G083.7071+03.2817 source is also directed away +from the massive star. +3.2 Moment Maps +In this section, we examine the m-0 (moment-0 or Inte- +grated Intensity), m-1 (moment-1 or Intensity-weighted +velocity), and linewidth (Intensity-weighted disper- +sion) maps of the molecular emission in the Sh2-112 +region. The examination is confined to only those re- +gions which are detected at ≥ 5σ detection level in the +spectral cubes (σ being the rms noise level of the re- +spective cubes). To achieve this, the clumpfind algo- +rithm of Williams et al. (1994) was utilised via starlink +software’s (Currie et al., 2014) cupid package (Berry et +al., 2007). The 5σ threshold was conservatively cho- +sen to eliminate false detections of clumps. As rec- +ommended by Williams et al. (1994), the gap between +contour levels was kept at 2σ. With these values, the +clumpfind algorithm was implemented on the mosaiced +CO(3-2), 13CO(3-2), and C18O(3-2) position-position- +velocity cubes. After the detection of clumps, we mask +those regions where no emission was detected at 5σ or +above to create a masked cube. Thereafter, the respec- +tive (masked) cubes were collapsed to create m-0, m-1, +and linewidth images. +The maps are shown in Figure 3 for CO(3-2), +13CO(3-2), and C18O(3-2) molecular transitions. The +massive star BD +45 3216, and the two RMS sources +(G083.7962+03.3058 and G083.7071+03.2817) have +been marked on each of the images. In the m-0 maps, +CO(3-2) emission appears to be diffused with no par- +ticular discernible structure, which is expected given +the ubiquitous nature of CO molecule and compara- +tively lower critical density than the other two tran- +sitions. +However, it is still worth noting that much +of the CO(3-2) emission still “faces away” from the +massive star BD +45 3216. The 13CO(3-2) m-0 map +most clearly traces the molecular filamentary struc- +tures which were examined in the channel maps (Fig- +ure 2). While the cavity to the west of the massive star +shows no 13CO(3-2) emission, the connecting filamen- + +#### Page 4 of 1 +J. Astrophys. Astr. (0000) 000: #### +[-8.0,-7.0] +[-7.0,-6.0] +Arc +[-6.0,-5.0] +[-5.0,-4.0] +[-4.0,-3.0] +Cavity +Connecting +Filament +[-3.0,-2.0] +83°50' +45' +40' +3°25' +20' +15' +10' +Galactic Longitude +Galactic Latitude +Arc +[-2.0,-1.0] +[-1.0,0.0] +[0.0,1.0] +0 +2 +4 +6 +8 +10 +12 +14 +K km s +1 +Figure 2. Channel maps of 13CO(3-2) emission. Plus symbol marks the massive star BD +45 3216. Circle and diamond +denote the locations of the two RMS sources, G083.7962+03.3058 and G083.7071+03.2817, respectively. An arc like +filamentary structure, a cavity like structure, and a connecting filament have been indicated by arrows. + +J. Astrophys. Astr. (0000)000: #### +Page 5 of 1 #### +tary structure between the eastern and western regions +is prominent. Given that all emission here is at ≥ 5σ +level, this indicates that there is significant molecular +mass concentration in the connecting filament and it is +not some minor emission structure. The critical den- +sity is highest for C18O(3-2) transition, and thus only +the densest clumps are detected here. Unsurprisingly, +the bulk of C18O(3-2) emission is associated with the +positions of RMS sources. The prominent arc like fil- +amentary structure marked on the 13CO(3-2) channel +map (Figure 2) is also partly seen in C18O(3-2) m-0 +map, indicating the presence of dense gas along it. The +linewidth maps for all three transitions show a max- +ima towards the positions of the RMS sources as com- +pared to the outer parts, which could be due to outflow +activity and/or possibly suggest a convergence of dif- +ferent flows towards these sources. In the 13CO(3-2) +linewidth map, the connecting filament has a relatively +lower velocity dispersion, while the arc like filamentary +structure (Figure 2) displays a relatively larger disper- +sion along its length as compared to others. +3.3 Physical Parameters +Figure 4 shows the 13CO(3-2) m-0 map of the re- +gion overlaid with the massive star BD +45 3216; +the two RMS sources (G083.7962+03.3058 +and +G083.7071+03.2817); and the MSX sources (m1 to +m11) which lay in the regions of emission ≥ 5σ +level (see section 3.2). The local peaks of m-0 emis- +sion, based on visual examination of the contours have +also been marked on the image (p1 to p10). To ob- +tain the physical parameters for these sources (except +BD +45 3216 which did not have any molecular emis- +sion associated with it), we extracted the spectra at +their locations (from a 2×2 pixel area). +As high- +velocity resolution is required for the calculations, the +non-averaged cubes with the native channel width of +∼0.05 km s−1 were utilised for spectrum extraction. +For the RMS sources, both 13CO(3-2) and C18O(3- +2) spectra were available, and are shown in Figure 5. +However, for the MSX sources (m1-m11) and the lo- +cal 13CO(3-2) integrated emission peaks (p1-p10), only +13CO(3-2) spectra could be extracted, shown in Figures +6 and 7, respectively. The results from the gaussian fit- +ting are given in Table 2. The first thing to notice is +that at some of the locations, a simple gaussian is a +poor fit to the spectra. For example, in Figure 5, the +spectra have broad wings on both sides, which is usu- +ally an indication of some kind of likely outflow as- +sociated with the sources. According to Maud et al. +(2015b), while G083.7071+03.2817 is associated with +an outflow, G083.7962+03.3058 has a possibility of +the same. For some of the locations in Figures 6 and 7, +there appears to be excess emission on the red side. It +had been discussed in section 3.1 that the arc like struc- +ture in Figure 2 probably has a merger of blueshifted +and redshifted filaments. The excess redshifted emis- +sion seen at the locations which lie on this arc, i.e. m1 +(Figure 6), and p1, p2 (Figure 7) could be an indica- +tion of the same. Emission peak positions p8 and p9 +display significant self-absorption in their spectra. The +mean velocity for most of the fits lies in the range - +2.5 to -4.0 km s−1. It is notable that the locations to- +wards the eastern side (namely p1, p2, p3, and m1) +have significantly more blueshifted mean velocities as +opposed to other locations. The RMS sources, though +with wide wings, have nearly same mean velocities (∼ - +4.0 km s−1), which is in agreement with the radial ve- +locity for the Sh2-112 region in Blitz et al. (1982); Stark +& Brand (1989); Brand & Blitz (1993). Though our lo- +cations display a range of mean velocities, the overall +distribution is consistent with other values cited in liter- +ature within error limits (Dobashi et al., 1994; Urquhart +et al., 2008; Lumsden et al., 2013; Maud et al., 2015a,b; +Panja et al., 2022). +Using the FWHM (full width at half maxima) re- +turned by the fit, we calculate the standard deviation +(or observed velocity dispersion), the non-thermal ve- +locity dispersion, and the total velocity dispersion using +the following set of equations (Fuller & Myers, 1992; +Fiege & Pudritz, 2000) : +∆V2 +tot += +∆V2 +obs + 8 ln 2 kT +� 1 +¯m − +1 +mobs +� +(1) +⇒ ∆V2 +tot +8 ln 2 += +kT +¯m + +������ +∆V2 +obs +8 ln 2 − kT +mobs +������ +⇒ σ2 +tot += +c2 +s + +� +σ2 +obs − σ2 +t +� +(2) += +c2 +s + σ2 +nt . +(3) +In the above equations, ∆Vobs is the FWHM of the +fit; σobs (= ∆Vobs/ +√ +8ln 2 for gaussian fits) is the stan- +dard deviation (or dispersion); σt(= √kT/mobs) is the +thermal velocity dispersion; mobs is the mass of the rele- +vant molecule (29 amu and 30 amu for 13CO and C18O +respectively); σnt is the non-thermal velocity disper- +sion; cs(= √kT/ ¯m) is the speed of sound; ¯m is the av- +erage molecular weight of the medium (2.37 amu); and +T is the excitation or gas kinetic temperature. +The FWHM shows a wide range at these locations, +ranging from ∼ 1.5-3.0 km s−1 for most of the sources, +with some of the highest values associated with lo- +cations in the vicinity of the two RMS sources, such +as m2, m3, m4, m11, and p10. This tallies with the +linewidth map in Figure 3(h). For the above calcula- +tions, we take the excitation temperature T as 20 K (see +Figure 15 in Panja et al., 2022). Furthermore, we also + +#### Page 6 of 1 +J. Astrophys. Astr. (0000) 000: #### +3°25' +20' +15' +10' +Galactic Latitude +a). CO(3-2) +m-0 +0 +50 +100 +150 +K km s +1 +3°25' +20' +15' +10' + +b). 13CO(3-2) +m-0 +0 +20 +40 +K km s +1 +3°25' +20' +15' +10' + +c). C18O(3-2) +m-0 +0 +5 +10 +K km s +1 +3°25' +20' +15' +10' +Galactic Latitude +d). CO(3-2) m-1 +4 +2 +0 +km s +1 +3°25' +20' +15' +10' + +e). 13CO(3-2) m-1 +4 +2 +0 +km s +1 +3°25' +20' +15' +10' + +f). C18O(3-2) m-1 +4 +2 +0 +km s +1 +83°50' +45' +40' +3°25' +20' +15' +10' +Galactic Longitude +Galactic Latitude +g). CO(3-2) linewidth +0 +2 +4 +km s +1 +83°50' +45' +40' +3°25' +20' +15' +10' +Galactic Longitude + +h). 13CO(3-2) linewidth +1 +2 +km s +1 +83°50' +45' +40' +3°25' +20' +15' +10' +Galactic Longitude + +i). C18O(3-2) linewidth +0.2 +0.4 +0.6 +0.8 +km s +1 +Figure 3. Row-wise : moment-0 (Integrated intensity), moment-1 (Intensity-weighted velocity), and linewidth (Intensity- +weighted dispersion) collapsed images for three cubes – CO(3-2), 13CO(3-2), C18O(3-2) in first, second, and third columns, +respectively. The symbols are same as Figure 2. + +J. Astrophys. Astr. (0000)000: #### +Page 7 of 1 #### +calculate Mach number (= σnt/cs) and the ratio of ther- +mal to non-thermal pressure (Ptnt = c2 +s/σ2 +nt) (Lada et +al., 2003) for each of the locations. The results of these +two calculations show the presence of supersonic mo- +tion, and a dominance of non-thermal pressure in the +cloud. The mach number and Ptnt values are inversely +correlated, as expected from their dependence on σnt. +Such values would suggest that the emission mecha- +nism is likely some supersonic non-thermal phenom- +ena, and could be via turbulence and magnetic fields +(Myers & Goodman, 1988; Crutcher, 1999), to suggest +one such mechanism. +4. Discussion +Figure 8 shows the CGPS 21 cm velocity channel maps +of this region. The channel maps show a depression in +H i emission (see Figure 9), which is (anti-)correlated +with strong (i.e. ≥ 5σ) molecular emission. Such fea- +tures have been referred to as H i self absorption (or +HISA) features in literature, and have been found to +be an extensive presence in 21 cm surveys (Kerton, +2005; Wang et al., 2020). HISA regions indicate the +presence of cold H I gas in foreground against warm +H I emission from background. In our channel maps, +most intense depression in emission seems to be con- +fined to the north-eastern quadrant, in the vicinity of +the source G083.7962+03.3058 (circle). Channels - +2.3 km s−1 and -3.1 km s−1 display shell-like feature on +a smaller spatial scale, while the channels -4.8 km s−1 +and -5.6 km s−1 show large scale shell-like feature. +The anti-correlation of molecular emission and depres- +sion in H I emission can be most prominently seen in +the -4.8 km s−1 and -5.6 km s−1 channels. Such shell- +like features have been found in other regions as well, +such as Sh2-237 (Dewangan et al., 2017), W4 and W5 +(Hosokawa & Inutsuka, 2007), and in Southern Galac- +tic Plane Survey (SGPS) regions (McClure-Griffiths et +al., 2001). +Figure 10 shows the 13CO(3-2) m-0 image (over- +laid with NVSS contours), the Hα image of the region, +and the continuum-subtracted H2 emission map of the +region from Panwar et al. (2020). The radio emission +seems to have multiple peaks, and on a comparison of +Figures 4 and 10(a), the radio peaks appear to be as- +sociated with the locations of m3/m4, p5, and a loca- +tion to the east of p10. +According to Panwar et al. +(2020), the location to the east of p10 is the ionized +boundary layer. Based on the presence of this ionized +boundary layer and pressure calculations, they further +conjecture the possibility of triggered star formation to- +wards the location of RMS source G083.7071+03.2817 +due to O8V type star BD +45 3216 (also see Morgan +et al., 2004, 2009; Urquhart et al., 2009). It is pos- +sible that the expansion of the ionized gas could have +had a role in the formation of the cavity towards the +west/northwest of the massive star BD +45 3216. There +is significant Hα emission in the cavity, while the H2 +2.12µm emission seems to trace the southern boundary +of the molecular cloud. All along the cavity perime- +ter, there are finger-like filamentary structures protrud- +ing into it, reminiscent of bright-rimmed clouds which +are seen in optical and infrared emission (Chauhan et +al., 2011; Sharma et al., 2016). Here these structures +are seen in absorption in optical (Figure 10(b)) and in +emission in 13CO(3-2) integrated intensity map (Figure +10(a)). Two out of three of these finger-like structures +seem to be hosting molecular peaks at their ends, i.e. +p5 and p10. If there is a possible case of triggering to- +wards the source G083.7071+03.2817, then based on +size considerations of the cavity, there could be possi- +ble triggering all along the cavity perimeter. The MSX +sources m3, m4, m5, m7, m8, m9, and m11 seem to +be distributed along the cavity boundary where there is +seemingly compression of molecular material, as evi- +denced by the sharp change in 13CO(3-2) contour lev- +els (see Figure 4). +Lastly we present the position-velocity maps in +Figure 11 along the different line segments marked +L1-L8 on the central image. +While the lines L1- +L5 trace the filamentary structures which converge on +the source G083.7962+03.3058 (circle), L6-L8 are +cuts along the filamentary structures associated with +G083.7071+03.2817 (diamond symbol). The position- +velocity maps show a complex structure with clumping +along their length. Significant velocity gradients can +also be seen in most of them, and especially in L1- +L5 in the vicinity of the source G083.7962+03.3058 +(i.e. bright clump towards the right in L1-L5). Here we +note that such a gradient could indicate gas being chan- +neled towards the source, and such a flow of gas along +the filamentary structures towards the RMS sources +– G083.7962+03.3058 and G083.7071+03.2817 – as +also discussed in section 3.2, is similar to the longi- +tudinal flow in filaments towards hubs in hub-filament +systems (Dewangan et al., 2017, 2020; Williams et al., +2018). Therefore, this region could present a good ex- +ample of exploring various star formation frameworks +such as global hierarchical collapse (GHC V´azquez- +Semadeni et al., 2019), conveyor belt model (Longmore +et al., 2014; Krumholz & McKee, 2020), filaments to +clusters model (Kumar et al., 2020), to name a few. +According to Panja et al. (2022), the column den- +sity of the molecular emission region has been calcu- +lated to be ∼1022 cm−2. Such regions have been desig- +nated as “hubs” in the context of hub-filament systems +in literature (Myers, 2009). The column density maps + +#### Page 8 of 1 +J. Astrophys. Astr. (0000) 000: #### +83°51' +48' +45' +42' +39' +3°22' +20' +18' +16' +14' +Galactic Longitude +Galactic Latitude +13CO(3-2) m-0 +m1 +m2 +m3 +m4 +m5 +m6 +m7 +m8 +m9 +m10 +m11 +p1 +p2 +p3 +p4 +p5 +p6 +p7 +p8 +p9 +p10 +1 pc +Figure 4. +13CO(3-2) integrated intensity map (contours at 2, 3.5, 6, 8, 10, 12, 14, 17, 21, 25, 30, 34, 40, and 45 K km s−1). +Plus symbol marks the massive star BD +45 3216. The two RMS sources, G083.7962+03.3058 and G083.7071+03.2817, +have been shown by circle and diamond symbols, respectively. The MSX sources have been marked in green boxes and la- +belled m1 to m11. Cyan boxes (labelled p1 to p10) are the locations of local contour peaks where also we extracted the spectra. +−15 +−10 +−5 +0 +5 +10 +VRAD (km s-1) +0 +4 +8 +12 +K +G083.7071+03.2817 +−15 +−10 +−5 +0 +5 +10 +VRAD (km s-1) +0 +4 +8 +12 +G083.7962+03.3058 +Figure 5. +13CO(3-2) (blue) and C18O(3-2) (red) spectra at positions of RMS sources marked in Figure 4. Green and +yellow curves show the gaussian fits to the respective spectra, with blue dashed line marking the velocity of the peak of the +13CO(3-2) gaussian fit. + +J. Astrophys. Astr. (0000)000: #### +Page 9 of 1 #### +0 +2 +4 +6 +m1 +0 +2 +4 +6 +m2 +0 +2 +4 +6 +m3 +0 +2 +4 +6 +m4 +0 +2 +4 +6 +m5 +0 +2 +4 +6 +m6 +0 +2 +4 +6 +m7 +0 +2 +4 +6 +m8 +-7.5 +-2.5 +2.5 +0 +2 +4 +6 +m9 +-7.5 +-2.5 +2.5 +VRAD (km s-1) +0 +2 +4 +6 +K +m10 +-7.5 +-2.5 +2.5 +0 +2 +4 +6 +m11 +Figure 6. +13CO(3-2) spectra at MSX source positions marked in Figure 4 (green boxes labelled m1-m11). Green curve +depicts the gaussian fit to the spectra, with blue dashed line marking the velocity of the peak of the gaussian fit. +VRAD (km s-1) +0 +5 +10 +K +p1 +0 +5 +10 +p2 +0 +5 +10 +p3 +0 +5 +10 +p4 +0 +5 +10 +p5 +−10 +−5 +0 +5 +VRAD (km s-1) +0 +5 +10 +K +p6 +−10 +−5 +0 +5 +0 +5 +10 +p7 +−10 +−5 +0 +5 +0 +5 +10 +p8 +−10 +−5 +0 +5 +0 +5 +10 +p9 +−10 +−5 +0 +5 +0 +5 +10 +p10 +Figure 7. +13CO(3-2) spectra at peak positions marked in Figure 4 (cyan boxes labelled p1-p10). Green curve depicts the +gaussian fit to the spectra, with blue dashed line marking the velocity of the peak of the gaussian fit. + +#### Page 10 of 1 +J. Astrophys. Astr. (0000) 000: #### +Table 2. Parameters derived from 13CO(3-2) spectra at the locations marked in Figure 4. For G083.7071+03.2817 and +G083.7962+03.3058, C18O(3-2) spectrum was also available and thus used for calculation as well. +clump +Mean +FWHM +Amplitude +σNT +Mach +PTNT +l +b +(km s−1) +(km s−1) +(K) +(km s−1) +Number +(deg) +(deg) +G083.7071+03.2817 +C18O +-3.88 +1.79 +3.18 +0.76 +2.86 +0.12 +83.7071 +03.2817 +13CO +-4.07 +2.48 +11.00 +1.05 +3.97 +0.06 +- +- +G083.7962+03.3058 +C18O +-4.11 +2.69 +2.62 +1.14 +4.30 +0.05 +83.7962 +3.3058 +13CO +-4.00 +3.58 +10.99 +1.52 +5.73 +0.03 +- +- +MSX Sources +m1 +-4.41 +1.92 +4.78 +0.81 +3.06 +0.11 +83.8330 +3.2881 +m2 +-3.53 +2.24 +6.48 +0.95 +3.59 +0.08 +83.7978 +3.3207 +m3 +-1.91 +2.81 +4.52 +1.19 +4.49 +0.05 +83.7835 +3.3079 +m4 +-2.99 +3.09 +1.75 +1.31 +4.95 +0.04 +83.7803 +3.3174 +m5 +-3.35 +1.47 +6.74 +0.62 +2.34 +0.18 +83.7721 +3.3279 +m6 +-2.54 +2.09 +3.75 +0.89 +3.34 +0.09 +83.7607 +3.3313 +m7 +-3.62 +1.25 +4.38 +0.53 +1.99 +0.25 +83.7502 +3.3258 +m8 +-3.56 +1.72 +3.68 +0.73 +2.74 +0.13 +83.7359 +3.3156 +m9 +-3.84 +1.97 +4.88 +0.83 +3.15 +0.10 +83.7310 +3.3121 +m10 +-2.54 +1.89 +4.20 +0.80 +3.02 +0.11 +83.7112 +3.3107 +m11 +-2.74 +2.87 +4.45 +1.22 +4.60 +0.05 +83.7237 +3.2822 +m-0 contour peaks +p1 +-4.32 +1.11 +5.14 +0.47 +1.76 +0.32 +83.8544 +3.2968 +p2 +-4.87 +2.28 +6.74 +0.96 +3.64 +0.08 +83.8336 +3.2942 +p3 +-4.77 +2.27 +5.49 +0.96 +3.62 +0.08 +83.8170 +3.3171 +p4 +-3.22 +1.97 +10.01 +0.83 +3.15 +0.10 +83.7776 +3.3288 +p5 +-2.39 +1.67 +3.89 +0.70 +2.66 +0.14 +83.7634 +3.3152 +p6 +-3.63 +1.91 +6.85 +0.81 +3.05 +0.11 +83.7530 +3.3291 +p7 +-3.34 +1.97 +5.46 +0.83 +3.14 +0.10 +83.7405 +3.3299 +p8 +-3.87 +2.66 +4.38 +1.12 +4.25 +0.05 +83.7053 +3.3157 +p9 +-3.11 +3.01 +8.72 +1.28 +4.82 +0.04 +83.7194 +3.3005 +p10 +-2.80 +3.30 +10.26 +1.40 +5.29 +0.04 +83.7219 +3.2869 + +J. Astrophys. Astr. (0000)000: #### +Page 11 of 1 #### +1 pc +0.2 km s +1 +1 pc +-0.6 km s +1 +1 pc +-1.5 km s +1 +1 pc +-2.3 km s +1 +1 pc +-3.1 km s +1 +1 pc +-3.9 km s +1 +83°51' +48' +45' +42' +39' +3°22' +20' +18' +16' +14' +Galactic Longitude +Galactic Latitude +1 pc +-4.8 km s +1 +1 pc +-5.6 km s +1 +1 pc +-6.4 km s +1 +60 +80 +100 +120 +K +Figure 8. +Velocity channel maps for the CGPS H I 21 cm emission. Green contour marks the 2 K km s−1 level of the +13CO(3-2) m-0 (Figure 4) image. In -2.3 and -3.1 km s−1 channels, the red contours have been drawn at (outer to inner) 65, +60, 55, and 50 K; while in -4.8 and -5.6 km s−1 channels, the red contours are at 81 (outer) and 65 (inner) K. The dashed +green circle in -3.1 km s−1 channel shows the location where spectrum was extracted (see Figure 9). The rest of the symbols +are same as Figure 4. + +#### Page 12 of 1 +J. Astrophys. Astr. (0000) 000: #### +-15 +-10 +-5 +0 +5 +10 +15 +velocity (km s +1) +0 +20 +40 +60 +80 +100 +K +Figure 9. +13CO(3-2) spectrum (in red) and CGPS H i +spectrum (in blue) at the location marked in Figure 8 (green +circle in -3.1 km s−1 +channel) – demonstrating the H i +self-absorption feature centered at ∼ [-5,-4] km s−1 and its +anti-correlation with the molecular emission. The 13CO(3-2) +spectrum has been scaled up by a factor of 30× for better +visibility. +by Panja et al. (2022) also suggest such a hub-filament +configuration for the larger region of the order of a few +10s of parsec encompassing Sh2-112. Thus, what we +have explored here seems to be the detailed structure +of a hub region where massive star formation is go- +ing on. However, given the importance of filamentary +structures in star formation and expositions of various +filament types in literature (see Hacar et al., 2022, for a +detailed review), it is essential to further study such re- +gions via high spectral and spatial resolution molecular +transitions of other species, magnetic fields, and so on +so as to understand the evolution of molecular clouds +as they form stars. +5. Summary and Conclusions +We have carried out an analysis of the Sh2-112 region +in CO(3-2), 13CO(3-2), and C18O(3-2) molecular line +transitions from the JCMT, supported by archival data +from CGPS H I line, MSX, NVSS, and IPHAS Hα for +visual examination. Our main conclusions are as fol- +lows : +1. The molecular emission appears filamentary in +channel maps and seems to be directed away +from the massive star BD +45 3216, which is also +located at the edge of a cavity-like structure. +83°51' +48' +45' +42' +39' +3°22' +20' +18' +16' +14' + +Galactic Latitude +13CO(3-2) m-0 +(a). +1 pc +83°51' +48' +45' +42' +39' +3°22' +20' +18' +16' +14' + +Galactic Latitude +H +(b). +1 pc +83°51' +48' +45' +42' +39' +3°22' +20' +18' +16' +14' +Galactic Longitude +Galactic Latitude +H2 +(c). +1 pc +Figure 10. +(a) 13CO(3-2) m-0 map. Cyan contours are +NVSS levels at 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, +0.08, 0.09, and 0.1 Jy/beam. (b) Hα image of the region. +(c) Continuum-subtracted H2 emission (2.12 µm) map from +Panwar et al. (2020). Green contour marks the 2 K km s−1 +level of the 13CO(3-2) m-0 (Figure 4) image. The rest of the +symbols are same as Figure 4. + +J. Astrophys. Astr. (0000)000: #### +Page 13 of 1 #### +83°51' +48' +45' +42' +39' +3°22' +20' +18' +16' +14' +Galactic Longitude +Galactic Latitude +L1 +L2 +L3 +L4 +L5 +L6 +L7 +L8 +1 pc +0.00 +0.02 +0.04 +0.06 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L1 +0.00 +0.02 +0.04 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L2 +0.00 +0.02 +0.04 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L3 +0.00 +0.02 +0.04 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L4 +0.00 +0.02 +0.04 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L5 +0.00 +0.02 +0.04 +0.06 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L6 +0.00 +0.02 +0.04 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L7 +0.00 +0.02 +0.04 +-10 +-5 +0 +5 +OFFSET (deg) +VRAD (km/s) +L8 +0 +2 +4 +6 +8 +10 +12 +T* +A (K) +Figure 11. Position-velocity diagrams for the line segments marked L1-L8 (in red) on the central image (13CO(3-2) m-0 +from Figure 4). Green contour marks the 2 K km s−1 level of the 13CO(3-2) m-0 (Figure 4) image. The rest of the symbols +are same as Figure 4. On the L1-L8 p-v maps, the contour levels are at 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, and 10 K. + +#### Page 14 of 1 +J. Astrophys. Astr. (0000) 000: #### +2. Multiple local peaks were found associated +with the molecular emission in the +13CO(3- +2) +integrated +intensity +emission +map +(m- +0) which was generated using clumps de- +tected above 5σ level in the position-position- +velocity space. +The linewidth map shows +high dispersion associated with the positions +of the RMS sources (G083.7962+03.3058 and +G083.7071+03.2817). +3. Analysis of CGPS 21 cm H i line emission re- +veals the presence of shell-like HISA feature, +where the molecular emission is nearly coinci- +dent with the depression in H i emission. +4. 13CO(3-2) spectra was extracted at the loca- +tions of RMS sources, MSX sources, and the +local peaks of emission. For the RMS sources, +C18O(3-2) spectra was also extracted. Spectral +profile fitting suggests significant deviation from +a gaussian profile for many sources. All the loca- +tions were found to have significant non-thermal +dispersions; large mach numbers (∼ 2–6) indicat- +ing dominance of supersonic motions within the +clumps; and a small thermal to non-thermal pres- +sure ratio (∼ 0.03–0.3). +Acknowledgements +We thank the anonymous referee for a critical reading +of the manuscript and for the suggestions for the im- +provement of this paper. DKO acknowledges the sup- +port of the Department of Atomic Energy, Government +of India, under project Identification No. RTI 4002. +The James Clerk Maxwell Telescope has historically +been operated by the Joint Astronomy Centre on be- +half of the Science and Technology Facilities Council +of the United Kingdom, the National Research Council +of Canada and the Netherlands Organisation for Sci- +entific Research. 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W. 2007, ARA&A, 45, 481. +doi:10.1146/annurev.astro.44.051905.092549 + diff --git a/btA0T4oBgHgl3EQfGf-a/content/tmp_files/load_file.txt b/btA0T4oBgHgl3EQfGf-a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe945723a4995017cb243ca4a67e616516c4bbcb --- /dev/null +++ b/btA0T4oBgHgl3EQfGf-a/content/tmp_files/load_file.txt @@ -0,0 +1,1234 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf,len=1233 +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000) 000: #### DOI Investigating the morphology and CO gas kinematics of Sh2-112 region Kshitiz K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Mallick1, Saurabh Sharma1, Lokesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Dewangan2, Devendra K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Ojha3, Neelam Panwar1, Tapas Baug4 1Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital, 263002, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 2Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 3Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mum- bai 400005, India 4S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, West Bengal, India Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' E-mail: kshitiz@aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='in MS received 1 January 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' accepted 1 January 2015 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' We present a study of the molecular cloud in Sh2-112 massive star forming region using the 3–2 transition of CO isotopologues - CO, 13CO, and C18O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' supplemented in part by CGPS H I line emission and MSX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Sh2-112 is an optically visible region powered by an O8V type massive star BD +45 3216, and hosts two Red MSX Survey sources – G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 – classified as H II region and young stellar object, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Reduced spectral data products from the James Clerk Maxwell Telescope archive, centered on the two RMS objects with ∼7’×7’ field of view each, were utilised for the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The 13CO(3-2) channel map of the region shows the molecular cloud to have filamentary extensions directed away from the massive star, which also seems to be at the edge of a cavity like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Multiple molecular cloud protrusions into this cavity structure host local peaks of emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The integrated emission map of the region constructed from only those emission clumps detected above 5σ level in the position-position-velocity space affirms the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' MSX sources were found distributed along the cavity boundary where the gas has the been compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Spectral extraction at these positions yielded high Mach numbers and low ratios of thermal to non-thermal pressure, suggesting a dominance of supersonic and non-thermal motion in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Interstellar filaments—H II regions—Millimeter astronomy—Star formation—Massive stars 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Introduction Formation and evolution of massive stars is a significant area of research (Zinnecker & Yorke, 2007), as such stars can affect the evolution of the Galaxy via their im- mense matter and radiation output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Though it has been difficult to establish an evolutionary sequence for mas- sive stars, due to their faster evolution and relatively rarer occurrences, various high-mass precursors have been sought to be established, such as massive cold molecular cores, massive starless clumps, infrared dark clouds (IRDCs) (Motte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2018), massive young stellar objects (MYSOs) (Hoare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2005), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The Lyman continuum radiation from high-mass stars ionizes the surrounding medium, and the subsequent expanding H II regions have been subjects of investi- gation for understanding the impact of radiation output in triggering or quenching further star formation in the natal cloud (Elmegreen, 1998, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Ogura, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Sh2-112 is an optically visible H II region (Dickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 1969) (Figure 1) (l ∼ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='758°, b ∼ +03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='275°) at a distance of 2 kpc (Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2020) in the north- ern Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' It is associated with the Cygnus superbubble, and is one of the many Sharpless (Sharp- less, 1959) H II regions in Cygnus (Uyanıker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The region appears nearly circular at optical wavelengths, and displays a blister morphology in radio emission (Israel, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Figure 1 displays a prominent dark lane against the optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The massive star BD +45 3216 – which has been estimated to be of spec- tral type O8V (Lahulla, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2020) – lies close to this lane, offcenter from the optical nebulos- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The Red MSX Source (RMS) survey by Lumsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2013) – which aims to catalog massive young stellar population in the Galaxy – has identified two sources associated with the Sh2-112 region, namely G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817, classi- fied as H II region and YSO, respectively, in the cata- log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Though the molecular cloud in this region has been subject to investigations in literature in various molec- © Indian Academy of Sciences 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02048v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='GA] 5 Jan 2023 #### Page 2 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000) 000: #### 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='900 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='800 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='400 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='200 Galactic longitude Galactic latitude 2 pc Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' DSS2-Red image of the Sh2-112 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Plus symbol marks the massive star BD +45 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Circle and diamond denote the locations of the two RMS sources, G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Dashed boxes show the field of view of the two JCMT fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' ular transitions, such as HCN (J=1-0) (Burov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 1988), CO (Blitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 1982), 13CO (J=1-0) (Dobashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 1994, 1996), and CO isotopologues (Urquhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Maud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Panja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2022), they have either been at a low resolution, and/or as a part of a large statistical study or a larger region encompass- ing Sh2-112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' As such, a detailed examination of the molecular cloud associated with Sh2-112 H II region is pending, a void we try to fill in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The organisation of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', we list the datasets used and any processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' This is followed by section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', where we present the analysis results for the molecular gas kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Finally we discuss our results in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', followed by a summary and conclusions section in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='. 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Data Used 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 Archival Spectral Data Products We obtained the archival reduced and calibrated JCMT (James Clerk Maxwell Telescope) spectral cubes from the Canadian Astronomy Data Centre1 (CADC), for 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='cadc-ccda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='hia-iha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='nrc-cnrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='ca/en/ CO(3-2) (rest frequency = 345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='79599 GHz), 13CO(3- 2) (rest frequency = 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='587960 GHz),and C18O(3-2) (rest frequency = 329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='330545 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The J=3–2 tran- sition traces gas at a critical density of ∼ 104−5 cm−3 (Buckle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In the direction of the Sh2-112 region, two fields have been observed by the JCMT, namely G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 (see Figure 1), using the HARP/ACSIS (Heterodyne Array Receiver Programme/Auto-Correlation Spectral Imaging System;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Buckle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2009) spectral imag- ing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Cubes for both of these fields were re- trieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The temperature scale used for the pixel bright- ness units is T∗ A (antenna temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' A basic pro- cessing using the starlink kappa (Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2014) package was carried out, wherein the downloaded re- duced cubes for the two fields were mosaiced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' the spec- tral axis was converted to LSRK velocity scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' and the coordinate system was transformed to Galactic from FK5 for ease of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Thereafter the cubes were re- binned along spectral axis to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 km s−1 channel width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Table 1 provides the details of the JCMT data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' While the rebinned cubes were used for the detection of spatial structures (sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2) as they have lower RMS noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' for the calculation of physical pa- rameters (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3) the native channel width (13CO and/or C18O) cubes were used as high velocity resolu- tion is required for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For the Sh2-112 region, we also procured archival spectral cube for 21 cm H I line emission produced by the Canadian Galactic Plane Survey (CGPS) Consor- tium (Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' We use the cube on an as is basis for examining the morphology of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The 21 cm cube has an angular resolution of ∼1’×1’ cosecδ, channel width of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='8 km s−1, and a velocity resolution of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2 Archival Imaging Data Products We obtained the stellar sources present in MSX (Mid- course Space Experiment) Catalog 6 (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2003) for our JCMT field of view, via the NASA/IPAC In- frared Science Archive2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Though observations by the MSX satellite were carried out in 6 bands ranging from 4-21 µm, the most sensitive observation has been in the mid-infrared (MIR) A band (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='28 µm) (Lumsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The other three MIR bands used by MSX are C (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='13 µm), D (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='65 µm), and E (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' MSX has surveyed the Galactic plane at MIR bands with a spa- tial resolution of ∼ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3” (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2001), and has helped in uncovering its MIR population, especially the deeply embedded stellar objects (Lumsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 2https://irsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='edu/ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000)000: #### Page 3 of 1 #### Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' JCMT Data Used Line/Wavelength Spatial Native Channel Rebinned Channel Program Resolution Width Width (& Noise) IDs Used CO(3-2) ∼14” ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='42 km s−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 km s−1 M08AU19 (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='37 K) 13CO(3-2) ∼14” ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='05 km s−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 km s−1 M08AU19, (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='35 K) M08BU18 C18O(3-2) ∼14” ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='05 km s−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 km s−1 M08AU19, (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='45 K) M08BU18 Besides the above, Hα image from the IPHAS3 (Drew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Barentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2014) survey (Obs date:2003-08-14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' the DSS2-Red image4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' and the NVSS (NRAO VLA Sky Survey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 1998) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='4 GHz continuum image was downloaded for a visual examination of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 Channel Maps Figure 2 shows the channel map for the Sh2-112 region in 13CO(3-2) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The massive star BD +45 3216, and the two RMS sources (G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817) have been marked on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The filamentary nature of the molecular emission is clearly brought out in the channel maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' While in the eastern part, different filamentary structures seem to join at one end at the RMS source G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 (circle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' in the western portion such structures seem to converge on G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 (diamond symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In the velocity range -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0 to -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0 km s−1, the two parts seem to be connected by molecular filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' An- other noticeable feature is that in the eastern part, the molecular emission forms an arc like structure directed away from the massive star BD +45 3216 (plus sym- bol), best delineated in the [-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] km s−1 panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' While in the velocity range [-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] km s−1, this arc appears to be directed towards north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' at the blueshifted velocities, for example [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] km s−1 range, it ap- pears curved towards the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In the middle veloc- ity ranges, [-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] km s−1, no proper pattern can be made out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' It is possible that two separate filaments, seen at blueshifted and redshifted velocities, respec- tively, are merging at this juncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Towards the north- west of the massive star BD +45 3216, there appears to be a cavity where hardly any molecular emission is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Furthermore, the western emission centered on 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='iphas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='org/images/ 4https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='org/dss/dss the G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 source is also directed away from the massive star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2 Moment Maps In this section, we examine the m-0 (moment-0 or Inte- grated Intensity), m-1 (moment-1 or Intensity-weighted velocity), and linewidth (Intensity-weighted disper- sion) maps of the molecular emission in the Sh2-112 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The examination is confined to only those re- gions which are detected at ≥ 5σ detection level in the spectral cubes (σ being the rms noise level of the re- spective cubes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' To achieve this, the clumpfind algo- rithm of Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (1994) was utilised via starlink software’s (Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2014) cupid package (Berry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The 5σ threshold was conservatively cho- sen to eliminate false detections of clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' As rec- ommended by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (1994), the gap between contour levels was kept at 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' With these values, the clumpfind algorithm was implemented on the mosaiced CO(3-2), 13CO(3-2), and C18O(3-2) position-position- velocity cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' After the detection of clumps, we mask those regions where no emission was detected at 5σ or above to create a masked cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Thereafter, the respec- tive (masked) cubes were collapsed to create m-0, m-1, and linewidth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The maps are shown in Figure 3 for CO(3-2), 13CO(3-2), and C18O(3-2) molecular transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The massive star BD +45 3216, and the two RMS sources (G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817) have been marked on each of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In the m-0 maps, CO(3-2) emission appears to be diffused with no par- ticular discernible structure, which is expected given the ubiquitous nature of CO molecule and compara- tively lower critical density than the other two tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' However, it is still worth noting that much of the CO(3-2) emission still “faces away” from the massive star BD +45 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The 13CO(3-2) m-0 map most clearly traces the molecular filamentary struc- tures which were examined in the channel maps (Fig- ure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' While the cavity to the west of the massive star shows no 13CO(3-2) emission, the connecting filamen- #### Page 4 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000) 000: #### [-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] Arc [-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] [-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] [-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] Cavity Connecting Filament [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content="0] 83°50' 45' 40' 3°25' 20' 15' 10' Galactic Longitude Galactic Latitude Arc [-2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0] 0 2 4 6 8 10 12 14 K km s 1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Channel maps of 13CO(3-2) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Plus symbol marks the massive star BD +45 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Circle and diamond denote the locations of the two RMS sources, G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' An arc like filamentary structure, a cavity like structure, and a connecting filament have been indicated by arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000)000: #### Page 5 of 1 #### tary structure between the eastern and western regions is prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Given that all emission here is at ≥ 5σ level, this indicates that there is significant molecular mass concentration in the connecting filament and it is not some minor emission structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The critical den- sity is highest for C18O(3-2) transition, and thus only the densest clumps are detected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Unsurprisingly, the bulk of C18O(3-2) emission is associated with the positions of RMS sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The prominent arc like fil- amentary structure marked on the 13CO(3-2) channel map (Figure 2) is also partly seen in C18O(3-2) m-0 map, indicating the presence of dense gas along it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The linewidth maps for all three transitions show a max- ima towards the positions of the RMS sources as com- pared to the outer parts, which could be due to outflow activity and/or possibly suggest a convergence of dif- ferent flows towards these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In the 13CO(3-2) linewidth map, the connecting filament has a relatively lower velocity dispersion, while the arc like filamentary structure (Figure 2) displays a relatively larger disper- sion along its length as compared to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3 Physical Parameters Figure 4 shows the 13CO(3-2) m-0 map of the re- gion overlaid with the massive star BD +45 3216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' the two RMS sources (G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' and the MSX sources (m1 to m11) which lay in the regions of emission ≥ 5σ level (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The local peaks of m-0 emis- sion, based on visual examination of the contours have also been marked on the image (p1 to p10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' To ob- tain the physical parameters for these sources (except BD +45 3216 which did not have any molecular emis- sion associated with it), we extracted the spectra at their locations (from a 2×2 pixel area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' As high- velocity resolution is required for the calculations, the non-averaged cubes with the native channel width of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='05 km s−1 were utilised for spectrum extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For the RMS sources, both 13CO(3-2) and C18O(3- 2) spectra were available, and are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' However, for the MSX sources (m1-m11) and the lo- cal 13CO(3-2) integrated emission peaks (p1-p10), only 13CO(3-2) spectra could be extracted, shown in Figures 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The results from the gaussian fit- ting are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The first thing to notice is that at some of the locations, a simple gaussian is a poor fit to the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For example, in Figure 5, the spectra have broad wings on both sides, which is usu- ally an indication of some kind of likely outflow as- sociated with the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' According to Maud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2015b), while G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 is associated with an outflow, G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 has a possibility of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For some of the locations in Figures 6 and 7, there appears to be excess emission on the red side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' It had been discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 that the arc like struc- ture in Figure 2 probably has a merger of blueshifted and redshifted filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The excess redshifted emis- sion seen at the locations which lie on this arc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' m1 (Figure 6), and p1, p2 (Figure 7) could be an indica- tion of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Emission peak positions p8 and p9 display significant self-absorption in their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The mean velocity for most of the fits lies in the range - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 to -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' It is notable that the locations to- wards the eastern side (namely p1, p2, p3, and m1) have significantly more blueshifted mean velocities as opposed to other locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The RMS sources, though with wide wings, have nearly same mean velocities (∼ - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0 km s−1), which is in agreement with the radial ve- locity for the Sh2-112 region in Blitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (1982);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Stark & Brand (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Brand & Blitz (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Though our lo- cations display a range of mean velocities, the overall distribution is consistent with other values cited in liter- ature within error limits (Dobashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Urquhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Lumsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Maud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2015a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Panja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Using the FWHM (full width at half maxima) re- turned by the fit, we calculate the standard deviation (or observed velocity dispersion), the non-thermal ve- locity dispersion, and the total velocity dispersion using the following set of equations (Fuller & Myers, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Fiege & Pudritz, 2000) : ∆V2 tot = ∆V2 obs + 8 ln 2 kT � 1 ¯m − 1 mobs � (1) ⇒ ∆V2 tot 8 ln 2 = kT ¯m + ������ ∆V2 obs 8 ln 2 − kT mobs ������ ⇒ σ2 tot = c2 s + � σ2 obs − σ2 t � (2) = c2 s + σ2 nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (3) In the above equations, ∆Vobs is the FWHM of the fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' σobs (= ∆Vobs/ √ 8ln 2 for gaussian fits) is the stan- dard deviation (or dispersion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' σt(= √kT/mobs) is the thermal velocity dispersion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' mobs is the mass of the rele- vant molecule (29 amu and 30 amu for 13CO and C18O respectively);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' σnt is the non-thermal velocity disper- sion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' cs(= √kT/ ¯m) is the speed of sound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' ¯m is the av- erage molecular weight of the medium (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='37 amu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' and T is the excitation or gas kinetic temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The FWHM shows a wide range at these locations, ranging from ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='0 km s−1 for most of the sources, with some of the highest values associated with lo- cations in the vicinity of the two RMS sources, such as m2, m3, m4, m11, and p10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' This tallies with the linewidth map in Figure 3(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For the above calcula- tions, we take the excitation temperature T as 20 K (see Figure 15 in Panja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Furthermore, we also #### Page 6 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" (0000) 000: #### 3°25' 20' 15' 10' Galactic Latitude a)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" CO(3-2) m-0 0 50 100 150 K km s 1 3°25' 20' 15' 10' b)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" 13CO(3-2) m-0 0 20 40 K km s 1 3°25' 20' 15' 10' c)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" C18O(3-2) m-0 0 5 10 K km s 1 3°25' 20' 15' 10' Galactic Latitude d)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" CO(3-2) m-1 4 2 0 km s 1 3°25' 20' 15' 10' e)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" 13CO(3-2) m-1 4 2 0 km s 1 3°25' 20' 15' 10' f)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" C18O(3-2) m-1 4 2 0 km s 1 83°50' 45' 40' 3°25' 20' 15' 10' Galactic Longitude Galactic Latitude g)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" CO(3-2) linewidth 0 2 4 km s 1 83°50' 45' 40' 3°25' 20' 15' 10' Galactic Longitude h)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" 13CO(3-2) linewidth 1 2 km s 1 83°50' 45' 40' 3°25' 20' 15' 10' Galactic Longitude i)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' C18O(3-2) linewidth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='8 km s 1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Row-wise : moment-0 (Integrated intensity), moment-1 (Intensity-weighted velocity), and linewidth (Intensity- weighted dispersion) collapsed images for three cubes – CO(3-2), 13CO(3-2), C18O(3-2) in first, second, and third columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The symbols are same as Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000)000: #### Page 7 of 1 #### calculate Mach number (= σnt/cs) and the ratio of ther- mal to non-thermal pressure (Ptnt = c2 s/σ2 nt) (Lada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2003) for each of the locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The results of these two calculations show the presence of supersonic mo- tion, and a dominance of non-thermal pressure in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The mach number and Ptnt values are inversely correlated, as expected from their dependence on σnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Such values would suggest that the emission mecha- nism is likely some supersonic non-thermal phenom- ena, and could be via turbulence and magnetic fields (Myers & Goodman, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Crutcher, 1999), to suggest one such mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Discussion Figure 8 shows the CGPS 21 cm velocity channel maps of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The channel maps show a depression in H i emission (see Figure 9), which is (anti-)correlated with strong (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' ≥ 5σ) molecular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Such fea- tures have been referred to as H i self absorption (or HISA) features in literature, and have been found to be an extensive presence in 21 cm surveys (Kerton, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' HISA regions indicate the presence of cold H I gas in foreground against warm H I emission from background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In our channel maps, most intense depression in emission seems to be con- fined to the north-eastern quadrant, in the vicinity of the source G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 (circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Channels - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3 km s−1 and -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 km s−1 display shell-like feature on a smaller spatial scale, while the channels -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='8 km s−1 and -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='6 km s−1 show large scale shell-like feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The anti-correlation of molecular emission and depres- sion in H I emission can be most prominently seen in the -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='8 km s−1 and -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='6 km s−1 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Such shell- like features have been found in other regions as well, such as Sh2-237 (Dewangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2017), W4 and W5 (Hosokawa & Inutsuka, 2007), and in Southern Galac- tic Plane Survey (SGPS) regions (McClure-Griffiths et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Figure 10 shows the 13CO(3-2) m-0 image (over- laid with NVSS contours), the Hα image of the region, and the continuum-subtracted H2 emission map of the region from Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The radio emission seems to have multiple peaks, and on a comparison of Figures 4 and 10(a), the radio peaks appear to be as- sociated with the locations of m3/m4, p5, and a loca- tion to the east of p10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' According to Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2020), the location to the east of p10 is the ionized boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Based on the presence of this ionized boundary layer and pressure calculations, they further conjecture the possibility of triggered star formation to- wards the location of RMS source G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 due to O8V type star BD +45 3216 (also see Morgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2004, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Urquhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' It is pos- sible that the expansion of the ionized gas could have had a role in the formation of the cavity towards the west/northwest of the massive star BD +45 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' There is significant Hα emission in the cavity, while the H2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='12µm emission seems to trace the southern boundary of the molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' All along the cavity perime- ter, there are finger-like filamentary structures protrud- ing into it, reminiscent of bright-rimmed clouds which are seen in optical and infrared emission (Chauhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Here these structures are seen in absorption in optical (Figure 10(b)) and in emission in 13CO(3-2) integrated intensity map (Figure 10(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Two out of three of these finger-like structures seem to be hosting molecular peaks at their ends, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' p5 and p10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' If there is a possible case of triggering to- wards the source G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817, then based on size considerations of the cavity, there could be possi- ble triggering all along the cavity perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The MSX sources m3, m4, m5, m7, m8, m9, and m11 seem to be distributed along the cavity boundary where there is seemingly compression of molecular material, as evi- denced by the sharp change in 13CO(3-2) contour lev- els (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Lastly we present the position-velocity maps in Figure 11 along the different line segments marked L1-L8 on the central image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' While the lines L1- L5 trace the filamentary structures which converge on the source G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 (circle), L6-L8 are cuts along the filamentary structures associated with G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 (diamond symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The position- velocity maps show a complex structure with clumping along their length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Significant velocity gradients can also be seen in most of them, and especially in L1- L5 in the vicinity of the source G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' bright clump towards the right in L1-L5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Here we note that such a gradient could indicate gas being chan- neled towards the source, and such a flow of gas along the filamentary structures towards the RMS sources – G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 – as also discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2, is similar to the longi- tudinal flow in filaments towards hubs in hub-filament systems (Dewangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2017, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Therefore, this region could present a good ex- ample of exploring various star formation frameworks such as global hierarchical collapse (GHC V´azquez- Semadeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2019), conveyor belt model (Longmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Krumholz & McKee, 2020), filaments to clusters model (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2020), to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' According to Panja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2022), the column den- sity of the molecular emission region has been calcu- lated to be ∼1022 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Such regions have been desig- nated as “hubs” in the context of hub-filament systems in literature (Myers, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The column density maps #### Page 8 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" (0000) 000: #### 83°51' 48' 45' 42' 39' 3°22' 20' 18' 16' 14' Galactic Longitude Galactic Latitude 13CO(3-2) m-0 m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 1 pc Figure 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 13CO(3-2) integrated intensity map (contours at 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5, 6, 8, 10, 12, 14, 17, 21, 25, 30, 34, 40, and 45 K km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Plus symbol marks the massive star BD +45 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The two RMS sources, G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817, have been shown by circle and diamond symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The MSX sources have been marked in green boxes and la- belled m1 to m11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Cyan boxes (labelled p1 to p10) are the locations of local contour peaks where also we extracted the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' −15 −10 −5 0 5 10 VRAD (km s-1) 0 4 8 12 K G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 −15 −10 −5 0 5 10 VRAD (km s-1) 0 4 8 12 G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 13CO(3-2) (blue) and C18O(3-2) (red) spectra at positions of RMS sources marked in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Green and yellow curves show the gaussian fits to the respective spectra, with blue dashed line marking the velocity of the peak of the 13CO(3-2) gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000)000: #### Page 9 of 1 #### 0 2 4 6 m1 0 2 4 6 m2 0 2 4 6 m3 0 2 4 6 m4 0 2 4 6 m5 0 2 4 6 m6 0 2 4 6 m7 0 2 4 6 m8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 0 2 4 6 m9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 VRAD (km s-1) 0 2 4 6 K m10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 0 2 4 6 m11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 13CO(3-2) spectra at MSX source positions marked in Figure 4 (green boxes labelled m1-m11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Green curve depicts the gaussian fit to the spectra, with blue dashed line marking the velocity of the peak of the gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' VRAD (km s-1) 0 5 10 K p1 0 5 10 p2 0 5 10 p3 0 5 10 p4 0 5 10 p5 −10 −5 0 5 VRAD (km s-1) 0 5 10 K p6 −10 −5 0 5 0 5 10 p7 −10 −5 0 5 0 5 10 p8 −10 −5 0 5 0 5 10 p9 −10 −5 0 5 0 5 10 p10 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 13CO(3-2) spectra at peak positions marked in Figure 4 (cyan boxes labelled p1-p10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Green curve depicts the gaussian fit to the spectra, with blue dashed line marking the velocity of the peak of the gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' #### Page 10 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000) 000: #### Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Parameters derived from 13CO(3-2) spectra at the locations marked in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058, C18O(3-2) spectrum was also available and thus used for calculation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' clump Mean FWHM Amplitude σNT Mach PTNT l b (km s−1) (km s−1) (K) (km s−1) Number (deg) (deg) G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 C18O 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='12 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817 13CO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='48 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='06 G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 C18O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='05 83.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2869 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000)000: #### Page 11 of 1 #### 1 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2 km s 1 1 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='6 km s 1 1 pc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5 km s 1 1 pc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3 km s 1 1 pc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 km s 1 1 pc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content="9 km s 1 83°51' 48' 45' 42' 39' 3°22' 20' 18' 16' 14' Galactic Longitude Galactic Latitude 1 pc 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='8 km s 1 1 pc 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='6 km s 1 1 pc 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='4 km s 1 60 80 100 120 K Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Velocity channel maps for the CGPS H I 21 cm emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Green contour marks the 2 K km s−1 level of the 13CO(3-2) m-0 (Figure 4) image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' In -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3 and -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 km s−1 channels, the red contours have been drawn at (outer to inner) 65, 60, 55, and 50 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' while in -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='8 and -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='6 km s−1 channels, the red contours are at 81 (outer) and 65 (inner) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The dashed green circle in -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 km s−1 channel shows the location where spectrum was extracted (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The rest of the symbols are same as Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' #### Page 12 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000) 000: #### 15 10 5 0 5 10 15 velocity (km s 1) 0 20 40 60 80 100 K Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 13CO(3-2) spectrum (in red) and CGPS H i spectrum (in blue) at the location marked in Figure 8 (green circle in -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 km s−1 channel) – demonstrating the H i self-absorption feature centered at ∼ [-5,-4] km s−1 and its anti-correlation with the molecular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The 13CO(3-2) spectrum has been scaled up by a factor of 30× for better visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' by Panja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2022) also suggest such a hub-filament configuration for the larger region of the order of a few 10s of parsec encompassing Sh2-112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Thus, what we have explored here seems to be the detailed structure of a hub region where massive star formation is go- ing on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' However, given the importance of filamentary structures in star formation and expositions of various filament types in literature (see Hacar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', 2022, for a detailed review), it is essential to further study such re- gions via high spectral and spatial resolution molecular transitions of other species, magnetic fields, and so on so as to understand the evolution of molecular clouds as they form stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Summary and Conclusions We have carried out an analysis of the Sh2-112 region in CO(3-2), 13CO(3-2), and C18O(3-2) molecular line transitions from the JCMT, supported by archival data from CGPS H I line, MSX, NVSS, and IPHAS Hα for visual examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Our main conclusions are as fol- lows : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The molecular emission appears filamentary in channel maps and seems to be directed away from the massive star BD +45 3216, which is also located at the edge of a cavity-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" 83°51' 48' 45' 42' 39' 3°22' 20' 18' 16' 14' Galactic Latitude 13CO(3-2) m-0 (a)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" 1 pc 83°51' 48' 45' 42' 39' 3°22' 20' 18' 16' 14' Galactic Latitude H (b)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" 1 pc 83°51' 48' 45' 42' 39' 3°22' 20' 18' 16' 14' Galactic Longitude Galactic Latitude H2 (c)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 1 pc Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (a) 13CO(3-2) m-0 map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Cyan contours are NVSS levels at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='09, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1 Jy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (b) Hα image of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (c) Continuum-subtracted H2 emission (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='12 µm) map from Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Green contour marks the 2 K km s−1 level of the 13CO(3-2) m-0 (Figure 4) image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The rest of the symbols are same as Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=" (0000)000: #### Page 13 of 1 #### 83°51' 48' 45' 42' 39' 3°22' 20' 18' 16' 14' Galactic Longitude Galactic Latitude L1 L2 L3 L4 L5 L6 L7 L8 1 pc 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='06 10 5 0 5 OFFSET (deg) VRAD (km/s) L1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 10 5 0 5 OFFSET (deg) VRAD (km/s) L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 10 5 0 5 OFFSET (deg) VRAD (km/s) L3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 10 5 0 5 OFFSET (deg) VRAD (km/s) L4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 10 5 0 5 OFFSET (deg) VRAD (km/s) L5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='06 10 5 0 5 OFFSET (deg) VRAD (km/s) L6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 10 5 0 5 OFFSET (deg) VRAD (km/s) L7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='04 10 5 0 5 OFFSET (deg) VRAD (km/s) L8 0 2 4 6 8 10 12 T* A (K) Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Position-velocity diagrams for the line segments marked L1-L8 (in red) on the central image (13CO(3-2) m-0 from Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Green contour marks the 2 K km s−1 level of the 13CO(3-2) m-0 (Figure 4) image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The rest of the symbols are same as Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' On the L1-L8 p-v maps, the contour levels are at 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='5, 2, 3, 4, 5, 6, 7, 8, 9, and 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' #### Page 14 of 1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' (0000) 000: #### 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Multiple local peaks were found associated with the molecular emission in the 13CO(3- 2) integrated intensity emission map (m- 0) which was generated using clumps de- tected above 5σ level in the position-position- velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The linewidth map shows high dispersion associated with the positions of the RMS sources (G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7962+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3058 and G083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='7071+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2817).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Analysis of CGPS 21 cm H i line emission re- veals the presence of shell-like HISA feature, where the molecular emission is nearly coinci- dent with the depression in H i emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 13CO(3-2) spectra was extracted at the loca- tions of RMS sources, MSX sources, and the local peaks of emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' For the RMS sources, C18O(3-2) spectra was also extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Spectral profile fitting suggests significant deviation from a gaussian profile for many sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' All the loca- tions were found to have significant non-thermal dispersions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' large mach numbers (∼ 2–6) indicat- ing dominance of supersonic motions within the clumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' and a small thermal to non-thermal pres- sure ratio (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='03–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Acknowledgements We thank the anonymous referee for a critical reading of the manuscript and for the suggestions for the im- provement of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' DKO acknowledges the sup- port of the Department of Atomic Energy, Government of India, under project Identification No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' RTI 4002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' The James Clerk Maxwell Telescope has historically been operated by the Joint Astronomy Centre on be- half of the Science and Technology Facilities Council of the United Kingdom, the National Research Council of Canada and the Netherlands Organisation for Sci- entific Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' This research made use of data products from the Mid- course Space Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' Processing of the data was funded by the Ballistic Missile Defense Organization with additional support from NASA Office of Space Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' This research has also made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronau- tics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' This 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1086/300337 Crutcher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 1999, ApJ, 520, 706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='1086/307483 Currie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='15585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content='x Motte, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', Bontemps, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=', & Louvet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} +page_content=' 2018, ARA&A, 56, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btA0T4oBgHgl3EQfGf-a/content/2301.02048v1.pdf'} 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0000000000000000000000000000000000000000..e8400614ca29b3631dda5b4c3f707a14829572d1 --- /dev/null +++ b/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f20326755fd86221bb83e8a0e9281a96f90c3548f307cc6ede4dff3467bd18c4 +size 194834 diff --git a/dNE0T4oBgHgl3EQfWgC_/content/tmp_files/2301.02280v1.pdf.txt b/dNE0T4oBgHgl3EQfWgC_/content/tmp_files/2301.02280v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..90ec3ca620bb8f0b9c5baaf2b5439b8141d2dc32 --- /dev/null +++ b/dNE0T4oBgHgl3EQfWgC_/content/tmp_files/2301.02280v1.pdf.txt @@ -0,0 +1,2402 @@ +Filtering, Distillation, and Hard Negatives for Vision-Language Pre-Training +Filip Radenovic, Abhimanyu Dubey∗, Abhishek Kadian∗, Todor Mihaylov∗, Simon Vandenhende∗ +Yash Patel, Yi Wen, Vignesh Ramanathan and Dhruv Mahajan† +Meta AI +Abstract +Vision-language models trained with contrastive learn- +ing on large-scale noisy data are becoming increasingly +popular for zero-shot recognition problems. In this paper +we improve the following three aspects of the contrastive +pre-training pipeline: dataset noise, model initialization +and the training objective. First, we propose a straightfor- +ward filtering strategy titled Complexity, Action, and Text- +spotting (CAT) that significantly reduces dataset size, while +achieving improved performance across zero-shot vision- +language tasks. Next, we propose an approach titled Con- +cept Distillation to leverage strong unimodal representa- +tions for contrastive training that does not increase train- +ing complexity while outperforming prior work. Finally, we +modify the traditional contrastive alignment objective, and +propose an importance-sampling approach to up-sample +the importance of hard-negatives without adding additional +complexity. +On an extensive zero-shot benchmark of 29 +tasks, our Distilled and Hard-negative Training (DiHT) ap- +proach improves on 20 tasks compared to the baseline. Fur- +thermore, for few-shot linear probing, we propose a novel +approach that bridges the gap between zero-shot and few- +shot performance, substantially improving over prior work. +Models are available at github.com/facebookresearch/diht. +1. Introduction +An increasingly popular paradigm in multimodal learn- +ing is contrastive pre-training [11,28,40,42,61,74,83,85], +which involves training multimodal models on very large- +scale noisy datasets of image-text pairs sourced from the +web. It has been shown to be incredibly effective for a vari- +ety of vision-language tasks without any task-specific fine- +tuning (i.e., zero-shot), such as image classification [64], +text and image retrieval [44, 58], visual question answer- +ing [21], among several others. +In this paper, we study +the problem of contrastive pre-training for dual-encoder ar- +chitectures [61] with the objective of improving image-text +alignment for zero-shot tasks. We revisit three important +aspects of the contrastive pre-training pipeline – noise in +*Equal contribution, listed alphabetically. †Research Lead. +B/32 +B/16 +L/14 +@336 +60 +65 +70 +75 +80 +Model Complexity +Accuracy@1 +ImageNet1K +DiHT +CLIP +B/32 +B/16 +L/14 +@336 +30 +35 +40 +45 +50 +Model Complexity +Recall@1 +COCO (T2I) +B/32 +B/16 +L/14 +@336 +59 +64 +69 +74 +79 +Model Complexity +Recall@1 +Flickr (T2I) +Figure 1. +DiHT trained on 438M LAION-CAT samples vs. +CLIP [61] trained on 400M OpenAI samples. +datasets, model initialization, and contrastive training, and +present strategies that significantly improve model perfor- +mance on a variety of zero-shot benchmarks, see Figure 1. +Most image-text datasets are noisy and poorly-aligned. +Few recent efforts [27] have tried to clean the noise by fil- +tering samples based on alignment scores from an existing +model like CLIP [61]. However, this approach is limited by +the biases and flaws of the model itself. On the other hand, +momentum-based approaches [40] to reduce noise are in- +feasible for large-scale training due to their increased com- +pute and memory requirements. To this end, we provide a +scalable and effective approach titled Complexity, Action +and Text-spotting (CAT) filtering. CAT is a filtering strat- +egy to select only informative text-image pairs from noisy +web-scale datasets. We show that training on a CAT-filtered +version of large-scale noisy datasets such as LAION [65] +can provide up to 12% relative improvements across vision- +language tasks despite removing almost 80% of the training +data, see Section 4.2 and Table 1 for more details. +A common strategy [57, 87] to further improve multi- +modal training is to warm-start it with image and text mod- +els pre-trained at large scale on their respective modali- +ties. +However, due to the increased noise in image-text +data, fine-tuning the entire model undermines the benefits +of the warm-start. One can alternatively use model freezing +strategies like locked-image tuning [87], but they are un- +able to adapt to the complex queries present in multimodal +problems (e.g., cross-modal retrieval) and the models per- +form poorly on retrieval benchmarks (see Section 4.2). We +1 +arXiv:2301.02280v1 [cs.CV] 5 Jan 2023 + +propose an entirely different approach, concept distillation +(CD), to leverage strong pre-trained vision models. The key +idea behind concept distillation is to train a linear classifier +on the image encoder to predict the distilled concepts from +a pre-trained teacher model, inspired by results in weakly- +supervised large-scale classification [48,69]. +Finally, we revisit the training objective: almost all prior +work has utilized noise-contrastive estimation via the In- +foNCE loss [54], shortcomings have been identified in the +standard InfoNCE formulation [12, 30]. We demonstrate +that by using a model-based importance sampling technique +to emphasize harder negatives, one can obtain substantial +improvements in performance. +A summary of our pipeline is available in Figure 2. +Our combined approach obtains significant improvements +over the baseline for dual-encoder architectures on an elab- +orate benchmark of 29 tasks. Specifically, with the ViT- +B/16 [17] architecture, we improve zero-shot performance +on 20 out of 29 tasks, over CLIP training on the LAION- +2B dataset [27, 65], despite training on a subset that is +80% smaller, see Figure 4. Furthermore, we demonstrate +that even when trained with the smaller (but relatively less +noisy) pretraining dataset PMD, our performance is better +on 28 out of 29 tasks than CLIP trained on the same data, +often with a large margin, see Figure 5. +Additionally, we present a simple yet effective approach +to maintain the performance continuum as one moves from +zero-shot to few-shot learning in the low data regime. Prior +work [61] has shown a substantial drop in performance +as one moves from zero-shot to k-shot learning, which is +undesirable for practical scenarios. We propose an alter- +nate linear probing approach that initializes the linear clas- +sifier with zero-shot text prompts and ensures that final +weights do not drift away too much via projected gradient +descent [5]. On ImageNet1K, we show huge improvements +over prior work for small k values. For example, our ap- +proach improves 5-shot top-1 accuracy by an absolute mar- +gin of 7% (see Figure 6) compared to the baseline strategy +of linear probing with a random initialization. +2. Related work +Dataset curation for contrastive pretraining. +Large- +scale contrastive pretraining [11, 28, 40, 42, 61, 74, 83, 85] +typically requires dataset sizes of the order of hundreds +of millions to billions. Seminal works in this area, e.g., +CLIP [61] and ALIGN [28], have largely relied on image- +text pairs crawled from the web. Subsequently, versions +of large-scale image-text datasets have been created but +not released publicly, including WIT-400M [61], ALIGN- +1.8B [28], FILIP-340M [83], FLD-900M [85], BASIC- +6.6B [57], PaLI-10B [10]. +These datasets often use un- +clear or primitive cleaning strategies, e.g., removing sam- +ples with short or non-English captions. Recently, LAION- +Object & Attribute +Cross-Entropy +Cute kitten +Cute kitten +Cute kitten +Cute kitten +Cute kitten +My cute +baby +CAT +Filtering +1 +Gray cat with +red scarf +watching TV +SWAG-ViT +Image +Encoder + +Object & Attribute +Classifiers +Parser +Cat +Scarf +TV +... Gray Red +Train separately +Image +Encoder +Text +Encoder +I1 +I2 +I3 +I4 +T1 +T2 +T3 +T4 +Positive +Hard-negative +HN-NCE +3 +Inference +Concept +Distillation +2 + +Storage +Figure 2. Summary of our pipeline. We propose improvements to +the standard vision-language pre-training: (1) Complexity, Action +and Text-spotting (CAT) filtering that removes non-informative +text-image pairs; (2) Concept distillation from a frozen (�) pre- +trained image encoder; (3) Hard-negative contrastive loss. +400M [66] used CLIP-based scores to filter down a large +dataset. The authors later released an English-only LAION- +2B and a LAION-5B unfiltered dataset sourced from Com- +mon Crawl1. +Apart from LAION-400M and BLIP [39] +which uses the bootstrapped image-grounded text encoder +to filter out noisy captions, there has not been a signifi- +cant investment in systematic curation strategies to improve +zero-shot alignment performance on large-scale pretraining. +In contrast to the previous work, we use quality-motivated +filters that retain images whose captions are sufficiently +complex, contain semantic concepts (actions), and do not +contain text that can be spotted in the image [37]. +Distillation from pre-trained visual models. +Knowl- +edge distillation [25] aims to transfer knowledge from +one model to another and has been used in many con- +texts ranging from improving performance and efficiency +[6, 7, 41, 63, 72, 79] to improving generalization capabili- +ties [16, 42, 43]. Several approaches use self-distillation to +improve performance with lower computational overhead +[23,80,86]. For vision and language pre-training, [2,40] use +soft-labels computed using embeddings from a moving av- +erage momentum model with the goal to reduce the adverse +effects of noisy image-text pairs in the training data. Our +concept distillation approach is a cheaper and more effec- +tive alternative, since it does not require us to run the expen- +sive teacher model throughout the training2 while retaining +the most useful information from the visual concepts. +Another approach to take advantage of pre-trained vi- +sual models is to use them to initialize the image encoder, +and continue pre-training either by locking the image en- +coder [57, 87] or fine-tuning [57]. +However, these ap- +proaches lack the ability to align complex text to a fully- +trained image encoder, and thus perform poorly on multi- +modal tasks, e.g. cross-modal retrieval (see Section 4.3). +1commoncrawl.org +2Distillation targets can be pre-computed and stored. +2 + +Contrastive +training +with +hard +negatives. +Noise- +contrastive estimation (NCE) [22] is the typical objective +for vision-text learning, with applications across large-scale +multimodal alignment [11,28, 42,61] and unsupervised vi- +sual representation learning [24, 52]. Several lines of work +have studied the shortcomings of the original InfoNCE ob- +jective [54], specifically, the selection and importance of +negative samples. +Chuang et al. [12] present a debias- +ing approach to account for false negatives at very large +batch sizes, typical in large-scale pretraining. Kalantidis et +al. [30] present a MixUp approach to improve the qual- +ity of hard negative samples for unsupervised alignment. +Using model-specific hard negatives in the training objec- +tive is proven to reduce the estimation bias of the model as +well [88]. Contrary to prior semi-supervised work, we ex- +tend the model-based hard negative objective, first proposed +in Robinson et al. [62] to multimodal alignment. +3. Method +Background. +We consider the task of contrastive image- +text pretraining. +Given a dataset D = {(Ii, Ti)}N +i=1 of +image-text pairs, we want to learn a dual encoder model +φ = {φimage, φtext}, where φimage represents the image en- +coder, and φtext denotes the text encoder. We use the short- +hand x = φimage(I) and t = φtext(T) to denote the encoded +images and texts, respectively, for an image-text pair (I, T). +We will now describe the three crucial components of our +approach followed by the final training objective. +3.1. Complexity, Action, and Text (CAT) filtering +Our complexity, action, and text spotting (CAT) filtering +is a combination of two filters: a caption complexity filter +that removes image-caption pairs if a caption is not suffi- +ciently complex, and an image filter that removes pairs if the +image contains text matching the caption to prevent poly- +semy during alignment. We use the LAION-2B pre-cleaned +obtained after using filters3 in [67] as the base dataset. +Filtering captions via complexity & actions. +Noisy +web-scale datasets do not have any semantic-based cura- +tion, and hence captions can be irrelevant, ungrammatical +and unaligned. Our motivation is to decrease such noise by +simply selecting captions that possess sufficient complex- +ity, so that the training distribution matches the target tasks. +To this end, we build a fast rule-based parser that extracts +objects, attributes and action relations (see Figure 3 for an +example) from text and we use the resulting semantic graph +to estimate the complexity of the image captions. Specifi- +cally, we define the complexity of a caption as the maximum +number of relations to any object present in the parse graph. +For example, in the caption “A black cat is chasing a small +3Not-suitable-for-view images and toxic captions. +Caption: A black cat is chasing a small brown bird. +chasing +ACTION +bird +OBJECT +cat +OBJECT +small +ATTRIBUTE +brown +ATTRIBUTE +black +ATTRIBUTE +has attribute +has attribute +has attribute +has object +has subject +Figure 3. An example caption and its parse. The caption has C3 +complexity (due to bird) and has 1 action (chasing). +brown bird,” the object “bird” has the attributes “small”, +“brown” and “A black cat is chasing”, and hence, the com- +plexity of the caption is C3. We only retain samples that at +least have a C1 caption complexity. To further remove pairs +likely containing products, we filter out captions if they do +not contain at least one action (as obtained from the parser). +Filtering images via text-spotting. +Image-caption pairs +in web-scale datasets often display part of the caption as text +in the image (on visual inspection, we found up to ∼30% +such examples for LAION [65]). Minimizing the objective, +in these cases, can correspond to spotting text (e.g., optical +character recognition) rather than the high-level visual se- +mantics (e.g., objects, attributes) we would like the model +to align to. This will reduce performance on object-centric +and scene-centric downstream zero-shot tasks, and hence +we remove such images from the training set using an off- +the-shelf text spotter [37]. We remove image-text pairs with +a text spotting confidence of at least 0.8 and at least 5 pre- +dicted characters matching the caption in a sliding window. +We observe (by inspection) that this approach is efficient at +identifying images with text, and failure cases are primarily +in non-English text. Filtering with multilingual text spotters +trained can fix this issue, however, we leave this as future +work. Filtering statistics can be found in the supplement. +3.2. Concept distillation +Recognizing visual concepts in images that correspond +to objects and attributes in corresponding captions is cru- +cial for alignment. We therefore propose to distill these +concepts from a pre-trained teacher model to our image en- +coder. Specifically, we add two auxiliary linear classifiers +on top of the encoded image embeddings x to predict (i) +objects and (ii) visual attributes and use the teacher model +to generate the pseudo-labels for training them. These clas- +sifiers are trained jointly with the contrastive loss. +We parse image captions using a semantic parser that +extracts objects and attributes from text (Section 3.1) and +use these as pseudo-labels. We then train the linear clas- +sifiers on the teacher model embeddings with a soft-target +cross-entropy loss [20], after square-root upsampling low- +frequency concepts [48]. It is important to freeze the back- +bone of the teacher model to make sure we retain the ad- +vantages of using a stronger model for distillation. +For +3 + +each image, we then use these trained linear classifiers to +generate two softmax probability vectors – pobj for objects, +and pattr for attributes, respectively. To minimize the stor- +age overhead, we further sparsify them by retaining only +the top-k predicted class values and re-normalizing them to +generate the final pseudo-labels. During multimodal train- +ing, we use the cross-entropy loss with these pseudo-label +vectors as targets. Unless specified otherwise, we use the +ViT-H/14 [17] architecture pretrained from SWAG [69] as +the teacher model. See Section 4.2 and the supplementary +material for ablations on the effect of different backbones +and retaining top-k predictions, and further details. +There are several advantages of our concept distillation +approach. First, the teacher predictions capture correlations +from the strong vision encoding, making them more infor- +mative as labels compared to the captions themselves. The +captions are limited to a few objects and attributes, while +the teacher predictions yield a more exhaustive list. More- +over, our approach reaps the benefits of the recently pro- +posed and publicly-available strong unimodal vision mod- +els more effectively than other distillation approaches, as +training linear classifiers on a frozen teacher model is inex- +pensive. After predictions are stored, we discard the teacher +model and thus bypass the memory and compute limitations +of simultaneously running the student and teacher model +in standard distillation approaches [25, 72], which is criti- +cal for large teacher models. We demonstrate empirically +(see Section 4.2) that our strategy works better than distill- +ing teacher embeddings directly. Additionally, compared +to approaches that warm-start the image encoder with pre- +trained models, our method can leverage higher capacity +teacher models without difficulty and unlike locked-image +tuning [57,87], our approach gives the flexibility of training +the image encoder for better alignment, while retaining the +strength of the pre-trained visual features. +3.3. Multimodal alignment with hard negatives +Contrastive learning [54] has quickly become the de- +facto approach for multimodal alignment, where most prior +work focuses on the multimodal InfoNCE [54] objective, +given for any batch X = {(xi, ti)}n +i=1 of featurized image- +text pairs as (for some learnable temperature τ > 0), +LNCE(X) = − +n +� +i=1 +� +�log +ex⊤ +i ti/τ +� +j ex⊤ +i tj/τ + log +ex⊤ +i ti/τ +� +j ex⊤ +j ti/τ +� +� . +While this approach has enjoyed immense success in +multimodal alignment [28, 61], when learning from large- +scale noisy datasets, uniform sampling as applied in noise- +contrastive estimation can often provide negative sam- +ples that are not necessarily discriminative, necessitating +very large batch sizes. +For the problem of contrastive +self-supervised learning, Robinson et al. [62] propose an +importance-sampling approach to reweight negative sam- +ples within a batch so that “harder” negatives are up- +sampled in proportion to their difficulty. We present a sim- +ilar strategy for multimodal alignment. +Specifically, for +some α ∈ (0, 1], β ≥ 0, we propose the following hard- +negative noise contrastive multimodal alignment objective: +LHN-NCE(X) = − +n +� +i=1 +log +� +�� +ex⊤ +i ti/τ +α · ex⊤ +i ti/τ + � +j̸=i +ex⊤ +i tj/τwi→t +xi,tj +� +�� +− +n +� +i=1 +log +� +��� +ex⊤ +i ti/τ +α · ex⊤ +i ti/τ + � +j̸=i +ex⊤ +j ti/τwt→i +xj,ti +� +��� . +Where the weighing functions are given as4: +wi→t +xi,tj = (n − 1) · eβx⊤ +i tj/τ +� +k̸=i eβx⊤ +i tk/τ +, wt→i +xj,ti = (n − 1) · eβx⊤ +j ti/τ +� +k̸=i eβx⊤ +k ti/τ +. +The weights wβ are designed such that difficult negative +pairs (with higher similarity) are emphasized, and easier +pairs are ignored. Furthermore, α rescales the normaliza- +tion with the positive terms to account for the case when +false negatives are present within the data. The form of +weights wβ is an unnormalized von Mises-Fisher distribu- +tion [49] with concentration parameter β. Observe that we +obtain the original objective when setting α = 1 and β = 0. +There are several key differences with the original formu- +lation of [62] and the HN-NCE objective presented above. +First, we utilize only cross-modal alignment terms, instead +of the unimodal objective presented in [62]. Next, we em- +ploy separate penalties for text-to-image and image-to-text +alignment. Finally, we incorporate a learnable temperature +parameter τ to assist in the learning process. We discuss our +design choices in more detail with additional theoretical and +experimental justifications in the supplementary material. +3.4. Training objective +For any batch X = {(xi, ti)n +i=1} of n image-text pairs, +we minimize the following objective: +LHN-NCE(X) + LCE-O(X) + LCE-A(X), where, +LCE-O(X) = +n +� +i=1 +CROSS-ENTROPY(pobj +i , fobj(xi)), and, +LCE-A(X) = +n +� +i=1 +CROSS-ENTROPY(pattr +i , fattr(xi)). +Here, both fobj and fattr are linear classifiers, the vectors +pobj, pattr are the top-k predicted objects and attributes from +the teacher model (Section 3.2), and LHN-NCE is the hard- +negative contrastive alignment loss (Section 3.3). +4We normalize by n − 1 as this is the number of negatives. +4 + +4. Experiments +Here we evaluate our approach across a broad range +of vision and vision-language tasks. +We provide exten- +sive ablations on 29 tasks over the design choices in Sec- +tion 4.2, and compare with state-of-the-art approaches on +popular zero-shot benchmarks in Section 4.3. Finally, we +present an alternate approach to do few-shot classification +with prompt-based initialization in Section 4.4. +4.1. Experimental setup +Training datasets. +We use a 2.1B English caption subset +of the LAION-5B dataset [65]. Prior to training, we fil- +ter out sample pairs with NSFW images, toxic words in the +text, or images with a watermark probability larger than 0.5, +following [67]. This leaves us with 1.98B images, which we +refer to throughout the paper as the LAION-2B dataset. Ad- +ditionally, we explore training our models on a collection of +Public Multimodal Datasets (PMD) from [68]. PMD con- +tains training splits of various public datasets. After down- +loading5 the data we are left with 63M (vs. 70M reported +in [68]) image-text pairs due to missing samples and SBU +Captions [55] (originally in PMD) going offline. +Training details. +For our model architecture, we closely +follow CLIP by Radford et al. [61]. We utilize Vision Trans- +formers (ViT) [17] for images and Text Transformers [73] +for captions. We experiment with 3 different architectures, +denoted as B/32, B/16, and L/14, where 32, 16, and 14 de- +note the input image patch size. See the supplementary for +architecture details. For distillation and fine-tuning experi- +ments, we utilize the public SWAG-ViT models [69], pre- +trained with weak supervision from hashtags. +We use the Adam [32] optimizer with a decoupled +weight decay [47] and a cosine learning rate schedule [46]. +The input image size is 224×224 pixels. To accelerate train- +ing and save memory, we use mixed-precision training [50]. +All hyperparameters are presented in the supplementary. +They are selected by training B/32 on a small scale setup, +and reused for all architectures. For objects and attributes +classifiers, we found that scaling the learning rate by 10.0 +and weight decay by 0.01 gave better results. We train our +models on 4B, 8B, 16B, and 32B total samples. For ViT- +L/14, we further train the model at a higher 336px resolu- +tion for 400M samples, denoting this model as L/14@336. +Evaluation benchmarks. +We evaluate our models on a +zero-shot benchmark of 29 tasks: (i) 17 image classifica- +tion, (ii) 10 cross-modal retrieval, (iii) 2 visual question an- +swering. Dataset details are presented in the supplement. +5Downloaded following huggingface.co/datasets/facebook/pmd. +Table 1. Evaluating effect of using LAION-2B subset filtered on +complexity (C), actions (A), and text-spotting (T). CLIP denotes +filtering pairs with CLIP score bellow 0.35. Evaluation performed +on ViT-B/32 model architecture trained for 4B processed samples. +# +Filter +Size +IN +COCO +Flickr +CLIP +C +A +T +T2I +I2T +T2I +I2T +1 +1.98B +60.8 +33.7 +52.1 +59.3 +77.7 +2 +✓ +440M +52.5 +29.8 +46.1 +54.8 +72.0 +3 +✓ +1.71B +60.8 +33.9 +52.5 +60.8 +77.8 +4 +✓ ✓ +642M +58.7 +35.9 +53.8 +64.3 +82.0 +5 +✓ ✓ ✓ 438M +61.5 +37.6 +55.9 +66.5 +83.2 +4.2. Ablations on zero-shot benchmarks +In this section, we ablate our three pretraining contri- +butions: dataset filtering, distillation from objects and at- +tributes predictions, and, hard negative contrastive objec- +tive. Ablations are performed over zero-shot Accuracy@1 +on the ImageNet1K [64] (IN) validation set, text-to-image +(T2I) and image-to-text (I2T) zero-shot Recall@1 on the +COCO [59] and Flickr [58] test sets. We also report the +change in accuracy (%) over 29 zero-shot tasks between our +model and baselines. For a fair comparison, we train all ap- +proaches presented in this section (including baselines). +Effect of dataset filtering. +We apply our filters, as well as +filtering based on CLIP [61] alignment score (<0.35), and +ablate the performance in Table 1 for ViT-B/32 model ar- +chitecture. All models see 4B total samples during training, +while the number of unique samples drops after each filter- +ing step. Complexity filter (C) in row (3) reduces the dataset +size by around 270M, while slightly increasing image-text +alignment as observed on T2I task. Next, action filter (A) +in row (4) reduces the size by more than 1B, while it has +a large benefit in aligning complex text. However, as ex- +pected, it hurts performance on object-centric ImageNet. +Finally, text-spotting (T) filter in row (5) boosts alignment +across the board, due to the fact that it removes the need +to learn a bimodal visual representation of the text. We +also compare with filtering based on CLIP score in row (2), +which was selected such that the dataset size is comparable +to ours, and show that it is too strict and removes plenty of +useful training pairs, thus hurting the performance. Finally, +LAION-CAT, with only 22% of the original dataset size, +significantly boosts image-text zero-shot performance. We +also observed that gains hold as we train for longer training +schedules. See the supplementary for details. +Effect of distillation approach. +To understand the effect +of direct distillation from a pre-trained SWAG-ViT visual +encoder [69], we investigate two baseline approaches: +(1) Embedding distillation (ED) borrows from SimSiam [9] +and uses an auxiliary negative cosine similarity loss be- +5 + +tween the image representation from the student visual en- +coder and the pre-trained SWAG model. +(2) Distribution distillation (DD) borrows ideas from mo- +mentum distillation in ALBEF [40] and computes the cross- +modal similarities between the SWAG image representation +and the student text representation and uses them as soft- +labels for student image representation and text alignment. +The soft-labels are linearly combined with the hard 0 − 1 +labels before applying the InfoNCE [54] loss. +A comparison of our distillation from predicted con- +cepts (CD) with the aforementioned distillation approaches +is presented in Table 2 (upper section). Note that for a fair +comparison, we do not use our hard-negative contrastive +loss for these experiments. Our distillation approach per- +forms the best, even though it has virtually no training over- +head as the predicted concepts are pre-computed, while, +e.g., ED is 60% slower with an 8% increase in GPU mem- +ory due to the need of running an additional copy of the vi- +sion tower. One could pre-compute embeddings for ED and +DD as well, but that increases dataset size by 1.2TB and +creates a data loading bottleneck, while our pre-computed +predictions take only 32.6GB additional storage space when +saving the top-10 predictions (see supplementary). We ad- +ditionally show that our approach is robust to the number of +top-k predictions used, details in the supplementary. +One could also use an external unimodal image model +and fine-tune it on the image-text alignment task instead of +using distillation. We follow [87] and explore three fine- +tuning options as baselines: (i) locked-image tuning (LiT) +where the image encoder is locked, and only the text en- +coder is trained, (ii) fine-tuning (FT) where the image en- +coder is trained with a learning rate scaled by 0.01 com- +pared to the text encoder, (iii) fine-tuning with delay (FT- +delay) where the image encoder is locked for half of the +pre-training epochs following (i), and then fine-tuned for +the rest following (ii). Results of these setups are ablated +in Table 2 (lower section). LiT vs. FT is a trade-off be- +tween strong performance on image recognition tasks (as +measured with ImageNet1K) and better image-text align- +ment (as measured by COCO and Flickr). Locking the im- +age encoder makes the alignment very hard to achieve, but +fine-tuning it hurts its original image recognition power. On +the other hand, we show that our concept distillation is the +best of both worlds, it surpasses LiT or FT in 4 out of 5 met- +rics. Another drawback of FT is that it requires the same +architecture in the final setup, while CD can be effortlessly +combined with any architecture or training setup, by using +stored predictions as metadata. To conclude, unlike related +approaches, our proposed distillation: (i) has almost no cost +at training, (ii) is architecture agnostic, (iii) improves both +image recognition and complex image-text alignment. +Table 2. Evaluating effect of using different initialization or dis- +tillation approaches. Evaluation performed on ViT-B/16 model +architecture trained for 16B processed samples on LAION-CAT. +Init: Initialization with random or SWAG-B/16 weights. ED: Em- +bedding distillation. DD: Distribution distillation. LiT: Locked +image tuning. FT: Fine-tuning. FT-delay: Locked image tuning +for 50% followed by fine-tuning for the rest. CD: Our concept +distillation using teacher-predicted objects and attributes. +Init +Method +SWAG +IN +COCO +Flickr +(teacher) +T2I +I2T +T2I +I2T +Random +Baseline +— +68.7 +42.8 +60.5 +72.8 +89.7 +ED +B/16 +69.2 +42.6 +59.4 +72.8 +86.8 +DD +B/16 +68.6 +41.8 +57.4 +71.7 +87.0 +CD (ours) +B/16 +71.0 +42.8 +59.5 +72.3 +86.5 +CD (ours) +H/14 +72.3 +43.4 +60.4 +73.8 +87.6 +SWAG +LiT +— +73.0 +32.5 +50.6 +60.8 +79.6 +FT +— +71.2 +43.1 +60.3 +73.1 +87.7 +FT-delay +— +72.0 +42.7 +60.7 +72.5 +86.2 +Table 3. Evaluating effect of using hard negative contrastive loss. +Evaluation performed on ViT-B/16 model architecture trained for +16B processed samples on LAION-CAT. CD: Our concept distilla- +tion using SWAG-H/14 predicted objects and attributes. HN: Our +proposed hard negative contrastive loss. +# +Method +IN +COCO +Flickr +CD +HN +T2I +I2T +T2I +I2T +1 +68.7 +42.8 +60.5 +72.8 +89.7 +2 +✓ +72.3 +43.4 +60.4 +73.8 +87.6 +3 +✓ +✓ +72.0 +43.7 +62.0 +73.2 +89.5 +Effect of hard negative contrastive training. +We present +the ablation when using hard negative contrastive objective +(HN-NCE) in Table 3. Performance on popular benchmarks +suggests that using the newly proposed loss is beneficial +compared to the vanilla InfoNCE, and that its positive ef- +fects are complementary to the gains from the proposed dis- +tillation from objects and attributes predictions. Please see +the supplementary for ablations on the effect of the hyper- +parameters α and β. +Effect when pre-training on PMD. +Finally, we analyze +our proposed recipes when training visual-language mod- +els on a much smaller dataset, i.e. PMD with 63M training +samples. Results are shown in Table 4. All contributions +improve the performance over baseline significantly, hence +we conclude that using the proposed pipeline is very benefi- +cial in low-resource training regimes6. Note that, the PMD +dataset contains COCO and Flickr training samples, hence, +it is not strictly zero-shot evaluation. For that reason, we +do not compare our models trained on PMD dataset with +state-of-the-art models in the following section. However, +we believe these strong findings will motivate usage of our +approach on smaller and cleaner datasets, as well. +6PMD is smaller and relatively much cleaner dataset compared to +LAION. Hence, we observed that our filtering step is not needed for it. +6 + +Table 4. Evaluating effect when pre-training on PMD using our ap- +proaches. Evaluation performed on ViT-B/32 and ViT-B/16 mod- +els trained for 4B processed samples. CD: Our concept distillation +using SWAG-H/14 predicted objects (-O) and attributes (-A). HN: +Our proposed hard negative contrastive loss. +Arch. +# +Method +IN +COCO +Flickr +CD-O CD-A +HN +T2I +I2T +T2I +I2T +B/32 +1 +49.0 +28.9 +50.2 +62.0 +80.3 +2 +✓ +57.8 +32.2 +54.0 +65.6 +85.7 +3 +✓ +✓ +59.7 +34.4 +55.7 +68.3 +87.8 +4 +✓ +✓ +✓ +62.4 +37.3 +60.4 +71.8 +89.9 +B/16 +5 +54.6 +33.1 +55.7 +67.4 +85.5 +6 +✓ +✓ +65.5 +37.4 +59.9 +72.4 +88.7 +7 +✓ +✓ +✓ +67.8 +42.7 +65.5 +77.6 +92.5 +[18] PascalVOC +[19] Caltech101 +[34] StanfordCars +[77] SUN397 +[3] Birdsnap +[61] Country211 +[36] CIFAR10 +[38] OpenImages +[53] Flowers102 +−20−15−10 −5 +0 +5 +10 +15 +20 +CIFAR100 [36] +STL10 [15] +COCO I2T [44] +HatefulMemes [31] +OxfordPets [56] +VQAv2 [21] +DTD [13] +LN-COCO I2T [59] +SNLI-VE [78] +Winoground T2I [71] +Food101 [4] +LN-COCO T2I [59] +COCO T2I [44] +ImageNet1K [64] +UCF101 [70] +Flickr T2I [58] +LN-Flickr I2T [59] +Flickr I2T [58] +Winoground I2T [71] +LN-Flickr T2I [59] +−7.62 +−5.63 +−3.07 +−2.81 +−2.22 +−1.94 +−1.86 +−1.38 +−1.04 +0.25 +0.25 +0.42 +0.49 +0.57 +0.71 +0.96 +0.96 +0.99 +1.00 +1.26 +1.40 +1.81 +1.89 +3.23 +3.26 +3.40 +3.80 +4.13 +7.33 +Score Difference (%) +Figure 4. DiHT-B/16 trained on LAION-CAT with 438M sam- +ples vs. CLIP-B/16 trained on LAION-2B with 2B samples. Both +models trained by us with 32B total processed samples. +[19] Caltech101 +−20−15−10 −5 +0 +5 +10 +15 +20 +SNLI-VE [78] +VQAv2 [21] +PascalVOC [18] +HatefulMemes [31] +STL10 [15] +Winoground T2I [71] +LN-Flickr I2T [59] +Country211 [61] +LN-COCO I2T [59] +Flickr I2T [58] +Winoground I2T [71] +LN-COCO T2I [59] +OpenImages [38] +COCO T2I [44] +LN-Flickr T2I [59] +COCO I2T [44] +Food101 [4] +Flickr T2I [58] +SUN397 [77] +CIFAR10 [36] +ImageNet1K [64] +UCF101 [70] +DTD [13] +OxfordPets [56] +StanfordCars [34] +Birdsnap [3] +Flowers102 [53] +CIFAR100 [36] +−0.61 +1.37 +2.16 +2.42 +2.95 +3.00 +4.76 +5.50 +6.26 +6.70 +7.00 +7.13 +7.80 +9.00 +9.57 +9.71 +9.78 +9.90 +10.14 +10.57 +11.88 +13.21 +13.22 +13.46 +13.52 +15.32 +15.73 +16.83 +18.13 +Score Difference (%) +Figure 5. DiHT-B/16 vs. CLIP-B/16. Both models trained by us +on PMD with 63M images and 4B total processed samples. +Zero-shot benchmarks. +We denote model trained with +our proposed concept distillation and hard-negative loss as +DiHT. To showcase our model’s performance in more de- +tail, we report our DiHT-B/16 trained on LAION-CAT with +438M samples vs. CLIP-B/16 baseline trained by us on +LAION-2B with 2B samples in Figure 4. +Additionally, +we report DiHT-B/16 vs. CLIP-B/16 baseline, where both +models are trained on PMD dataset with 63M samples in +Figure 5. When trained on LAION-CAT or LAION-2B, re- +spectively, DiHT wins on 20 out of 29 benchmark tasks. +Impressively, when trained on PMD, DiHT wins on 28 out +of 29 benchmarks tasks, usually with a very large margin. +4.3. Comparison with zero-shot state of the art +We compare our DiHT models against state-of-the-art +dual-encoder models in Table 5. +Given that all models +use different architectures, input image resolutions, train- +ing databases, and number of processed samples at training, +we outline those details in the table, for easier comparison. +Our approach is most similar to CLIP [61] and Open- +CLIP [27], and has same training complexity and inference +complexity. +We outperform models with same architec- +ture by substantial margins, even when our training dataset +is much smaller. Our best models DiHT-L/14 and DiHT- +L/14@336 trained at higher 336px resolution for additional +400M samples outperform models with significantly more +complexity on popular text-image COCO and Flickr bench- +marks. Compared to ALIGN [28] that has approximately +twice the number of parameters compared to our DiHT- +L/14 model and is trained on 4x bigger data, we improve +the performance substantially for all the retrieval bench- +marks. Our model also performs better than FILIP [83] +which utilizes token-wise similarity to compute the final +alignment, thus noticeably increasing the training speed and +memory cost. We also outperform Florence [85] on all 4 re- +trieval benchmarks. Note that Florence [85] utilizes a more +recent and powerful Swin-H Vision Transformer architec- +ture [45] with convolutional embeddings [76], and a unified +contrastive objective [82]. Our proposed contributions are +complementary to FILIP [83] and Florence [85], and we +believe additional gains can be achieved when combined. +Finally, LiT [87] and BASIC [57] first pre-train model on +an large-scale image annotation dataset with cross-entropy +before further training with contrastive loss on an image- +text dataset. Though this strategy results in state-of-the-art +performance on ImageNet1K [64] and image classification +benchmarks, it has severe downsides on multi-modal tasks +such as cross-modal retrieval. Our ablation in Section 4.2 +also confirms this issue. On the other hand, our approach +does not suffer from such negative effects. +4.4. Few-shot linear probing +The ideal scenario for leveraging zero-shot recognition +models is to warm start the task without training data and +then improve the performance (by training a linear probe) +via few-shot learning as more and more data is seen. How- +ever, in practice, few-shot models perform significantly +worse than zero-shot models in the low-data regime. +We present an alternate approach to do few-shot classi- +fication with prompt-based initialization. The key idea of +our approach is to initialize the classifier with the zero-shot +text prompts for each class, but to also ensure that the fi- +nal weights do not drift much from the prompt using pro- +jected gradient descent (PGD) [5]. While few-shot models +have been initialized with prompt priors in the past with +naive L2 penalties for weight to prevent catastrophic for- +7 + +Table 5. Comparison with zero-shot state-of-the-art dual-encoder +models. +px: input image size; #P: model size; #D: training +dataset size; #S: total samples processed at training. We evaluate +CLIP [61] and OpenCLIP [27] using our codebase, other numbers +are copied from respective papers. Grouped models (e.g., ViT- +B/32) share same vision and language architecture as our model, +following CLIP [61], others have different architectures and we +outline the vision one. ∗FILIP uses token-wise similarity, which is +more expensive than global-token similarity and requires adapting +the architecture, hence we put it in “Other”. +Method +px +#P +#D +#S +IN +COCO +Flickr +T2I +I2T +T2I +I2T +ViT-B/32 +CLIP [61] +224 151M 400M 12.8B 63.4 31.4 49.0 59.5 79.9 +OpenCLIP [27] 224 151M 400M 12.8B 62.9 34.8 52.3 61.7 79.2 +OpenCLIP [27] 224 151M 2.3B +34B +66.6 39.0 56.7 65.7 81.7 +DiHT +224 151M 438M 16B +67.5 40.3 56.3 67.9 83.8 +DiHT +224 151M 438M 32B +68.0 40.6 59.3 68.6 84.4 +ViT-B/16 +CLIP [61] +224 150M 400M 12.8B 68.4 33.7 51.3 63.3 81.9 +OpenCLIP [27] 224 150M 400M 12.8B 67.1 37.8 55.4 65.2 84.1 +OpenCLIP [27] 240 150M 400M 12.8B 69.2 40.5 57.8 67.7 85.3 +DiHT +224 150M 438M 16B +71.9 43.7 62.0 73.2 89.5 +DiHT +224 150M 438M 32B +72.2 43.3 60.3 72.9 89.8 +ViT-L/14 +CLIP [61] +224 428M 400M 12.8B 75.6 36.5 54.9 66.1 84.5 +CLIP [61] +336 428M 400M 13.2B 76.6 37.7 57.1 68.6 86.6 +OpenCLIP [27] 224 428M 400M 12.8B 72.8 42.1 60.1 70.4 86.8 +OpenCLIP [27] 224 428M 2.3B +32B +75.2 46.2 64.3 75.4 90.4 +DiHT +224 428M 438M 16B +77.0 48.0 65.1 76.7 92.0 +DiHT +336 428M 438M 16.4B 77.9 49.3 65.3 78.2 91.1 +Other +ALIGN [28] +EfficientNet-L2 +289 820M 1.8B 19.7B 76.4 45.6 58.6 75.7 88.6 +FILIP [83]∗ +ViT-L/14 +224 428M 340M 10.2B 77.1 45.9 61.3 75.0 89.8 +OpenCLIP [27] +ViT-H/14 +224 986M 2.3B +32B +77.9 49.0 67.5 76.8 91.3 +Florence [85] +CoSwin-H +384 893M 900M 31B +83.7 47.2 64.7 76.7 90.9 +LiT [87] +ViT-g/14 +288 +2.0B +3.6B 18.2B 85.2 41.9 59.3 +— +— +BASIC [57] +CoAtNet-7 +224 +3.1B +6.6B 32.8B 85.7 +— +— +— +— +getting [33], these approaches do not improve performance +and the model simply ignores the supervision. +In con- +trast, for any target dataset Dtarget = {(xi, yi)n +i=1}, where +xi = φimage(Ii) denotes the image features from the trained +image tower, we solve the following optimization problem, +for some δ, δb > 0: +min +∥W∥2≤δ,∥b∥2≤δb +n +� +i=1 +LCE +� +yi, x⊤ +i (W + W0) + b +� +. +Here W0 ∈ Rd×nc denotes the prompt initialization from +the text encoder. To optimize the objective, one can use pro- +jected gradient descent [5]. We observe that our approach is +able to bridge the gap between zero-shot and 1-shot classi- +fication, a common issue in prior linear probe evaluations. +1 +2 +5 +10 +25 +40 +45 +50 +55 +60 +65 +70 +75 +80 +85 +k-shot +Accuracy@1 +DiHT-B/16 (PGD) +DiHT-B/16 (SGD) +DiHT-B/16 (0-shot) +1 +2 +5 +10 +25 +40 +45 +50 +55 +60 +65 +70 +75 +80 +85 +k-shot +Accuracy@1 +DiHT-L/14 (PGD) +DiHT-L/14 (SGD) +DiHT-L/14 (0-shot) +CLIP-L/14 (SGD) +SWAG-H/14 (SGD) +Figure 6. k-shot linear probing performance on ImageNet1K. +Figure 6 presents the full summary of results on the Ima- +geNet1K [64] k-shot classification task. Hyperparameters δ +and δb for our approach, and weight decay for the baseline +approach of training linear probes from scratch are found +using grid search. Note that compared to the baseline, our +method performs substantially better at very low values of +k and maintains the performance continuum from zero-shot +to 1-shot, and so on. At large k values, both approaches +perform similarly, since there are sufficient data samples +to render the zero-shot initialization ineffective. +To fur- +ther showcase the strength of our approach, we also com- +pare our performance with linear probes trained on powerful +SWAG [69] models that are especially suited for this task. +Note that our approach outperforms the much larger SWAG +ViT-H/14 model up to 25-shot classification. We would like +to emphasize that this albeit straightforward approach is one +of the first to resolve this discontinuity problem between +zero-shot and few-shot learning. +5. Conclusion and future work +In this paper, we demonstrate that with careful dataset +filtering and simple but effective modeling changes, it is +possible to achieve substantial improvements in zero-shot +performance on retrieval and classification tasks through +large-scale pre-training. Our CAT filtering approach can be +applied generically to any large-scale dataset for improved +performance with smaller training schedules. +Moreover, +our concept distillation approach presents a compute and +storage efficient way of leveraging very large capacity pre- +trained image models for multimodal training. Finally, our +simple projected gradient approach covers the crucial per- +formance gap between zero-shot and few-shot learning. +In future, we would like to extend our approach to +multi-modal encoder/decoder [1,10,40,81,84] architectures +that although expensive, have better zero-shot performance +compared to dual encoders. 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The aim of the parser is high speed with high pre- +cision of common object relations such as ‘has attribute‘ +and ‘has part‘ and basic ‘action‘ support. Below, we de- +scribe the structured relations that we extract from natural +language text. +We support the following semantic relations: +Object ( obj). +We extract objects that are supposedly pre- +sented in an image. +We consider nouns that are not at- +tributes of another noun (not part of a noun phrase). E.g. in +birthday cake and baby stroller, the nouns cake and stroller +are parsed as objects, and the nouns birthday and baby are +considered attributes. We do not consider proper nouns. +Attribute (has attr). +Denotes attributes that characterize +an object or another attribute. For example, dark green, +would result in a fact green - has attr - dark, and yellow +candles results in candles - has attr - yellow. +Part (has part). +Characterizes a visual part of an object. +E.g. cake with 21 yellow candles would result in a part fact +cake - has part - candles. +Action ( act). +Verbs that do not entail attributes or parts +(e.g. forms of be, looks, seems, and have are excluded) are +considered actions. For actions, we also parse the subject +and object arguments. +Subject of an action (act has subj, is act subj). +We use +the act has subj and is act subj relation to represent argu- +ments (nouns) that are the subject of an action. E.g. for the +text a person is eating an apple, we add the object-centric +and corresponding action-centric symmetric facts: person - +is subj act - eating and eating act has subj person. +Object of an action (act has obj, is act obj). +We also +include the relations that specify the object arguments of +an action. E.g. for the text a person is eating an apple, +we add the object-centric and corresponding action-centric +symmetric facts: apple - is obj act - eating and eating +act has obj apple. +We recognize the following limitations of ours approach: +Semantic attributes. +In this work, we focus on object- +centric visual and action characteristics and we do not pro- +cess spatial relations ( X next to Y) or additional action ar- +guments (read a book *in* the library). Spatial relations +and additional arguments of verbs usually involve more +complex semantic reasoning and require more robust ap- +proaches and task-specific models such as one trained on +Semantic Role Labeling which are usually compute-heavy. +We leave these for future work. +Dependency parser errors. +In the current version of the +parser, we also parse potential attributes as actions, which +are not likely to be always visual. E.g. In the phrase “run- +ning person”, running is an action and an attribute, and we +parse them as such. However, sometimes the underlying +parser would also parse attributes in phrases such as “striped +mug” as verbs, where we process the attribute “striped” as +both an attribute and an action (without arguments). +A.1.2 +Concept distillation +The teacher model is built by training linear classifiers - +which predict objects and attributes - on top of a frozen +SWAG [69] backbone. +SWAG is trained in a weakly- +supervised manner by predicting hashtags from Instagram +images. We use the publicly available weights, and adopt +a training procedure that is similar to the one from SWAG +for learning the linear classifiers. The procedure for train- +ing the object classifier is as follows. First, we parse the +captions to extract nouns. Next, we canonicalize the nouns +via WordNet [51] synsets and remove ones which occur less +than 250 times in the dataset. The resulting vocabulary con- +tains ∼10K unique synsets. Finally, we optimize the linear +layer’s weights through a cross-entropy loss. Each entry in +the target distribution of the cross-entropy is either 1/K or +0 depending on whether the corresponding synset is present +or not, where K is the number of synsets for that image. We +apply inverse square-root resampling of images to upsam- +ple the tail classes following [69]. The target length of the +dataset is set to 50 million samples during resampling . We +train the linear layer using SGD with momentum 0.9 and +weight decay 1e-4. The learning rate is set following the lin- +ear scaling rule: lr=0.001· bs +256. To speedup training, we use +64 GPUs with batch size of 256 per GPU. The attribute clas- +sifiers are build in a similar way, but the WordNet adjective +synsets require additional filtering to remove non-visual at- +tributes, e.g., claustrophobic, experienced. Following [60], +we select the attributes based on their sharedness and visu- +alness. We rank the attributes based on the aforementioned +scores, and keep ∼1200 attributes. +12 + +Table A.1. DiHT architecture hyperparameters. +Model +Dim +Vision +Language +layers +width +heads +layers +width +heads +B/32 +512 +12 +768 +12 +12 +512 +8 +B/16 +512 +12 +768 +12 +12 +512 +8 +L/14 +768 +24 +1024 +16 +12 +768 +12 +Table A.2. DiHT common hyperparameters. +Shared +Learning rate (LR) +1e-3 +Warm-up +1% +Vocabulary size +49408 +Temperature (init, max) +( +1 +0.07, 100.0) +Adam (β1, β2) +(0.9, 0.98) +Adam ϵ +1e-6 +High resolution LR +1e-4 +Dataset specific +LAION +PMD +CD learning rate scale +10.0 +1.0 +CD weight decay scale +0.01 +1.0 +HN-NCE α +1.0 +0.999 +HN-NCE β +0.25 +0.5 +LAION +PMD +Model specific +L/14 +B/16,B/32 +B/16,B/32 +Batch size +98304 +49152 +32768 +Weight decay +0.2 +0.1 +0.1 +A.2. Training details +For our model architecture, we closely follow CLIP +by Radford et al. [61]. +We utilize Vision Transformers +(ViT) [17] for images and Text Transformers [73] for cap- +tions. We experiment with 3 different architectures, denoted +as B/32, B/16, and L/14, where 32, 16, and 14 denote the +input image patch size. Other architecture scaling param- +eters are in Table A.1. For distillation and fine-tuning ex- +periments, we utilize the public SWAG-ViT models [69], +pre-trained with weak supervision from hashtags. +We use the Adam [32] optimizer with a decoupled +weight decay [47] and a cosine learning rate schedule [46]. +Input image size is 224×224 pixels, for pre-training runs. +All hyperparameters are presented in Table A.2. They are +selected by training on a small scale setup, and reused for +other experiments. For objects and attributes classifiers in +concept distillation (CD), we found that scaling the learning +rate by 10.0 and weight decay by 0.01 gave better results. +We pre-train the models on 4B, 8B, 16B, or 32B pro- +cessed samples, depending on the experiment. For L/14 we +train at a higher 336px resolution for additional 400M sam- +ples, denoting this models as L/14@336. We trained L/14 +for 6 days on 512 A100 GPUs with 16B processed samples +for a total of 7.4 × 104 GPU hours. +To accelerate training and save memory, we use mixed- +precision training [50]. For L/14 we use grad checkpoint- +ing [8] and BFLOAT16 [14,29] format, all the other models +are trained using FP16 [50] format. Contrastive loss is com- +puted on the local subset of the pairwise similarities [61]. +A.3. Evaluation details +We evaluate our models on a zero-shot benchmark of +24 datasets: (i) 17 image classification: Birdsnap [3], +CIFAR10 [36], CIFAR100 [36], Caltech101 [19], Coun- +try211 [61], DTD [13], Flowers102 [53], Food101 [4], Ima- +geNet1K [64], OxfordPets [56], STL10 [15], SUN397 [77], +StanfordCars [34], UCF101 [70], HatefulMemes [31], +PascalVOC2007 [18], OpenImages [38]; +(ii) 5 cross- +modal retrieval (text-to-image T2I, image-to-text I2T): +COCO [44], Flickr [58], LN-COCO [59], LN-Flickr [59], +Winoground [71]; +(iii) 2 visual question answering: +SNLI-VE [78], VQAv2 [21]. Note that, cross-modal re- +trieval datasets have 2 tasks (T2I and I2T), so in total we +evaluate across 29 tasks. +We follow zero-shot CLIP benchmark7 implementation +for most of the datasets, and implement the ones that are +missing. +For most image classification tasks we com- +pute Accuracy@1, except HatefulMemes where we com- +pute AUROC because it is binary classification, OpenIm- +ages where we compute FlatHit@1 following [75], and +PascalVOC2007 where we compute mean average preci- +sion (mAP) because it is multi-label classification. We use +the same prompt ensembling method as CLIP [61] to im- +prove zero-shot image classification. For cross-modal re- +trieval (T2I and I2T), we compute Recall@1. For COCO +and Flickr we apply a simple prompt pretext “a photo of +{caption}”, for LN-COCO, LN-Flickr, and Winoground +no prompt is applied. We cast visual question answering +(VQA) as binary prediction task and compute AP on the +cosine similarity between an image and a text (a hypothe- +sis or a question). For SNLI-VE, we take a subset which +has agreement among annotators, we use “entailement” and +“contradiction” as binary classes, and drop the “neutral” +class. For VQAv2, we take the subset with yes/no ques- +tions. No prompt is applied for SNLI-VE and VQAv2. +A.4. Additional ablations +Effect of dataset filtering. +In Figure A.1 we observe +that gains from our proposed complexity, action, and text- +spotting (CAT) dataset filtering hold as we train for longer +training schedules. We ran small scale experiments with +several complexity filters (see Table A.3) and we found that +CAT with minimum complexity C1 performed the best. +Effect of top-k predicted objects and attributes. +In Ta- +ble A.4, we show that our concept distillation approach is +quite robust to the choice of the number of predicted ob- +jects and attributes. For k = 10 strong accuracy is achieved +with a small increase in dataset memory. +7github.com/LAION-AI/CLIP benchmark +13 + +4B +8B +16B +32B +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +Num Samples +Accuracy@1 +ImageNet-1K +LAION-2B +LAION-CAT +4B +8B +16B +32B +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +Num Samples +Recall@1 +COCO (T2I) +4B +8B +16B +32B +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +Num Samples +Recall@1 +COCO (I2T) +4B +8B +16B +32B +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +Num Samples +Recall@1 +Flickr (T2I) +4B +8B +16B +32B +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +Num Samples +Recall@1 +Flickr (I2T) +Figure A.1. Evaluating effect of using our LAION-CAT subset filtered on complexity (C), actions (A), and text spotting (T). Evaluation +performed on ViT-B/32 architecture trained for a varying number of processed samples. +Table A.3. Number of examples after filtering with different filters. +Filter +# examples % of full +[67] C0 +C1 +C2 +A +T +2,121,505,329 +100.00 +✓ +1,983,345,180 +93.49 +✓ +✓ +1,891,725,045 +89.17 +✓ +✓ +1,709,522,548 +80.58 +✓ +✓ +1,143,660,096 +53.91 +✓ +✓ +691,535,901 +32.60 +✓ +✓ +✓ +642,162,957 +30.27 +✓ +✓ +✓ +487,493,190 +22.98 +✓ +✓ +✓ +✓ +438,358,791 +20.66 +Table A.4. Evaluating effect of using different number of top-k +predicted objects and attributes. Evaluation on ViT-B/16 model +architecture trained for 8B processed samples on LAION-CAT. +Memory denotes storage needed to store predicted concepts. +top-k +Memory +IN +COCO +Flickr +T2I +I2T +T2I +I2T +5 +16.3GB +71.4 +42.9 +59.4 +72.2 +86.5 +10 +32.6GB +71.9 +42.9 +60.3 +73.3 +87.0 +25 +81.6GB +71.4 +43.1 +60.0 +72.9 +87.9 +Effect of α and β on HN-NCE. +From intuition, one can +see that the term α controls the mass of the positive align- +ment term in the loss function, and the term β controls the +difficulty of the negatives. The need for the term α can +be attributed as follows. If there are false negatives within +the dataset, dampening the positive alignment term can pre- +vent the model from becoming overly discriminative with +the true and false positive pairs. Hence, we would like to re- +duce α as the likelihood of having false positives increases +(e.g., smaller datasets, less noisy training). The need for β +is straightforward: higher β pushes the weighing function +to be “sharper”, with more mass on the hardest negatives. +Table A.5 shows the effect of different values of α and β on +LAION-CAT. +Table A.5. Evaluating effect of different hyperparameters α and β +for the HN-NCE loss. Evaluation on ViT-B/16 model architecture +trained for 16B processed samples on LAION-CAT. +α +β +IN +COCO +Flickr +T2I +I2T +T2I +I2T +1 +0 +68.7 +42.8 +60.5 +72.8 +87.6 +1 +0.25 +69.2 +42.9 +61.2 +72.6 +87.8 +1 +0.5 +66.5 +40.3 +59.7 +71.4 +84.9 +0.999 +0.25 +69.0 +42.6 +60.9 +72.3 +87.9 +0.9 +0.25 +68.6 +42.1 +59.2 +71.2 +85.5 +Table A.6. Evaluating linear probing with the complete training +set for ImageNet1K on the ViT-L/14 architecture. +Model +Optimizer +ImageNet-1K +Accuracy (%) +CLIP-L/14 @ 224px +SGD +83.60 +DiHT-L/14 @ 224px +SGD +85.40 +DiHT-L/14 @ 224px +PGD +85.41 +CLIP-L/14 @ 336px +SGD +85.40 +DiHT-L/14 @ 336px +SGD +85.87 +DiHT-L/14 @ 336px +PGD +85.89 +Additional results on few-shot probing. +We examine the +performance of our models on linear probing with the full +training set for ImageNet1K [64]. We compare the perfor- +mance of DiHT-L/14 and CLIP-L/14 [61] architectures for +both the 224px and 336px input sizes in Table A.6. We +observe that the PGD approach with the DiHT model out- +performs prior work, and also find that there is no notable +difference in performance between SGD-trained and PGD- +trained models, as there is no need for regularization when +training with the full dataset. We reproduce the reported +numbers for CLIP [61] and train our models with a learning +rate of 24, no weight decay, and batch size of 96,000 for +160 epochs. +14 + +Table A.7. Zero-shot state-of-the-art dual-encoder models comparison. We evaluate CLIP [61] and OpenCLIP [27] using our codebase. +Method +Birdsnap +CIFAR10 +CIFAR100 +Caltech101 +Country211 +DTD +Flowers102 +Food101 +ImageNet1K +OxfordPets +STL10 +SUN397 +StanfordCars +UCF101 +HatefulMemes +PascalVOC +OpenImages +COCO T2I +COCO I2T +Flickr T2I +Flickr I2T +LN-COCO T2I +LN-COCO I2T +LN-Flickr T2I +LN-Flickr I2T +Winoground T2I +Winoground I2T +SNLI-VE +VQAv2 +ViT-B/32 @ 224 +CLIP +40.3 89.8 65.1 83.9 17.2 43.8 66.6 83.9 63.4 87.4 97.2 62.3 59.7 64.2 58.1 84.2 27.8 31.4 49.0 59.5 79.9 16.8 24.6 30.2 38.1 28.1 27.4 77.6 57.3 +OpenCLIP +50.5 93.6 75.8 86.4 16.7 56.1 71.7 82.7 66.6 90.6 96.6 68.5 86.0 66.1 53.4 85.4 34.6 39.0 56.7 65.7 81.7 29.5 35.1 44.0 51.4 32.0 30.2 78.6 59.3 +DiHT +46.5 92.0 73.6 80.4 16.3 55.3 69.8 84.1 68.0 91.7 97.2 66.5 79.6 68.3 53.5 78.9 32.4 40.6 59.3 68.6 84.4 29.8 35.7 46.1 54.0 30.9 33.0 79.1 59.9 +ViT-B/16 @ 224 +CLIP +43.2 90.8 68.3 84.7 22.8 44.9 71.2 88.7 68.4 89.1 98.3 64.4 64.7 69.5 59.3 85.3 29.3 33.7 51.3 63.3 81.9 18.7 25.2 31.3 37.4 31.0 30.2 77.9 57.7 +OpenCLIP +52.1 91.7 71.4 86.2 18.1 50.8 69.3 86.1 67.1 89.4 97.0 69.6 83.8 67.7 55.7 84.2 35.2 37.8 55.4 65.2 84.1 26.1 33.1 43.5 46.9 30.5 30.2 78.4 59.3 +DiHT +54.5 92.7 77.5 81.2 19.1 59.4 70.5 89.1 72.2 92.7 98.2 68.4 86.0 70.3 56.2 79.5 34.6 43.3 60.3 72.9 89.8 32.4 38.2 52.9 57.7 32.0 33.4 80.8 60.3 +ViT-L/14 @ 224 +CLIP +52.5 95.6 78.2 86.7 31.9 55.5 79.1 93.1 75.6 93.5 99.4 67.6 77.8 77.0 60.4 85.5 30.6 36.5 54.9 66.1 84.5 20.8 28.6 36.2 44.2 31.9 32.0 78.2 58.4 +OpenCLIP +62.9 96.6 83.4 88.0 26.3 62.9 75.5 91.0 75.2 93.2 98.9 74.3 92.6 75.2 55.1 87.5 38.0 46.2 64.3 75.4 90.4 34.6 39.9 50.9 57.7 33.4 36.4 80.8 60.0 +DiHT +60.4 91.7 81.3 81.6 26.0 60.3 77.6 92.7 77.0 93.8 98.0 70.2 91.1 77.9 56.5 79.3 35.0 48.0 65.1 76.7 92.0 35.6 40.7 52.7 60.3 31.8 33.4 81.3 61.0 +ViT-L/14 @ 336 +CLIP +53.7 95.0 77.0 87.2 34.4 56.0 78.6 93.8 76.6 93.8 99.5 68.7 79.2 77.6 61.6 86.2 31.8 37.7 57.1 68.6 86.6 20.2 28.6 38.1 45.7 32.3 21.4 78.7 58.5 +DiHT +62.0 92.2 81.2 82.4 27.8 61.1 77.0 92.9 77.9 94.0 98.2 71.2 91.5 77.7 56.3 81.0 36.5 49.3 65.3 78.2 91.1 36.7 41.2 54.5 61.6 35.0 38.5 81.7 61.4 +Additional results on zero-shot benchmark. +We report +performance of CLIP [61], OpenCLIP [27], and DiHT on +all 29 zero-shot tasks in Table A.7. +A.5. Contrastive Alignment with Hard Negatives +Convergence guarantees +Proposition 1. Let L⋆(φi, φt) = supq∈Π L(φi, φt, q). +Then for any measurable φi, φt +: +X +→ +Sd−1 and +τ = O(1) we observe the convergence L(φi, φt, q) → +L⋆(φi, φt) as β → ∞. +Proof. Follows from Proposition 6 of [62] with the loss +function L(φi, φt, qβ) defined as follows for any β. +L(φi, φt, qβ) = +log +� +eφi(x)⊤φt(x)/τ +eφi(x)⊤φt(x)/τ + Q · Ey∼qβ +� +eφi(x)⊤φt(y)/τ� +� ++ log +� +eφi(x)⊤φt(x)/τ +eφi(x)⊤φt(x)/τ + Q · Ey∼qβ +� +eφi(x)⊤φt(y)/τ� +� +. +15 + diff --git a/dNE0T4oBgHgl3EQfWgC_/content/tmp_files/load_file.txt b/dNE0T4oBgHgl3EQfWgC_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1e340e0eb1c4ea8e9ed4a3d4bc96fa7660896cf --- /dev/null +++ b/dNE0T4oBgHgl3EQfWgC_/content/tmp_files/load_file.txt @@ -0,0 +1,1713 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf,len=1712 +page_content='Filtering, Distillation, and Hard Negatives for Vision-Language Pre-Training Filip Radenovic, Abhimanyu Dubey∗, Abhishek Kadian∗, Todor Mihaylov∗, Simon Vandenhende∗ Yash Patel, Yi Wen, Vignesh Ramanathan and Dhruv Mahajan† Meta AI Abstract Vision-language models trained with contrastive learn- ing on large-scale noisy data are becoming increasingly popular for zero-shot recognition problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In this paper we improve the following three aspects of the contrastive pre-training pipeline: dataset noise, model initialization and the training objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' First, we propose a straightfor- ward filtering strategy titled Complexity, Action, and Text- spotting (CAT) that significantly reduces dataset size, while achieving improved performance across zero-shot vision- language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Next, we propose an approach titled Con- cept Distillation to leverage strong unimodal representa- tions for contrastive training that does not increase train- ing complexity while outperforming prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, we modify the traditional contrastive alignment objective, and propose an importance-sampling approach to up-sample the importance of hard-negatives without adding additional complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' On an extensive zero-shot benchmark of 29 tasks, our Distilled and Hard-negative Training (DiHT) ap- proach improves on 20 tasks compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Fur- thermore, for few-shot linear probing, we propose a novel approach that bridges the gap between zero-shot and few- shot performance, substantially improving over prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Models are available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='com/facebookresearch/diht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Introduction An increasingly popular paradigm in multimodal learn- ing is contrastive pre-training [11,28,40,42,61,74,83,85], which involves training multimodal models on very large- scale noisy datasets of image-text pairs sourced from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' It has been shown to be incredibly effective for a vari- ety of vision-language tasks without any task-specific fine- tuning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', zero-shot), such as image classification [64], text and image retrieval [44, 58], visual question answer- ing [21], among several others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In this paper, we study the problem of contrastive pre-training for dual-encoder ar- chitectures [61] with the objective of improving image-text alignment for zero-shot tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We revisit three important aspects of the contrastive pre-training pipeline – noise in Equal contribution, listed alphabetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' †Research Lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' B/32 B/16 L/14 @336 60 65 70 75 80 Model Complexity Accuracy@1 ImageNet1K DiHT CLIP B/32 B/16 L/14 @336 30 35 40 45 50 Model Complexity Recall@1 COCO (T2I) B/32 B/16 L/14 @336 59 64 69 74 79 Model Complexity Recall@1 Flickr (T2I) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' DiHT trained on 438M LAION-CAT samples vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CLIP [61] trained on 400M OpenAI samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' datasets, model initialization, and contrastive training, and present strategies that significantly improve model perfor- mance on a variety of zero-shot benchmarks, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Most image-text datasets are noisy and poorly-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Few recent efforts [27] have tried to clean the noise by fil- tering samples based on alignment scores from an existing model like CLIP [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' However, this approach is limited by the biases and flaws of the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' On the other hand, momentum-based approaches [40] to reduce noise are in- feasible for large-scale training due to their increased com- pute and memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To this end, we provide a scalable and effective approach titled Complexity, Action and Text-spotting (CAT) filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CAT is a filtering strat- egy to select only informative text-image pairs from noisy web-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We show that training on a CAT-filtered version of large-scale noisy datasets such as LAION [65] can provide up to 12% relative improvements across vision- language tasks despite removing almost 80% of the training data, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 and Table 1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A common strategy [57, 87] to further improve multi- modal training is to warm-start it with image and text mod- els pre-trained at large scale on their respective modali- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' However, due to the increased noise in image-text data, fine-tuning the entire model undermines the benefits of the warm-start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' One can alternatively use model freezing strategies like locked-image tuning [87], but they are un- able to adapt to the complex queries present in multimodal problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', cross-modal retrieval) and the models per- form poorly on retrieval benchmarks (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='02280v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='CV] 5 Jan 2023 propose an entirely different approach, concept distillation (CD), to leverage strong pre-trained vision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The key idea behind concept distillation is to train a linear classifier on the image encoder to predict the distilled concepts from a pre-trained teacher model, inspired by results in weakly- supervised large-scale classification [48,69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, we revisit the training objective: almost all prior work has utilized noise-contrastive estimation via the In- foNCE loss [54], shortcomings have been identified in the standard InfoNCE formulation [12, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We demonstrate that by using a model-based importance sampling technique to emphasize harder negatives, one can obtain substantial improvements in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A summary of our pipeline is available in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our combined approach obtains significant improvements over the baseline for dual-encoder architectures on an elab- orate benchmark of 29 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Specifically, with the ViT- B/16 [17] architecture, we improve zero-shot performance on 20 out of 29 tasks, over CLIP training on the LAION- 2B dataset [27, 65], despite training on a subset that is 80% smaller, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Furthermore, we demonstrate that even when trained with the smaller (but relatively less noisy) pretraining dataset PMD, our performance is better on 28 out of 29 tasks than CLIP trained on the same data, often with a large margin, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Additionally, we present a simple yet effective approach to maintain the performance continuum as one moves from zero-shot to few-shot learning in the low data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Prior work [61] has shown a substantial drop in performance as one moves from zero-shot to k-shot learning, which is undesirable for practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We propose an alter- nate linear probing approach that initializes the linear clas- sifier with zero-shot text prompts and ensures that final weights do not drift away too much via projected gradient descent [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' On ImageNet1K, we show huge improvements over prior work for small k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For example, our ap- proach improves 5-shot top-1 accuracy by an absolute mar- gin of 7% (see Figure 6) compared to the baseline strategy of linear probing with a random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Related work Dataset curation for contrastive pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Large- scale contrastive pretraining [11, 28, 40, 42, 61, 74, 83, 85] typically requires dataset sizes of the order of hundreds of millions to billions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Seminal works in this area, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', CLIP [61] and ALIGN [28], have largely relied on image- text pairs crawled from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Subsequently, versions of large-scale image-text datasets have been created but not released publicly, including WIT-400M [61], ALIGN- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8B [28], FILIP-340M [83], FLD-900M [85], BASIC- 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6B [57], PaLI-10B [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' These datasets often use un- clear or primitive cleaning strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', removing sam- ples with short or non-English captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Recently, LAION- Object & Attribute Cross-Entropy Cute kitten Cute kitten Cute kitten Cute kitten Cute kitten My cute baby CAT Filtering 1 Gray cat with red scarf watching TV SWAG-ViT Image Encoder Object & Attribute Classifiers Parser Cat Scarf TV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Gray Red Train separately Image Encoder Text Encoder I1 I2 I3 I4 T1 T2 T3 T4 Positive Hard-negative HN-NCE 3 Inference Concept Distillation 2 Storage Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Summary of our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We propose improvements to the standard vision-language pre-training: (1) Complexity, Action and Text-spotting (CAT) filtering that removes non-informative text-image pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' (2) Concept distillation from a frozen (�) pre- trained image encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' (3) Hard-negative contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 400M [66] used CLIP-based scores to filter down a large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The authors later released an English-only LAION- 2B and a LAION-5B unfiltered dataset sourced from Com- mon Crawl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Apart from LAION-400M and BLIP [39] which uses the bootstrapped image-grounded text encoder to filter out noisy captions, there has not been a signifi- cant investment in systematic curation strategies to improve zero-shot alignment performance on large-scale pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In contrast to the previous work, we use quality-motivated filters that retain images whose captions are sufficiently complex, contain semantic concepts (actions), and do not contain text that can be spotted in the image [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Distillation from pre-trained visual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Knowl- edge distillation [25] aims to transfer knowledge from one model to another and has been used in many con- texts ranging from improving performance and efficiency [6, 7, 41, 63, 72, 79] to improving generalization capabili- ties [16, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Several approaches use self-distillation to improve performance with lower computational overhead [23,80,86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For vision and language pre-training, [2,40] use soft-labels computed using embeddings from a moving av- erage momentum model with the goal to reduce the adverse effects of noisy image-text pairs in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our concept distillation approach is a cheaper and more effec- tive alternative, since it does not require us to run the expen- sive teacher model throughout the training2 while retaining the most useful information from the visual concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Another approach to take advantage of pre-trained vi- sual models is to use them to initialize the image encoder, and continue pre-training either by locking the image en- coder [57, 87] or fine-tuning [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' However, these ap- proaches lack the ability to align complex text to a fully- trained image encoder, and thus perform poorly on multi- modal tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' cross-modal retrieval (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 1commoncrawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='org 2Distillation targets can be pre-computed and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 2 Contrastive training with hard negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Noise- contrastive estimation (NCE) [22] is the typical objective for vision-text learning, with applications across large-scale multimodal alignment [11,28, 42,61] and unsupervised vi- sual representation learning [24, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Several lines of work have studied the shortcomings of the original InfoNCE ob- jective [54], specifically, the selection and importance of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' [12] present a debias- ing approach to account for false negatives at very large batch sizes, typical in large-scale pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Kalantidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' [30] present a MixUp approach to improve the qual- ity of hard negative samples for unsupervised alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Using model-specific hard negatives in the training objec- tive is proven to reduce the estimation bias of the model as well [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Contrary to prior semi-supervised work, we ex- tend the model-based hard negative objective, first proposed in Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' [62] to multimodal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Method Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We consider the task of contrastive image- text pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Given a dataset D = {(Ii, Ti)}N i=1 of image-text pairs, we want to learn a dual encoder model φ = {φimage, φtext}, where φimage represents the image en- coder, and φtext denotes the text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the short- hand x = φimage(I) and t = φtext(T) to denote the encoded images and texts, respectively, for an image-text pair (I, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We will now describe the three crucial components of our approach followed by the final training objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Complexity, Action, and Text (CAT) filtering Our complexity, action, and text spotting (CAT) filtering is a combination of two filters: a caption complexity filter that removes image-caption pairs if a caption is not suffi- ciently complex, and an image filter that removes pairs if the image contains text matching the caption to prevent poly- semy during alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the LAION-2B pre-cleaned obtained after using filters3 in [67] as the base dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Filtering captions via complexity & actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Noisy web-scale datasets do not have any semantic-based cura- tion, and hence captions can be irrelevant, ungrammatical and unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our motivation is to decrease such noise by simply selecting captions that possess sufficient complex- ity, so that the training distribution matches the target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To this end, we build a fast rule-based parser that extracts objects, attributes and action relations (see Figure 3 for an example) from text and we use the resulting semantic graph to estimate the complexity of the image captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Specifi- cally, we define the complexity of a caption as the maximum number of relations to any object present in the parse graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For example, in the caption “A black cat is chasing a small 3Not-suitable-for-view images and toxic captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Caption: A black cat is chasing a small brown bird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' chasing ACTION bird OBJECT cat OBJECT small ATTRIBUTE brown ATTRIBUTE black ATTRIBUTE has attribute has attribute has attribute has object has subject Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' An example caption and its parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The caption has C3 complexity (due to bird) and has 1 action (chasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' brown bird,” the object “bird” has the attributes “small”, “brown” and “A black cat is chasing”, and hence, the com- plexity of the caption is C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We only retain samples that at least have a C1 caption complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To further remove pairs likely containing products, we filter out captions if they do not contain at least one action (as obtained from the parser).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Filtering images via text-spotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Image-caption pairs in web-scale datasets often display part of the caption as text in the image (on visual inspection, we found up to ∼30% such examples for LAION [65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Minimizing the objective, in these cases, can correspond to spotting text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', optical character recognition) rather than the high-level visual se- mantics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', objects, attributes) we would like the model to align to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' This will reduce performance on object-centric and scene-centric downstream zero-shot tasks, and hence we remove such images from the training set using an off- the-shelf text spotter [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We remove image-text pairs with a text spotting confidence of at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 and at least 5 pre- dicted characters matching the caption in a sliding window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We observe (by inspection) that this approach is efficient at identifying images with text, and failure cases are primarily in non-English text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Filtering with multilingual text spotters trained can fix this issue, however, we leave this as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Filtering statistics can be found in the supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Concept distillation Recognizing visual concepts in images that correspond to objects and attributes in corresponding captions is cru- cial for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We therefore propose to distill these concepts from a pre-trained teacher model to our image en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Specifically, we add two auxiliary linear classifiers on top of the encoded image embeddings x to predict (i) objects and (ii) visual attributes and use the teacher model to generate the pseudo-labels for training them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' These clas- sifiers are trained jointly with the contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We parse image captions using a semantic parser that extracts objects and attributes from text (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1) and use these as pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We then train the linear clas- sifiers on the teacher model embeddings with a soft-target cross-entropy loss [20], after square-root upsampling low- frequency concepts [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' It is important to freeze the back- bone of the teacher model to make sure we retain the ad- vantages of using a stronger model for distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For 3 each image, we then use these trained linear classifiers to generate two softmax probability vectors – pobj for objects, and pattr for attributes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To minimize the stor- age overhead, we further sparsify them by retaining only the top-k predicted class values and re-normalizing them to generate the final pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' During multimodal train- ing, we use the cross-entropy loss with these pseudo-label vectors as targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Unless specified otherwise, we use the ViT-H/14 [17] architecture pretrained from SWAG [69] as the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 and the supplementary material for ablations on the effect of different backbones and retaining top-k predictions, and further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' There are several advantages of our concept distillation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' First, the teacher predictions capture correlations from the strong vision encoding, making them more infor- mative as labels compared to the captions themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The captions are limited to a few objects and attributes, while the teacher predictions yield a more exhaustive list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' More- over, our approach reaps the benefits of the recently pro- posed and publicly-available strong unimodal vision mod- els more effectively than other distillation approaches, as training linear classifiers on a frozen teacher model is inex- pensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' After predictions are stored, we discard the teacher model and thus bypass the memory and compute limitations of simultaneously running the student and teacher model in standard distillation approaches [25, 72], which is criti- cal for large teacher models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We demonstrate empirically (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2) that our strategy works better than distill- ing teacher embeddings directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Additionally, compared to approaches that warm-start the image encoder with pre- trained models, our method can leverage higher capacity teacher models without difficulty and unlike locked-image tuning [57,87], our approach gives the flexibility of training the image encoder for better alignment, while retaining the strength of the pre-trained visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Multimodal alignment with hard negatives Contrastive learning [54] has quickly become the de- facto approach for multimodal alignment, where most prior work focuses on the multimodal InfoNCE [54] objective, given for any batch X = {(xi, ti)}n i=1 of featurized image- text pairs as (for some learnable temperature τ > 0), LNCE(X) = − n � i=1 � �log ex⊤ i ti/τ � j ex⊤ i tj/τ + log ex⊤ i ti/τ � j ex⊤ j ti/τ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' While this approach has enjoyed immense success in multimodal alignment [28, 61], when learning from large- scale noisy datasets, uniform sampling as applied in noise- contrastive estimation can often provide negative sam- ples that are not necessarily discriminative, necessitating very large batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For the problem of contrastive self-supervised learning, Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' [62] propose an importance-sampling approach to reweight negative sam- ples within a batch so that “harder” negatives are up- sampled in proportion to their difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We present a sim- ilar strategy for multimodal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Specifically, for some α ∈ (0, 1], β ≥ 0, we propose the following hard- negative noise contrastive multimodal alignment objective: LHN-NCE(X) = − n � i=1 log � �� ex⊤ i ti/τ α · ex⊤ i ti/τ + � j̸=i ex⊤ i tj/τwi→t xi,tj � �� − n � i=1 log � ��� ex⊤ i ti/τ α · ex⊤ i ti/τ + � j̸=i ex⊤ j ti/τwt→i xj,ti � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Where the weighing functions are given as4: wi→t xi,tj = (n − 1) · eβx⊤ i tj/τ � k̸=i eβx⊤ i tk/τ , wt→i xj,ti = (n − 1) · eβx⊤ j ti/τ � k̸=i eβx⊤ k ti/τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The weights wβ are designed such that difficult negative pairs (with higher similarity) are emphasized, and easier pairs are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Furthermore, α rescales the normaliza- tion with the positive terms to account for the case when false negatives are present within the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The form of weights wβ is an unnormalized von Mises-Fisher distribu- tion [49] with concentration parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Observe that we obtain the original objective when setting α = 1 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' There are several key differences with the original formu- lation of [62] and the HN-NCE objective presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' First, we utilize only cross-modal alignment terms, instead of the unimodal objective presented in [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Next, we em- ploy separate penalties for text-to-image and image-to-text alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, we incorporate a learnable temperature parameter τ to assist in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We discuss our design choices in more detail with additional theoretical and experimental justifications in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Training objective For any batch X = {(xi, ti)n i=1} of n image-text pairs, we minimize the following objective: LHN-NCE(X) + LCE-O(X) + LCE-A(X), where, LCE-O(X) = n � i=1 CROSS-ENTROPY(pobj i , fobj(xi)), and, LCE-A(X) = n � i=1 CROSS-ENTROPY(pattr i , fattr(xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Here, both fobj and fattr are linear classifiers, the vectors pobj, pattr are the top-k predicted objects and attributes from the teacher model (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2), and LHN-NCE is the hard- negative contrastive alignment loss (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 4We normalize by n − 1 as this is the number of negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Experiments Here we evaluate our approach across a broad range of vision and vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We provide exten- sive ablations on 29 tasks over the design choices in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2, and compare with state-of-the-art approaches on popular zero-shot benchmarks in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, we present an alternate approach to do few-shot classification with prompt-based initialization in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Experimental setup Training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1B English caption subset of the LAION-5B dataset [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Prior to training, we fil- ter out sample pairs with NSFW images, toxic words in the text, or images with a watermark probability larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5, following [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' This leaves us with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='98B images, which we refer to throughout the paper as the LAION-2B dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Ad- ditionally, we explore training our models on a collection of Public Multimodal Datasets (PMD) from [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' PMD con- tains training splits of various public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' After down- loading5 the data we are left with 63M (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 70M reported in [68]) image-text pairs due to missing samples and SBU Captions [55] (originally in PMD) going offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Training details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For our model architecture, we closely follow CLIP by Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We utilize Vision Trans- formers (ViT) [17] for images and Text Transformers [73] for captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We experiment with 3 different architectures, denoted as B/32, B/16, and L/14, where 32, 16, and 14 de- note the input image patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' See the supplementary for architecture details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For distillation and fine-tuning experi- ments, we utilize the public SWAG-ViT models [69], pre- trained with weak supervision from hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the Adam [32] optimizer with a decoupled weight decay [47] and a cosine learning rate schedule [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The input image size is 224×224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To accelerate train- ing and save memory, we use mixed-precision training [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' All hyperparameters are presented in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' They are selected by training B/32 on a small scale setup, and reused for all architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For objects and attributes classifiers, we found that scaling the learning rate by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 and weight decay by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='01 gave better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We train our models on 4B, 8B, 16B, and 32B total samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For ViT- L/14, we further train the model at a higher 336px resolu- tion for 400M samples, denoting this model as L/14@336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We evaluate our models on a zero-shot benchmark of 29 tasks: (i) 17 image classifica- tion, (ii) 10 cross-modal retrieval, (iii) 2 visual question an- swering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Dataset details are presented in the supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 5Downloaded following huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='co/datasets/facebook/pmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect of using LAION-2B subset filtered on complexity (C), actions (A), and text-spotting (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CLIP denotes filtering pairs with CLIP score bellow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation performed on ViT-B/32 model architecture trained for 4B processed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' # Filter Size IN COCO Flickr CLIP C A T T2I I2T T2I I2T 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='98B 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 2 ✓ 440M 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 3 ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='71B 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 4 ✓ ✓ 642M 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 5 ✓ ✓ ✓ 438M 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Ablations on zero-shot benchmarks In this section, we ablate our three pretraining contri- butions: dataset filtering, distillation from objects and at- tributes predictions, and, hard negative contrastive objec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Ablations are performed over zero-shot Accuracy@1 on the ImageNet1K [64] (IN) validation set, text-to-image (T2I) and image-to-text (I2T) zero-shot Recall@1 on the COCO [59] and Flickr [58] test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We also report the change in accuracy (%) over 29 zero-shot tasks between our model and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For a fair comparison, we train all ap- proaches presented in this section (including baselines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Effect of dataset filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We apply our filters, as well as filtering based on CLIP [61] alignment score (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='35), and ablate the performance in Table 1 for ViT-B/32 model ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' All models see 4B total samples during training, while the number of unique samples drops after each filter- ing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Complexity filter (C) in row (3) reduces the dataset size by around 270M, while slightly increasing image-text alignment as observed on T2I task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Next, action filter (A) in row (4) reduces the size by more than 1B, while it has a large benefit in aligning complex text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' However, as ex- pected, it hurts performance on object-centric ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, text-spotting (T) filter in row (5) boosts alignment across the board, due to the fact that it removes the need to learn a bimodal visual representation of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We also compare with filtering based on CLIP score in row (2), which was selected such that the dataset size is comparable to ours, and show that it is too strict and removes plenty of useful training pairs, thus hurting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, LAION-CAT, with only 22% of the original dataset size, significantly boosts image-text zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We also observed that gains hold as we train for longer training schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' See the supplementary for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Effect of distillation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To understand the effect of direct distillation from a pre-trained SWAG-ViT visual encoder [69], we investigate two baseline approaches: (1) Embedding distillation (ED) borrows from SimSiam [9] and uses an auxiliary negative cosine similarity loss be- 5 tween the image representation from the student visual en- coder and the pre-trained SWAG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' (2) Distribution distillation (DD) borrows ideas from mo- mentum distillation in ALBEF [40] and computes the cross- modal similarities between the SWAG image representation and the student text representation and uses them as soft- labels for student image representation and text alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The soft-labels are linearly combined with the hard 0 − 1 labels before applying the InfoNCE [54] loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A comparison of our distillation from predicted con- cepts (CD) with the aforementioned distillation approaches is presented in Table 2 (upper section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Note that for a fair comparison, we do not use our hard-negative contrastive loss for these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our distillation approach per- forms the best, even though it has virtually no training over- head as the predicted concepts are pre-computed, while, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', ED is 60% slower with an 8% increase in GPU mem- ory due to the need of running an additional copy of the vi- sion tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' One could pre-compute embeddings for ED and DD as well, but that increases dataset size by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2TB and creates a data loading bottleneck, while our pre-computed predictions take only 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6GB additional storage space when saving the top-10 predictions (see supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We ad- ditionally show that our approach is robust to the number of top-k predictions used, details in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' One could also use an external unimodal image model and fine-tune it on the image-text alignment task instead of using distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We follow [87] and explore three fine- tuning options as baselines: (i) locked-image tuning (LiT) where the image encoder is locked, and only the text en- coder is trained, (ii) fine-tuning (FT) where the image en- coder is trained with a learning rate scaled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='01 com- pared to the text encoder, (iii) fine-tuning with delay (FT- delay) where the image encoder is locked for half of the pre-training epochs following (i), and then fine-tuned for the rest following (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Results of these setups are ablated in Table 2 (lower section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' LiT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' FT is a trade-off be- tween strong performance on image recognition tasks (as measured with ImageNet1K) and better image-text align- ment (as measured by COCO and Flickr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Locking the im- age encoder makes the alignment very hard to achieve, but fine-tuning it hurts its original image recognition power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' On the other hand, we show that our concept distillation is the best of both worlds, it surpasses LiT or FT in 4 out of 5 met- rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Another drawback of FT is that it requires the same architecture in the final setup, while CD can be effortlessly combined with any architecture or training setup, by using stored predictions as metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To conclude, unlike related approaches, our proposed distillation: (i) has almost no cost at training, (ii) is architecture agnostic, (iii) improves both image recognition and complex image-text alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect of using different initialization or dis- tillation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation performed on ViT-B/16 model architecture trained for 16B processed samples on LAION-CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Init: Initialization with random or SWAG-B/16 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' ED: Em- bedding distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' DD: Distribution distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' LiT: Locked image tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' FT: Fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' FT-delay: Locked image tuning for 50% followed by fine-tuning for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CD: Our concept distillation using teacher-predicted objects and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Init Method SWAG IN COCO Flickr (teacher) T2I I2T T2I I2T Random Baseline — 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 ED B/16 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 DD B/16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 CD (ours) B/16 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 CD (ours) H/14 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 SWAG LiT — 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 FT — 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 FT-delay — 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect of using hard negative contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation performed on ViT-B/16 model architecture trained for 16B processed samples on LAION-CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CD: Our concept distilla- tion using SWAG-H/14 predicted objects and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' HN: Our proposed hard negative contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' # Method IN COCO Flickr CD HN T2I I2T T2I I2T 1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 2 ✓ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 3 ✓ ✓ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 Effect of hard negative contrastive training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We present the ablation when using hard negative contrastive objective (HN-NCE) in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Performance on popular benchmarks suggests that using the newly proposed loss is beneficial compared to the vanilla InfoNCE, and that its positive ef- fects are complementary to the gains from the proposed dis- tillation from objects and attributes predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Please see the supplementary for ablations on the effect of the hyper- parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Effect when pre-training on PMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, we analyze our proposed recipes when training visual-language mod- els on a much smaller dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' PMD with 63M training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' All contributions improve the performance over baseline significantly, hence we conclude that using the proposed pipeline is very benefi- cial in low-resource training regimes6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Note that, the PMD dataset contains COCO and Flickr training samples, hence, it is not strictly zero-shot evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For that reason, we do not compare our models trained on PMD dataset with state-of-the-art models in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' However, we believe these strong findings will motivate usage of our approach on smaller and cleaner datasets, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 6PMD is smaller and relatively much cleaner dataset compared to LAION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Hence, we observed that our filtering step is not needed for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect when pre-training on PMD using our ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation performed on ViT-B/32 and ViT-B/16 mod- els trained for 4B processed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CD: Our concept distillation using SWAG-H/14 predicted objects (-O) and attributes (-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' HN: Our proposed hard negative contrastive loss.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='83 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='13 Score Difference (%) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' DiHT-B/16 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CLIP-B/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Both models trained by us on PMD with 63M images and 4B total processed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Zero-shot benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We denote model trained with our proposed concept distillation and hard-negative loss as DiHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To showcase our model’s performance in more de- tail, we report our DiHT-B/16 trained on LAION-CAT with 438M samples vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CLIP-B/16 baseline trained by us on LAION-2B with 2B samples in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Additionally, we report DiHT-B/16 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' CLIP-B/16 baseline, where both models are trained on PMD dataset with 63M samples in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' When trained on LAION-CAT or LAION-2B, re- spectively, DiHT wins on 20 out of 29 benchmark tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Impressively, when trained on PMD, DiHT wins on 28 out of 29 benchmarks tasks, usually with a very large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Comparison with zero-shot state of the art We compare our DiHT models against state-of-the-art dual-encoder models in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Given that all models use different architectures, input image resolutions, train- ing databases, and number of processed samples at training, we outline those details in the table, for easier comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our approach is most similar to CLIP [61] and Open- CLIP [27], and has same training complexity and inference complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We outperform models with same architec- ture by substantial margins, even when our training dataset is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our best models DiHT-L/14 and DiHT- L/14@336 trained at higher 336px resolution for additional 400M samples outperform models with significantly more complexity on popular text-image COCO and Flickr bench- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Compared to ALIGN [28] that has approximately twice the number of parameters compared to our DiHT- L/14 model and is trained on 4x bigger data, we improve the performance substantially for all the retrieval bench- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our model also performs better than FILIP [83] which utilizes token-wise similarity to compute the final alignment, thus noticeably increasing the training speed and memory cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We also outperform Florence [85] on all 4 re- trieval benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Note that Florence [85] utilizes a more recent and powerful Swin-H Vision Transformer architec- ture [45] with convolutional embeddings [76], and a unified contrastive objective [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our proposed contributions are complementary to FILIP [83] and Florence [85], and we believe additional gains can be achieved when combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, LiT [87] and BASIC [57] first pre-train model on an large-scale image annotation dataset with cross-entropy before further training with contrastive loss on an image- text dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Though this strategy results in state-of-the-art performance on ImageNet1K [64] and image classification benchmarks, it has severe downsides on multi-modal tasks such as cross-modal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our ablation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 also confirms this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' On the other hand, our approach does not suffer from such negative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Few-shot linear probing The ideal scenario for leveraging zero-shot recognition models is to warm start the task without training data and then improve the performance (by training a linear probe) via few-shot learning as more and more data is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' How- ever, in practice, few-shot models perform significantly worse than zero-shot models in the low-data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We present an alternate approach to do few-shot classi- fication with prompt-based initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The key idea of our approach is to initialize the classifier with the zero-shot text prompts for each class, but to also ensure that the fi- nal weights do not drift much from the prompt using pro- jected gradient descent (PGD) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' While few-shot models have been initialized with prompt priors in the past with naive L2 penalties for weight to prevent catastrophic for- 7 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Comparison with zero-shot state-of-the-art dual-encoder models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' px: input image size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' #P: model size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' #D: training dataset size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' #S: total samples processed at training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We evaluate CLIP [61] and OpenCLIP [27] using our codebase, other numbers are copied from respective papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Grouped models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', ViT- B/32) share same vision and language architecture as our model, following CLIP [61], others have different architectures and we outline the vision one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' ∗FILIP uses token-wise similarity, which is more expensive than global-token similarity and requires adapting the architecture, hence we put it in “Other”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Method px #P #D #S IN COCO Flickr T2I I2T T2I I2T ViT-B/32 CLIP [61] 224 151M 400M 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8B 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 OpenCLIP [27] 224 151M 400M 12.' metadata={'source': 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+page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 ViT-L/14 CLIP [61] 224 428M 400M 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8B 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 84.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 DiHT 336 428M 438M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4B 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 Other ALIGN [28] EfficientNet-L2 289 820M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8B 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7B 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 FILIP [83]∗ ViT-L/14 224 428M 340M 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2B 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 OpenCLIP [27] ViT-H/14 224 986M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3B 32B 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 Florence [85] CoSwin-H 384 893M 900M 31B 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 LiT [87] ViT-g/14 288 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0B 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6B 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2B 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 — — BASIC [57] CoAtNet-7 224 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1B 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6B 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8B 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 — — — — getting [33], these approaches do not improve performance and the model simply ignores the supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In con- trast, for any target dataset Dtarget = {(xi, yi)n i=1}, where xi = φimage(Ii) denotes the image features from the trained image tower, we solve the following optimization problem, for some δ, δb > 0: min ∥W∥2≤δ,∥b∥2≤δb n � i=1 LCE � yi, x⊤ i (W + W0) + b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Here W0 ∈ Rd×nc denotes the prompt initialization from the text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To optimize the objective, one can use pro- jected gradient descent [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We observe that our approach is able to bridge the gap between zero-shot and 1-shot classi- fication, a common issue in prior linear probe evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 1 2 5 10 25 40 45 50 55 60 65 70 75 80 85 k-shot Accuracy@1 DiHT-B/16 (PGD) DiHT-B/16 (SGD) DiHT-B/16 (0-shot) 1 2 5 10 25 40 45 50 55 60 65 70 75 80 85 k-shot Accuracy@1 DiHT-L/14 (PGD) DiHT-L/14 (SGD) DiHT-L/14 (0-shot) CLIP-L/14 (SGD) SWAG-H/14 (SGD) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' k-shot linear probing performance on ImageNet1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Figure 6 presents the full summary of results on the Ima- geNet1K [64] k-shot classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Hyperparameters δ and δb for our approach, and weight decay for the baseline approach of training linear probes from scratch are found using grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Note that compared to the baseline, our method performs substantially better at very low values of k and maintains the performance continuum from zero-shot to 1-shot, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' At large k values, both approaches perform similarly, since there are sufficient data samples to render the zero-shot initialization ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To fur- ther showcase the strength of our approach, we also com- pare our performance with linear probes trained on powerful SWAG [69] models that are especially suited for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Note that our approach outperforms the much larger SWAG ViT-H/14 model up to 25-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We would like to emphasize that this albeit straightforward approach is one of the first to resolve this discontinuity problem between zero-shot and few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Conclusion and future work In this paper, we demonstrate that with careful dataset filtering and simple but effective modeling changes, it is possible to achieve substantial improvements in zero-shot performance on retrieval and classification tasks through large-scale pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Our CAT filtering approach can be applied generically to any large-scale dataset for improved performance with smaller training schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Moreover, our concept distillation approach presents a compute and storage efficient way of leveraging very large capacity pre- trained image models for multimodal training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, our simple projected gradient approach covers the crucial per- formance gap between zero-shot and few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In future, we would like to extend our approach to multi-modal encoder/decoder [1,10,40,81,84] architectures that although expensive, have better zero-shot performance compared to dual encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We also observe that benefits of our hard-negatives loss are less on noisier LAION dataset compared to PMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' It would be interesting to explore how to make it more effective in these very noisy settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We hope that our improvements and extensive large-scale abla- tions will further advance the vision-language research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 8 References [1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, An- toine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Understanding hard neg- atives in noise contrastive estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In NAACL, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 3 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Method details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 Semantic parser To enable a rich complexity and semantic filtering, we built a fast custom semantic parser that converts a given tex- tual caption to a semantic graph similar to the one in Vi- sual Genome [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In particular, we extract objects, their parts, their attributes, and the actions that they are involved in (see Figure 3 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The parser is built on top of the English language dependency parser from Spacy [26] combined with multiple rules to infer common object rela- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The aim of the parser is high speed with high pre- cision of common object relations such as ‘has attribute‘ and ‘has part‘ and basic ‘action‘ support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Below, we de- scribe the structured relations that we extract from natural language text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We support the following semantic relations: Object ( obj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We extract objects that are supposedly pre- sented in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We consider nouns that are not at- tributes of another noun (not part of a noun phrase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' in birthday cake and baby stroller, the nouns cake and stroller are parsed as objects, and the nouns birthday and baby are considered attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We do not consider proper nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Attribute (has attr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Denotes attributes that characterize an object or another attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For example, dark green, would result in a fact green - has attr - dark, and yellow candles results in candles - has attr - yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Part (has part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Characterizes a visual part of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' cake with 21 yellow candles would result in a part fact cake - has part - candles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Action ( act).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Verbs that do not entail attributes or parts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' forms of be, looks, seems, and have are excluded) are considered actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For actions, we also parse the subject and object arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Subject of an action (act has subj, is act subj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the act has subj and is act subj relation to represent argu- ments (nouns) that are the subject of an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' for the text a person is eating an apple, we add the object-centric and corresponding action-centric symmetric facts: person - is subj act - eating and eating act has subj person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Object of an action (act has obj, is act obj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We also include the relations that specify the object arguments of an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' for the text a person is eating an apple, we add the object-centric and corresponding action-centric symmetric facts: apple - is obj act - eating and eating act has obj apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We recognize the following limitations of ours approach: Semantic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In this work, we focus on object- centric visual and action characteristics and we do not pro- cess spatial relations ( X next to Y) or additional action ar- guments (read a book *in* the library).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Spatial relations and additional arguments of verbs usually involve more complex semantic reasoning and require more robust ap- proaches and task-specific models such as one trained on Semantic Role Labeling which are usually compute-heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We leave these for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Dependency parser errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In the current version of the parser, we also parse potential attributes as actions, which are not likely to be always visual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In the phrase “run- ning person”, running is an action and an attribute, and we parse them as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' However, sometimes the underlying parser would also parse attributes in phrases such as “striped mug” as verbs, where we process the attribute “striped” as both an attribute and an action (without arguments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 Concept distillation The teacher model is built by training linear classifiers - which predict objects and attributes - on top of a frozen SWAG [69] backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' SWAG is trained in a weakly- supervised manner by predicting hashtags from Instagram images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the publicly available weights, and adopt a training procedure that is similar to the one from SWAG for learning the linear classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The procedure for train- ing the object classifier is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' First, we parse the captions to extract nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Next, we canonicalize the nouns via WordNet [51] synsets and remove ones which occur less than 250 times in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The resulting vocabulary con- tains ∼10K unique synsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Finally, we optimize the linear layer’s weights through a cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Each entry in the target distribution of the cross-entropy is either 1/K or 0 depending on whether the corresponding synset is present or not, where K is the number of synsets for that image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We apply inverse square-root resampling of images to upsam- ple the tail classes following [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The target length of the dataset is set to 50 million samples during resampling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We train the linear layer using SGD with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 and weight decay 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The learning rate is set following the lin- ear scaling rule: lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='001· bs 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To speedup training, we use 64 GPUs with batch size of 256 per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The attribute clas- sifiers are build in a similar way, but the WordNet adjective synsets require additional filtering to remove non-visual at- tributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', claustrophobic, experienced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Following [60], we select the attributes based on their sharedness and visu- alness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We rank the attributes based on the aforementioned scores, and keep ∼1200 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 12 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' DiHT architecture hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Model Dim Vision Language layers width heads layers width heads B/32 512 12 768 12 12 512 8 B/16 512 12 768 12 12 512 8 L/14 768 24 1024 16 12 768 12 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' DiHT common hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Shared Learning rate (LR) 1e-3 Warm-up 1% Vocabulary size 49408 Temperature (init, max) ( 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='07, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0) Adam (β1, β2) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='98) Adam ϵ 1e-6 High resolution LR 1e-4 Dataset specific LAION PMD CD learning rate scale 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 CD weight decay scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 HN-NCE α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='999 HN-NCE β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 LAION PMD Model specific L/14 B/16,B/32 B/16,B/32 Batch size 98304 49152 32768 Weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Training details For our model architecture, we closely follow CLIP by Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We utilize Vision Transformers (ViT) [17] for images and Text Transformers [73] for cap- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We experiment with 3 different architectures, denoted as B/32, B/16, and L/14, where 32, 16, and 14 denote the input image patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Other architecture scaling param- eters are in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For distillation and fine-tuning ex- periments, we utilize the public SWAG-ViT models [69], pre-trained with weak supervision from hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the Adam [32] optimizer with a decoupled weight decay [47] and a cosine learning rate schedule [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Input image size is 224×224 pixels, for pre-training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' All hyperparameters are presented in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' They are selected by training on a small scale setup, and reused for other experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For objects and attributes classifiers in concept distillation (CD), we found that scaling the learning rate by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 and weight decay by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='01 gave better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We pre-train the models on 4B, 8B, 16B, or 32B pro- cessed samples, depending on the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For L/14 we train at a higher 336px resolution for additional 400M sam- ples, denoting this models as L/14@336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We trained L/14 for 6 days on 512 A100 GPUs with 16B processed samples for a total of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 × 104 GPU hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' To accelerate training and save memory, we use mixed- precision training [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For L/14 we use grad checkpoint- ing [8] and BFLOAT16 [14,29] format, all the other models are trained using FP16 [50] format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Contrastive loss is com- puted on the local subset of the pairwise similarities [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation details We evaluate our models on a zero-shot benchmark of 24 datasets: (i) 17 image classification: Birdsnap [3], CIFAR10 [36], CIFAR100 [36], Caltech101 [19], Coun- try211 [61], DTD [13], Flowers102 [53], Food101 [4], Ima- geNet1K [64], OxfordPets [56], STL10 [15], SUN397 [77], StanfordCars [34], UCF101 [70], HatefulMemes [31], PascalVOC2007 [18], OpenImages [38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' (ii) 5 cross- modal retrieval (text-to-image T2I, image-to-text I2T): COCO [44], Flickr [58], LN-COCO [59], LN-Flickr [59], Winoground [71];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' (iii) 2 visual question answering: SNLI-VE [78], VQAv2 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Note that, cross-modal re- trieval datasets have 2 tasks (T2I and I2T), so in total we evaluate across 29 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We follow zero-shot CLIP benchmark7 implementation for most of the datasets, and implement the ones that are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For most image classification tasks we com- pute Accuracy@1, except HatefulMemes where we com- pute AUROC because it is binary classification, OpenIm- ages where we compute FlatHit@1 following [75], and PascalVOC2007 where we compute mean average preci- sion (mAP) because it is multi-label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We use the same prompt ensembling method as CLIP [61] to im- prove zero-shot image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For cross-modal re- trieval (T2I and I2T), we compute Recall@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For COCO and Flickr we apply a simple prompt pretext “a photo of {caption}”, for LN-COCO, LN-Flickr, and Winoground no prompt is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We cast visual question answering (VQA) as binary prediction task and compute AP on the cosine similarity between an image and a text (a hypothe- sis or a question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For SNLI-VE, we take a subset which has agreement among annotators, we use “entailement” and “contradiction” as binary classes, and drop the “neutral” class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For VQAv2, we take the subset with yes/no ques- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' No prompt is applied for SNLI-VE and VQAv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Additional ablations Effect of dataset filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 we observe that gains from our proposed complexity, action, and text- spotting (CAT) dataset filtering hold as we train for longer training schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We ran small scale experiments with several complexity filters (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3) and we found that CAT with minimum complexity C1 performed the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Effect of top-k predicted objects and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' In Ta- ble A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4, we show that our concept distillation approach is quite robust to the choice of the number of predicted ob- jects and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' For k = 10 strong accuracy is achieved with a small increase in dataset memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 7github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='com/LAION-AI/CLIP benchmark ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8B ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='Num Samples ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='78 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='79 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='83 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='84 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='86 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='Num Samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='Recall@1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='Flickr (I2T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect of using our LAION-CAT subset filtered on complexity (C), actions (A), and text spotting (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation performed on ViT-B/32 architecture trained for a varying number of processed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Number of examples after filtering with different filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Filter # examples % of full [67] C0 C1 C2 A T 2,121,505,329 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='00 ✓ 1,983,345,180 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='49 ✓ ✓ 1,891,725,045 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='17 ✓ ✓ 1,709,522,548 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='58 ✓ ✓ 1,143,660,096 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='91 ✓ ✓ 691,535,901 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='60 ✓ ✓ ✓ 642,162,957 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='27 ✓ ✓ ✓ 487,493,190 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='98 ✓ ✓ ✓ ✓ 438,358,791 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='66 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect of using different number of top-k predicted objects and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation on ViT-B/16 model architecture trained for 8B processed samples on LAION-CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Memory denotes storage needed to store predicted concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' top-k Memory IN COCO Flickr T2I I2T T2I I2T 5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3GB 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 10 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6GB 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 25 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6GB 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 Effect of α and β on HN-NCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' From intuition, one can see that the term α controls the mass of the positive align- ment term in the loss function, and the term β controls the difficulty of the negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The need for the term α can be attributed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' If there are false negatives within the dataset, dampening the positive alignment term can pre- vent the model from becoming overly discriminative with the true and false positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Hence, we would like to re- duce α as the likelihood of having false positives increases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=', smaller datasets, less noisy training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' The need for β is straightforward: higher β pushes the weighing function to be “sharper”, with more mass on the hardest negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 shows the effect of different values of α and β on LAION-CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating effect of different hyperparameters α and β for the HN-NCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluation on ViT-B/16 model architecture trained for 16B processed samples on LAION-CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' α β IN COCO Flickr T2I I2T T2I I2T 1 0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='25 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='25 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='25 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Evaluating linear probing with the complete training set for ImageNet1K on the ViT-L/14 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Model Optimizer ImageNet-1K Accuracy (%) CLIP-L/14 @ 224px SGD 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='60 DiHT-L/14 @ 224px SGD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='40 DiHT-L/14 @ 224px PGD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='41 CLIP-L/14 @ 336px SGD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='40 DiHT-L/14 @ 336px SGD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='87 DiHT-L/14 @ 336px PGD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='89 Additional results on few-shot probing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We examine the performance of our models on linear probing with the full training set for ImageNet1K [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We compare the perfor- mance of DiHT-L/14 and CLIP-L/14 [61] architectures for both the 224px and 336px input sizes in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We observe that the PGD approach with the DiHT model out- performs prior work, and also find that there is no notable difference in performance between SGD-trained and PGD- trained models, as there is no need for regularization when training with the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We reproduce the reported numbers for CLIP [61] and train our models with a learning rate of 24, no weight decay, and batch size of 96,000 for 160 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 14 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Zero-shot state-of-the-art dual-encoder models comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We evaluate CLIP [61] and OpenCLIP [27] using our codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Method Birdsnap CIFAR10 CIFAR100 Caltech101 Country211 DTD Flowers102 Food101 ImageNet1K OxfordPets STL10 SUN397 StanfordCars UCF101 HatefulMemes PascalVOC OpenImages COCO T2I COCO I2T Flickr T2I Flickr I2T LN-COCO T2I LN-COCO I2T LN-Flickr T2I LN-Flickr I2T Winoground T2I Winoground I2T SNLI-VE VQAv2 ViT-B/32 @ 224 CLIP 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 83.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 OpenCLIP 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 DiHT 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 55.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='4 Additional results on zero-shot benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' We report performance of CLIP [61], OpenCLIP [27], and DiHT on all 29 zero-shot tasks in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Contrastive Alignment with Hard Negatives Convergence guarantees Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Let L⋆(φi, φt) = supq∈Π L(φi, φt, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Then for any measurable φi, φt : X → Sd−1 and τ = O(1) we observe the convergence L(φi, φt, q) → L⋆(φi, φt) as β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' Follows from Proposition 6 of [62] with the loss function L(φi, φt, qβ) defined as follows for any β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' L(φi, φt, qβ) = log � eφi(x)⊤φt(x)/τ eφi(x)⊤φt(x)/τ + Q · Ey∼qβ � eφi(x)⊤φt(y)/τ� � + log � eφi(x)⊤φt(x)/τ eφi(x)⊤φt(x)/τ + Q · Ey∼qβ � eφi(x)⊤φt(y)/τ� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfWgC_/content/2301.02280v1.pdf'} diff --git a/ddAzT4oBgHgl3EQfn_0r/content/tmp_files/2301.01588v1.pdf.txt b/ddAzT4oBgHgl3EQfn_0r/content/tmp_files/2301.01588v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..237e4c7baceb71c094bf8a296cca48c68ecfc61f --- /dev/null +++ b/ddAzT4oBgHgl3EQfn_0r/content/tmp_files/2301.01588v1.pdf.txt @@ -0,0 +1,823 @@ + +Power Spectral Density-Based Resting-State EEG Classification of First-Episode +Psychosis +Sadi Md. Redwan1, Md Palash Uddin2,3, Anwaar Ulhaq4* and Muhammad Imran Sharif5 +1Department of Computer Science and Engineering, University of Rajshahi, Rajshahi 6205, +Bangladesh (e-mail: sadi.redwan@ru.ac.bd) +2Department of Computer Science and Engineering, Hajee Mohammad Danesh Science and +Technology University, Dinajpur 5200, Bangladesh (e-mail: palash_cse@hstu.ac.bd) +3School of Information Technology, Deakin University, Geelong, VIC 3220, Australia +4School of Computing. Mathematics and Engineering, Charles Sturt University, NSW, Australia +(e-mail: aulhaq@csu.edu.au) +5COMSATS University Islamabad, Wah Campus, Punjab 47040, Pakistan (e-mail: +mimraansharif@gmail.com) +*Corresponding author +Abstract +Historically, the analysis of stimulus-dependent time-frequency patterns has been the cornerstone +of most electroencephalography (EEG) studies. The abnormal oscillations in high-frequency +waves associated with psychotic disorders during sensory and cognitive tasks have been studied +many times. However, any significant dissimilarity in the resting-state low-frequency bands is +yet to be established. Spectral analysis of the alpha and delta band waves shows the effectiveness +of stimulus-independent EEG in identifying the abnormal activity patterns of pathological brains. +A generalized model incorporating multiple frequency bands should be more efficient in +associating potential EEG biomarkers with First-Episode Psychosis (FEP), leading to an accurate +diagnosis. We explore multiple machine-learning methods, including random-forest, support +vector machine, and Gaussian Process Classifier (GPC), to demonstrate the practicality of resting- +state Power Spectral Density (PSD) to distinguish patients of FEP from healthy controls. A +comprehensive discussion of our preprocessing methods for PSD analysis and a detailed +comparison of different models are included in this paper. The GPC model outperforms the other +models with a specificity of 95.78% to show that PSD can be used as an effective feature +extraction technique for analyzing and classifying resting-state EEG signals of psychiatric +disorders. +Keywords: First-Episode Psychosis, EEG, PSD, GPC, Machine-Learning +1. Introduction +Psychosis is a symptom commonly associated with an extended array of neurological and +psychiatric disorders, including schizophrenia spectrum (schizophreniform, schizoaffective, and + + +paranoid schizophrenia). The first episode of psychosis in schizophrenia can be hard to +distinguish from other forms of psychosis. An early diagnosis relies heavily on identifying trait +markers of schizophrenia in First-Episode Psychosis (FEP/First-Episode Schizophrenia/FESz) +patients. Electroencephalography (EEG) has been tremendously successful in the time-frequency +analysis of neural activation patterns during different cognitive and behavioral assessments. +Recent resting-state studies show that EEG can also be used to decode intrinsic brain activity in +a task-negative state. Multiple studies involving spectral analysis support the alterations in +resting-state delta/alpha activity in schizophrenia spectrum [1, 2, 36]. Several cortical alpha +networks have been shown to be pathological in FEP patients in a recent +magnetoencephalography (MEG) study [3]. Power Spectral Density (PSD) has been used in +analyzing the alpha band Default Mode Network (DMN) in schizophrenia in another MEG +analysis [4]. This raises the question of whether PSD can also be used for EEG analysis to identify +FEP patients accurately. +In contemporary EEG and MEG studies, delta and alpha powers have been affiliated with +attention and prolonged focus, signifying spontaneous resting-state brain activity. A more +generalized model using multiple robust feature extraction techniques for highly accurate +schizophrenia classification has also been proposed recently [5]. Several studies support the use +of PSD as an effective EEG feature extraction method for machine-learning classification [37, +38]. In another study, researchers used PSD of multiple frequency bands along with fuzzy +entropy and functional connectivity for Generalized Anxiety Disorder (GAD) classification with +97.83 (±0.4)% accuracy [6]. This signifies the potential utility of combining the spectral features +of multiple bands for EEG classification of FEP. The core objective of this work is to combine +the PSD of delta (0.5-4 Hz), theta (4-8 Hz), alpha (8-12 Hz), and sigma (12-16 Hz) bands of +resting-state EEG for the machine learning approaches. +Machine learning models for EEG classification have been popularized with the success of +Linear Discriminant Analysis (LDA), Support Vector Machine (SVM), and neural networks in +multiple EEG paradigms. A random forest classifier has been proposed for the classification and +analysis of mental states using single-channel EEG [7]. SVM has been successfully used in +multiple sclerosis [8] and epilepsy detection [9]. Gaussian Process Classifier (GPC) has also been +proposed for classifying mental states [10] and detecting neonatal seizures [11]. In this work, we +analyze the effectiveness of multiple methods, namely random forest, SVM, and GPC, for +classifying FEP patients and healthy controls based on the PSD of multiple EEG frequency +bands. A medium-sized dataset of 28 controls and 44 patients has been balanced using borderline- +SMOTE [12] for this work. With a very small number of parameters, the computationally +efficient GPC has performed very well, with an accuracy of 95.51 (±1.74)% and a specificity of +95.78 (±3.3)%. The proposed framework sets a baseline for FEP and control classification using +resting-state EEG, and we expect it to be improved upon in the future with more complex neural +network models and multiple feature extraction techniques based on time-frequency analysis. +2. Materials and Methods +2.1 Electroencephalography (EEG) +EEG is a waveform representation of the (electrical) brain signals measured by the fluctuations +of voltage induced by the neuronal ionic activity [13]. The effectiveness of EEG in decoding +neurological and emotional states of the brain is attributed to the high temporal resolution of the + + +signal [14] and our understanding of which frequency or pattern of the signal relates to a +particular task, stimulus, or emotion. Several visual, auditory, and task-based stimuli have been +developed over the years by researchers on account of EEG studies. These studies have +eventually built the foundation of modern EEG-based emotion recognition, seizure detection, +medical diagnosis, and Brain-Computer Interface (BCI) systems. In particular, EEG is currently +established as the primary method for seizure detection [15]. Most publicly available EEG +datasets are focused on diverse neural activation events of healthy and occasionally pathological +brains. That being said, the publication of resting-state EEG studies and datasets has also +increased in the past few years. Major depressive disorder [16], depression [17, 19], cognitive +states [18], and multiple other psychiatric disorders [19] have been studied using resting-state +EEG as of late, and some of them have been published as datasets. In addition to the MEG study +of resting-state cortical alpha networks of FEP/FESz [3], Salisbury et al. also published the +corresponding EEG datasets in 2022 [20, 21]. For our work, we use the Resting Task 1 dataset, +excluding the Resting Task 2 samples of 10 subjects that are also present in the Resting Task 1 +dataset. The subject population consists of 72 subjects (28 controls and 44 patients). The +demographic information of the subjects are presented in Table 1. +Table 1. Demographic information of the subject population. +Group +N (male, female) +Average age (SD) +Ethnicity – White, Black, Asian, +Mixed, Undisclosed +All subjects +72 (46, 26) +21.96 (4.66) +46, 17, 5, 3, 1 +Control +28 (16, 12) +21.33 (3.88) +21, 4, 3, 0, 0 +FEP +44 (30, 14) +22.36 (5.06) +25, 13, 2, 3, 1 +The dataset is obtained from OpenNeuro [22] (accession number: ds003944). It is available under +the Creative Commons License (CC0). The phenotypic information is also included in the +dataset. The cognitive and socio-economic assessments have been conducted using the +MATRICS score and SES score respectively, and the negative effects of FEP are evident in the +patient population. +2.2 Preprocessing +The initial step of every EEG study is preprocessing the data to reduce the effects of several +unwanted artifacts. The EEG signals used in this work are obtained in a 5-minute period using a +low-impedance 10-10 system 60-channel cap. Two additional electrooculogram (EOG) channels +and an electrocardiogram (ECG) channel are also included in the data. EOG channels are +particularly important as they capture the eye-blink artifacts that are also present in the EEG +signals. Much work has been done to establish a correct method for EOG-related artifact removal +based on Independent Component Analysis (ICA) and regression [23]. EEG signals also correlate +with the ECG signal (heartbeat artifacts), which can be removed using ICA [24] and Signal- +Space Projection (SSP). +ICA is a blind source separation (BSS) technique that has revolutionized signal separation from +mixed signals and has been used in numerous EEG and fMRI studies over the years. With the +success of a fast and efficient ICA implementation, fittingly named FastICA [25], it has become +much easier to remove artifacts from EEG signals. In this work, FasstICA is used to remove both +EOG and ECG artifacts separately. We apply temporal band-pass filtering of 0.5-35Hz before +applying ICA to remove low-frequency drifts and high-frequency components that are not + + +needed for this study. We extract 20 Independent Components (ICs) from all the channels to find +out which components correspond to EOG and ECG artifacts and remove those components. The +ICs for a sample subject are shown in Figure 1. + +Figure 1. All 20 ICs for a subject. From a cursory glance, the IC-001 and IC-002 appear to be +related to unwanted artifacts. IC-001 is close to the eyes, which indicates EOG-related +potential, and IC-002 appears to be incoherent compared to the other ICs. +We identify ICs that are related to EOG artifacts by correcting the baseline (0.2 seconds interval) +and averaging across the channels, as shown in Figure 2. + +Figure 2. The ICs identified to be EOG-related IC (-0.5s–0.5s range, 1000 time points). + +ICAcomponents +IC-000 +IC-001 +IC-002 +IC-003 +IC-004 +IC-005 +IC-006 +IC-007 +IC-008 +IC-009 +IC-010 +IC-011 +IC-012 +IC-013 +IC-014 +IC-015 +IC-016 +IC-017 +IC-018 +IC-019EEG (60 channels) +0.000s +0.376 s +0.439 $ +~20 +Ngve=97 +100 +80 +60 +40 +20 +-0.4 +-0.2 +0.0 +0.2 +0.4 +(s) auul +The ECG-related ICs are also identified using the same principle. Correlation is also applied to +identify the heartbeat artifacts, since these artifacts do not affect each EEG electrode with the +same potential due to the temporal properties of the ECG signal. Figure 3 shows the ICs that +correlate to the ECG signal, and Figure 4 shows the effect of EOG and ECG-related artifact +removal. + +Figure 3. IC(s) identified to be ECG-related IC (-0.5s–0.5s range, 1000 time points). + +Figure 4. Effect of artifact removal. The original signals are shown in the left panel, and the +processed signals are in the right panel. 20 out of 60 channels are shown with 0.5-16 Hz +bandpass filtering in a 10s window; EOG artifacts are visible at ~4s timestamp in the left panel. +2.3 Cross-Spectral Density (CSD) +Before proceeding to the feature extraction step, we verify sensor-to-sensor coherence by +calculating the CSD of the channels to justify using spectral features for further analysis. CSD +compares two signals by measuring the spectral power distribution. There are different ways this +can be achieved, for instance, using the Morlet wavelet (CWT/wavelet decomposition) and +Short-Time Fourier Transform (STFT). We decompose every signal into time-frequency +components using the Morlet wavelet to calculate the spectral correlation of the signals. For each +frequency band, eight equidistant values (frequency scales) are specified from lower-bound to + +EEG (60 channels) +-0.006s +0.279 s +0.419 s +-3 +-6 +9 +12 +-15 +Nave=220 +5.0 +2.5 +0.0 +~2.5 +5.0 +7.5 +10.0 +12.5 +0.4 +0.2 +0.0 +0.2 +0.4 +Time (s)EEG001 +UN +EEG001 +MwNn +EEG002 +/ +EEG002 +M +EEG003 +M +EEG003 +MWyyM, +MVw// +EEG004 +EEG004 +EEG005 +EEG005 +EEG006 +/ +Aw/ +EEG006 +EEG007 +IL,wy +w +VJ +M/w~ +EEG007 +wwMw +EEG008 +EEG008 +EEG009 +u_iuy +EEG009 +EEG010 +W +EEG010 +WMyyNr +EEG011 +EEG011 +wwM +EEG012 +EEG012 +EEG013 +EEG013 - +EEG014 +EEG014 +EEG015 +EEG015 +EEG016 +EEG016 +enry +EEG017 +EEG017 +EEG018 +Mnww +My +EEG018 +EEG019 +EEG019 +EEG020 +AMa +EEG020 +2 +6 +10 +2 +4 +D +8 +10 +50 +100 +150 +200 +250 +300 +o +50 +100 +150 +Time (s) +Time (s) +200 +250 +300 +upper-bound. The wavelet power spectrum can be defined as +(𝑊𝑃𝑆)𝑥(𝜏, 𝑠) = │𝑊𝑥(𝜏, 𝑠)│2, + + + + +(1) +where 𝑊𝑥 is the wavelet transform and 𝜏,𝑠 represent the position of the wavelet in the time and +frequency domain, respectively [26]. The Morlet wavelet is given by +𝜓(𝑥) = 𝑒𝑥𝑝 (− 𝑥2 +2 )𝑐𝑜𝑠(5𝑥). + + + + +(2) +By combining the correlation between the power spectrums for each pair of signals, we +eventually get a 60×60 matrix for all 60 channels. The average CSD matrices for a sample subject +across different frequency bands are presented in Figure 5. + +Figure 5. CSD analysis of a single subject. (a) delta, (b) theta, (c) alpha, and (d) sigma CSD +matrices denote coherence across channel signals. +2.4 Power Spectral Density (PSD) +PSD is an effective method to differentiate between noise and features in a signal by making a +spectral representation of the power distribution of its frequency components. We use Thomson’s +multitier spectral estimation [27] method to compute PSD. This method starts by calculating a +periodogram for each of the first 𝐾 ≈ 2𝑁𝑊 Discrete Prolate Spheroidal Sequences +(DPSS/Slepian tapers) [28] and then averaging these periodograms. Figure 6 shows the power +spectra of a sample subject’s preprocessed EEG data in 𝜇𝑉2/Hz (decibels). + +Cross-spectraldensity +Cross-spectral density +0.5-4.0Hz. +1e-10 +4.0-8.0 Hz. +1e-11 +1.4 +1.2 +1.2 +1.0 +1.0 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +(a) +(b) +Cross-spectraldensity +Cross-spectral density +8.0-12.0Hz. +1e-12 +12.0-16.0Hz. +1e-12 +6 +1.2 +5 +1.0 +4 +0.8 +3 +0.6 +2 +0.4 +1 +0.2 +(c) +(d) +0.0 + + +Figure 6. Power spectral representation of EEG data. Each frequency band shows the +characteristic PSD of the signal. +We divide the data into 30s segments and compute four PSD bands for each subject. The four +bands are then combined for the classification step. +2.5 Random Forest +Random forest is a tree-based ensemble learning technique [29] that has been used many times +in different classification tasks. The core idea of a random forest classifier is to combine multiple +decision trees using an ensemble (bagging) mechanism. The prediction of the random forest is +given by the averaged prediction of the decision trees combined with the extremely-randomized +method [30]. A random forest of 200 decision trees with a maximum depth of 30 per tree is used +in this work to classify PSD feature vectors. Figure 7 presents a simple diagram of the random +forest classifier. + +EEG +50 +45 +() ZH/zrl +40 +5 +30 +25 +20 +10 +15 +20 +25 +30 +Frequency (Hz)Delta (0.5-4 Hz) +Theta (4-8 Hz) +60616.279 +8085.297 +μV-/Hz +μV-/Hz +2448.479 +362.791 +Alpha(8-12Hz) +Sigma (12-16 Hz) +3643.581 +5470.014 +μV-/Hz +μV-/Hz +186.861 +134.537 + +Figure 7. Random forest classifier architecture for binary classification. +2.6 Gaussian Process Classifier (GPC) +The GPC for binary classification is based on Laplace approximation [31]. With the joint +probability 𝑝(𝑦)𝑝(𝑥|𝑦) derived from Bayes’ theorem, where 𝑦 denotes the class label, the +marginal likelihood 𝑝(𝑦|𝑋) is given by +𝑝(𝑦|𝑋) = ∫ 𝑝(𝑦|𝑓)𝑝(𝑓|𝑋)𝑑𝑓 = ∫ 𝑒𝑥𝑝(𝛹(𝑓))𝑑𝑓. + + +(3) +Using a Taylor expansion of 𝛹(𝑓) the approximation 𝑞(𝑦|𝑋) to the marginal likelihood is derived +as follows. +𝑝(𝑦|𝑋) ≃ 𝑞(𝑦|𝑋) = 𝑒𝑥𝑝 (𝛹(𝑓̂)) ∫ 𝑒𝑥𝑝 (− +1 +2 (𝑓 − 𝑓̂) +𝑇𝐴(𝑓 − 𝑓̂)) 𝑑𝑓. +(4) +An approximation to the log marginal likelihood is derived by analyzing this Gaussian integral. +𝑙𝑜𝑔𝑞(𝑦|𝑋, 𝜃) = − +1 +2 𝑓̂𝑇𝐾−1𝑓̂ + 𝑙𝑜𝑔𝑝(𝑦|𝑓̂) − +1 +2 𝑙𝑜𝑔│𝐵│, + + +(5) +where +│𝐵│ = │𝐾│. │𝐾−1 + 𝑊│ = │𝐼𝑛 + 𝑊 +1 +2𝐾𝑊 +1 +2│, + + + +(6) +and 𝜃 is a vector of hyperparameters of the covariance function. +We use a stationary covariance function, Radial Basis Function (RBF), as the Gaussian process +kernel. With 𝑟 = ‖𝑥 − 𝑥𝑖‖ and a specified shape parameter 𝜀, the Gaussian RBF is given as +follows while the schematic working procedure of GPC is illustrated in Figure 8. +𝜑(𝑟) = 𝑒𝑥𝑝(−(𝜀𝑟)2) + + + + + + +(7) + +Dataset +Tree-1 +Tree-2 +Tree-3 +Final +Prediction + +Figure 8. GPC architecture for binary classification. +2.7 Support Vector Machine (SVM) +Support vector machines (SVMs) are widely used for classification because they build a linear +decision surface from a very large feature space to which input vectors are mapped non-linearly +[32]. Based on the properties of the optimal hyperplane (feature map), the SVM algorithm can +be classified into linearly separable, linearly inseparable, and non-linearly separable. For non- +linear feature mapping, a kernel function is used to map the inputs implicitly. Similar to the GPC, +we use the Gaussian RBF as the kernel function for our SVM model. For Gaussian RBF, 𝜑 the +kernel function can be written as +𝐾(𝑥𝑖, 𝑥𝑗) = 𝜑(𝑥𝑖). 𝜑(𝑥𝑗). + + + + + + +(8) +Then the vector to the hyperplane (weight) is given by +𝑤 = ∑ α𝑖𝑦𝑖φ(𝑥𝑖) +𝑖 + + + + + + + + +(9) +The SVM classifier minimizes the following expression to separate the input feature vectors with +the parameter 𝜆 > 0, which denotes the tradeoff between the size and flexibility of the margin +for classification while the basic architecture for non-linear SVM is shown in Figure 9. +[ +1 +𝑛 ∑ +𝑚𝑎𝑥(0,1 − 𝑦𝑖(𝑤𝑇𝑥𝑖 − 𝑏)) +𝑛 +𝑖=1 +] + λ|𝑤|2 + + + +(10) + +Figure 9. Non-linear SVM with Gaussian RBF kernel for binary classification. + +y1 +y2 +y3 +yn +Prediction +Sigmoid +f3 +Kernel +X1 +X2 +X3 +Xn +InputsWeights +Input Vector (X) +Kernel K(Xi, xj) +Output (y) +3. Result and Discussion +The experiments were done using MATLAB R2022b and Python 3.10 in Microsoft Windows 11 +(22H2) platform on an AMD Ryzen 7 3750H computer. The performance of each model is +evaluated using 5-fold cross-validation. The final confusion matrix for each model is derived by +taking the average of all confusion matrices, as shown in Figure 10. + +Figure 10. Average confusion matrices for test data. +We use precision, recall, and F1-score to evaluate the classification accuracy for each class. The +mathematical expressions for precision, recall, and F1-score are as follows. +𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = +𝑇𝑃 +𝑇𝑃+𝐹𝑃, + + + + + +(11) +𝑅𝑒𝑐𝑎𝑙𝑙 = +𝑇𝑃 +𝑇𝑃+𝐹𝑁, + + + + + + +(12) +𝐹1 = +2×𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛×𝑅𝑒𝑐𝑎𝑙𝑙 +𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛+𝑅𝑒𝑐𝑎𝑙𝑙 , + + + + + +(13) +where 𝑇𝑃, 𝐹𝑃, and 𝐹𝑁 denote true-positive, false-positive, and false-negative predictions +respectively. Specificity or true negative rate is defined as the recall of the negative class +(control). The accuracy score, precision, recall, and F1 scores for the random forest, GPC, and +SVM models are discussed in Table 2, Table 3, and Table 4, respectively. +Table 2. Classification report for the random forest model. +Group +Precision (SD) +Recall (SD) +F1-score (SD) +Overall +Accuracy (SD) +Control +89.2 (±4.8) +83.93 (±1.9) +86.34 (±2.3) +85.84 (±2.72) +FEP +82.89 (±1.9) +89.2 (±4.9) +85.27 (±3.2) +Table 3. Classification report for the GPC model. +Group +Precision (SD) +Recall (SD) +F1-score (SD) +Overall +Accuracy (SD) +Control +95.93 (±3.5) +95.78 (±3.3) +95.72 (±1.7) +95.51 (±1.74) +FEP +95.56 (±3.5) +95.3 (±3.1) +95.26 (±1.8) + +11 +1 - +0 +39.6 +7.6 +0 +45.2 +2.0 +0. +39.2 +7.2 +1 +5.0 +36.8 +1 - +2.0 +39.8 +1 +8.6 +34.0 +Random forest +GPC +SVM +Table 4. Classification report for the SVM model. +Group +Precision (SD) +Recall (SD) +F1-score (SD) +Overall +Accuracy (SD) +Control +82.49 (±3.6) +84.69 (±4.2) +83.45 (±2.2) +82.25 (±2.18) +FEP +82.47 (±3.4) +79.45 (±5.3) +80.75 (2.5) +With an accuracy of 95.51 (±1.74)% and specificity of 95.78 (±3.3)%, the GPC model has +outperformed the other models (↑9.67% accuracy over random forest and ↑13.26% accuracy over +SVM) and thus, decided as the best model for PSD-based classification of FEP vs. control. The +proposed GPC model has a comparatively small number of parameters and can be considered a +‘shallow’ learning model. The high accuracy of GPC can be attributed to selecting a suitable +covariance function for the input features. Other RBF kernels should also be considered for +comparison. Deep recurrent neural network (RNN) models trained with time-frequency features, +much like the recently proposed models for epilepsy classification, age prediction, and +concussion classification [33, 34, 35], can hypothetically outperform this model. Another aspect +that requires further analysis is the method for computing PSD. Future studies should also +consider Welch's method for computing PSD to compare with the results of the DPSS method. +Combining the CSD features with the PSD features can also provide insight into which electrode +signals have the most significant impact on classification. This work can also be extended further +for a spectrum-wide analysis of the schizophrenia spectrum. +4. Conclusion +In this study, we have evaluated the use of machine learning methods for the classification of +patients with first-episode psychosis (FEP) and healthy controls based on the Power Spectral +Density (PSD) of resting-state EEG. We have reviewed various feature engineering techniques +and machine learning models to demonstrate that FEP patients can be accurately detected +utilizing resting-state EEG. In addition, we have demonstrated that low-to-medium frequency +(delta-to-sigma band) waves are pathological in FEP patients and can differentiate patients from +healthy persons with the same degree of accuracy as task/event-related high-frequency waves. +PSD is shown to be a reliable characteristic for the effective classification of FEP using machine +learning. We conclude that resting-state EEG studies can lead to an accurate diagnosis of +FEP/FESz and other psychiatric disorders and should be regarded as equally essential as +stimulus-based EEG studies. +Data Availability +The +denoised +and +preprocessed +data +used +in +this +work +is +available +at +https://zenodo.org/record/7315010 while the original EEG: First Episode Psychosis vs. Control +Resting Task 1 dataset is available at doi:10.18112/openneuro.ds003944.v1.0.1. +References +1. Dvey-Aharon, Z., Fogelson, N., Peled, A., and Intrator, N. (2015). Schizophrenia detection and +classification by advanced analysis of EEG recordings using a single electrode approach. PLoS One 10 (4): +e0123033. https://doi.org/10.1371/journal.pone.0123033. +2. Howells, F.M., Temmingh, H.S., Hsieh, J.H. et al. (2018). Electroencephalographic delta/ alpha frequency +activity differentiates psychotic disorders: a study of schizophrenia, bipolar disorder, and +methamphetamine-induced psychotic disorder. 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Spectral +Power Density analysis of the resting-state as a marker of the central effects of opioid use in fibromyalgia. +Scientific reports, 11(1), pp.1-13. + + diff --git a/ddAzT4oBgHgl3EQfn_0r/content/tmp_files/load_file.txt b/ddAzT4oBgHgl3EQfn_0r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ebf6d514ffbc96be29cd8c244f6ae3ccea79c640 --- /dev/null +++ b/ddAzT4oBgHgl3EQfn_0r/content/tmp_files/load_file.txt @@ -0,0 +1,709 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf,len=708 +page_content='Power Spectral Density-Based Resting-State EEG Classification of First-Episode Psychosis Sadi Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Redwan1, Md Palash Uddin2,3, Anwaar Ulhaq4* and Muhammad Imran Sharif5 1Department of Computer Science and Engineering, University of Rajshahi, Rajshahi 6205, Bangladesh (e-mail: sadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='redwan@ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='bd) 2Department of Computer Science and Engineering, Hajee Mohammad Danesh Science and Technology University, Dinajpur 5200, Bangladesh (e-mail: palash_cse@hstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='bd) 3School of Information Technology, Deakin University, Geelong, VIC 3220, Australia 4School of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Mathematics and Engineering, Charles Sturt University, NSW, Australia (e-mail: aulhaq@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='au) 5COMSATS University Islamabad, Wah Campus, Punjab 47040, Pakistan (e-mail: mimraansharif@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='com) *Corresponding author Abstract Historically, the analysis of stimulus-dependent time-frequency patterns has been the cornerstone of most electroencephalography (EEG) studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The abnormal oscillations in high-frequency waves associated with psychotic disorders during sensory and cognitive tasks have been studied many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' However, any significant dissimilarity in the resting-state low-frequency bands is yet to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Spectral analysis of the alpha and delta band waves shows the effectiveness of stimulus-independent EEG in identifying the abnormal activity patterns of pathological brains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' A generalized model incorporating multiple frequency bands should be more efficient in associating potential EEG biomarkers with First-Episode Psychosis (FEP), leading to an accurate diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We explore multiple machine-learning methods, including random-forest, support vector machine, and Gaussian Process Classifier (GPC), to demonstrate the practicality of resting- state Power Spectral Density (PSD) to distinguish patients of FEP from healthy controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' A comprehensive discussion of our preprocessing methods for PSD analysis and a detailed comparison of different models are included in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The GPC model outperforms the other models with a specificity of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='78% to show that PSD can be used as an effective feature extraction technique for analyzing and classifying resting-state EEG signals of psychiatric disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Keywords: First-Episode Psychosis, EEG, PSD, GPC, Machine-Learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Introduction Psychosis is a symptom commonly associated with an extended array of neurological and psychiatric disorders, including schizophrenia spectrum (schizophreniform, schizoaffective, and paranoid schizophrenia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The first episode of psychosis in schizophrenia can be hard to distinguish from other forms of psychosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' An early diagnosis relies heavily on identifying trait markers of schizophrenia in First-Episode Psychosis (FEP/First-Episode Schizophrenia/FESz) patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Electroencephalography (EEG) has been tremendously successful in the time-frequency analysis of neural activation patterns during different cognitive and behavioral assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Recent resting-state studies show that EEG can also be used to decode intrinsic brain activity in a task-negative state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Multiple studies involving spectral analysis support the alterations in resting-state delta/alpha activity in schizophrenia spectrum [1, 2, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Several cortical alpha networks have been shown to be pathological in FEP patients in a recent magnetoencephalography (MEG) study [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Power Spectral Density (PSD) has been used in analyzing the alpha band Default Mode Network (DMN) in schizophrenia in another MEG analysis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' This raises the question of whether PSD can also be used for EEG analysis to identify FEP patients accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In contemporary EEG and MEG studies, delta and alpha powers have been affiliated with attention and prolonged focus, signifying spontaneous resting-state brain activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' A more generalized model using multiple robust feature extraction techniques for highly accurate schizophrenia classification has also been proposed recently [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Several studies support the use of PSD as an effective EEG feature extraction method for machine-learning classification [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In another study, researchers used PSD of multiple frequency bands along with fuzzy entropy and functional connectivity for Generalized Anxiety Disorder (GAD) classification with 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='83 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4)% accuracy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' This signifies the potential utility of combining the spectral features of multiple bands for EEG classification of FEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The core objective of this work is to combine the PSD of delta (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5-4 Hz), theta (4-8 Hz), alpha (8-12 Hz), and sigma (12-16 Hz) bands of resting-state EEG for the machine learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Machine learning models for EEG classification have been popularized with the success of Linear Discriminant Analysis (LDA), Support Vector Machine (SVM), and neural networks in multiple EEG paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' A random forest classifier has been proposed for the classification and analysis of mental states using single-channel EEG [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' SVM has been successfully used in multiple sclerosis [8] and epilepsy detection [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Gaussian Process Classifier (GPC) has also been proposed for classifying mental states [10] and detecting neonatal seizures [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In this work, we analyze the effectiveness of multiple methods, namely random forest, SVM, and GPC, for classifying FEP patients and healthy controls based on the PSD of multiple EEG frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' A medium-sized dataset of 28 controls and 44 patients has been balanced using borderline- SMOTE [12] for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' With a very small number of parameters, the computationally efficient GPC has performed very well, with an accuracy of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='51 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='74)% and a specificity of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='78 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The proposed framework sets a baseline for FEP and control classification using resting-state EEG, and we expect it to be improved upon in the future with more complex neural network models and multiple feature extraction techniques based on time-frequency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='1 Electroencephalography (EEG) EEG is a waveform representation of the (electrical) brain signals measured by the fluctuations of voltage induced by the neuronal ionic activity [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The effectiveness of EEG in decoding neurological and emotional states of the brain is attributed to the high temporal resolution of the signal [14] and our understanding of which frequency or pattern of the signal relates to a particular task, stimulus, or emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Several visual, auditory, and task-based stimuli have been developed over the years by researchers on account of EEG studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' These studies have eventually built the foundation of modern EEG-based emotion recognition, seizure detection, medical diagnosis, and Brain-Computer Interface (BCI) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In particular, EEG is currently established as the primary method for seizure detection [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Most publicly available EEG datasets are focused on diverse neural activation events of healthy and occasionally pathological brains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' That being said, the publication of resting-state EEG studies and datasets has also increased in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Major depressive disorder [16], depression [17, 19], cognitive states [18], and multiple other psychiatric disorders [19] have been studied using resting-state EEG as of late, and some of them have been published as datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In addition to the MEG study of resting-state cortical alpha networks of FEP/FESz [3], Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' also published the corresponding EEG datasets in 2022 [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' For our work, we use the Resting Task 1 dataset, excluding the Resting Task 2 samples of 10 subjects that are also present in the Resting Task 1 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The subject population consists of 72 subjects (28 controls and 44 patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The demographic information of the subjects are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Demographic information of the subject population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Group N (male, female) Average age (SD) Ethnicity – White, Black, Asian, Mixed, Undisclosed All subjects 72 (46, 26) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='96 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='66) 46, 17, 5, 3, 1 Control 28 (16, 12) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='33 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='88) 21, 4, 3, 0, 0 FEP 44 (30, 14) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='36 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='06) 25, 13, 2, 3, 1 The dataset is obtained from OpenNeuro [22] (accession number: ds003944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' It is available under the Creative Commons License (CC0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The phenotypic information is also included in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The cognitive and socio-economic assessments have been conducted using the MATRICS score and SES score respectively, and the negative effects of FEP are evident in the patient population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 Preprocessing The initial step of every EEG study is preprocessing the data to reduce the effects of several unwanted artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The EEG signals used in this work are obtained in a 5-minute period using a low-impedance 10-10 system 60-channel cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Two additional electrooculogram (EOG) channels and an electrocardiogram (ECG) channel are also included in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' EOG channels are particularly important as they capture the eye-blink artifacts that are also present in the EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Much work has been done to establish a correct method for EOG-related artifact removal based on Independent Component Analysis (ICA) and regression [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' EEG signals also correlate with the ECG signal (heartbeat artifacts), which can be removed using ICA [24] and Signal- Space Projection (SSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' ICA is a blind source separation (BSS) technique that has revolutionized signal separation from mixed signals and has been used in numerous EEG and fMRI studies over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' With the success of a fast and efficient ICA implementation, fittingly named FastICA [25], it has become much easier to remove artifacts from EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In this work, FasstICA is used to remove both EOG and ECG artifacts separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We apply temporal band-pass filtering of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5-35Hz before applying ICA to remove low-frequency drifts and high-frequency components that are not needed for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We extract 20 Independent Components (ICs) from all the channels to find out which components correspond to EOG and ECG artifacts and remove those components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The ICs for a sample subject are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' All 20 ICs for a subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' From a cursory glance, the IC-001 and IC-002 appear to be related to unwanted artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' IC-001 is close to the eyes, which indicates EOG-related potential, and IC-002 appears to be incoherent compared to the other ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We identify ICs that are related to EOG artifacts by correcting the baseline (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 seconds interval) and averaging across the channels, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The ICs identified to be EOG-related IC (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5s–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5s range, 1000 time points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' ICAcomponents IC-000 IC-001 IC-002 IC-003 IC-004 IC-005 IC-006 IC-007 IC-008 IC-009 IC-010 IC-011 IC-012 IC-013 IC-014 IC-015 IC-016 IC-017 IC-018 IC-019EEG (60 channels) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='000s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='376 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='439 $ ~20 Ngve=97 100 80 60 40 20 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 (s) auul The ECG-related ICs are also identified using the same principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Correlation is also applied to identify the heartbeat artifacts, since these artifacts do not affect each EEG electrode with the same potential due to the temporal properties of the ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 3 shows the ICs that correlate to the ECG signal, and Figure 4 shows the effect of EOG and ECG-related artifact removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' IC(s) identified to be ECG-related IC (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5s–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5s range, 1000 time points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Effect of artifact removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The original signals are shown in the left panel, and the processed signals are in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 20 out of 60 channels are shown with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5-16 Hz bandpass filtering in a 10s window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' EOG artifacts are visible at ~4s timestamp in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3 Cross-Spectral Density (CSD) Before proceeding to the feature extraction step, we verify sensor-to-sensor coherence by calculating the CSD of the channels to justify using spectral features for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' CSD compares two signals by measuring the spectral power distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' There are different ways this can be achieved, for instance, using the Morlet wavelet (CWT/wavelet decomposition) and Short-Time Fourier Transform (STFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We decompose every signal into time-frequency components using the Morlet wavelet to calculate the spectral correlation of the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' For each frequency band, eight equidistant values (frequency scales) are specified from lower-bound to EEG (60 channels) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='006s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='279 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='419 s -3 -6 9 12 -15 Nave=220 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 Time (s)EEG001 UN EEG001 MwNn EEG002 / EEG002 M EEG003 M EEG003 MWyyM, MVw// EEG004 EEG004 EEG005 EEG005 EEG006 / Aw/ EEG006 EEG007 IL,wy w VJ M/w~ EEG007 wwMw EEG008 EEG008 EEG009 u_iuy EEG009 EEG010 W EEG010 WMyyNr EEG011 EEG011 wwM EEG012 EEG012 EEG013 EEG013 - EEG014 EEG014 EEG015 EEG015 EEG016 EEG016 enry EEG017 EEG017 EEG018 Mnww My EEG018 EEG019 EEG019 EEG020 AMa EEG020 2 6 10 2 4 D 8 10 50 100 150 200 250 300 o 50 100 150 Time (s) Time (s) 200 250 300 upper-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The wavelet power spectrum can be defined as (𝑊𝑃𝑆)𝑥(𝜏, 𝑠) = │𝑊𝑥(𝜏, 𝑠)│2, (1) where 𝑊𝑥 is the wavelet transform and 𝜏,𝑠 represent the position of the wavelet in the time and frequency domain, respectively [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The Morlet wavelet is given by 𝜓(𝑥) = 𝑒𝑥𝑝 (− 𝑥2 2 )𝑐𝑜𝑠(5𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' (2) By combining the correlation between the power spectrums for each pair of signals, we eventually get a 60×60 matrix for all 60 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The average CSD matrices for a sample subject across different frequency bands are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' CSD analysis of a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' (a) delta, (b) theta, (c) alpha, and (d) sigma CSD matrices denote coherence across channel signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 Power Spectral Density (PSD) PSD is an effective method to differentiate between noise and features in a signal by making a spectral representation of the power distribution of its frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We use Thomson’s multitier spectral estimation [27] method to compute PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' This method starts by calculating a periodogram for each of the first 𝐾 ≈ 2𝑁𝑊 Discrete Prolate Spheroidal Sequences (DPSS/Slepian tapers) [28] and then averaging these periodograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 6 shows the power spectra of a sample subject’s preprocessed EEG data in 𝜇𝑉2/Hz (decibels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Cross spectraldensity Cross spectral density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 1e 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 1e 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 (a) (b) Cross spectraldensity Cross spectral density 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 1e 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 1e 12 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Power spectral representation of EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Each frequency band shows the characteristic PSD of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We divide the data into 30s segments and compute four PSD bands for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The four bands are then combined for the classification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 Random Forest Random forest is a tree-based ensemble learning technique [29] that has been used many times in different classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The core idea of a random forest classifier is to combine multiple decision trees using an ensemble (bagging) mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The prediction of the random forest is given by the averaged prediction of the decision trees combined with the extremely-randomized method [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' A random forest of 200 decision trees with a maximum depth of 30 per tree is used in this work to classify PSD feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 7 presents a simple diagram of the random forest classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' EEG 50 45 () ZH/zrl 40 5 30 25 20 10 15 20 25 30 Frequency (Hz)Delta (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5 4 Hz) Theta (4 8 Hz) 60616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='279 8085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='297 μV /Hz μV /Hz 2448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='479 362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='791 Alpha(8 12Hz) Sigma (12 16 Hz) 3643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='581 5470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='014 μV /Hz μV /Hz 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='861 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='537 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Random forest classifier architecture for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 Gaussian Process Classifier (GPC) The GPC for binary classification is based on Laplace approximation [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' With the joint probability 𝑝(𝑦)𝑝(𝑥|𝑦) derived from Bayes’ theorem, where 𝑦 denotes the class label, the marginal likelihood 𝑝(𝑦|𝑋) is given by 𝑝(𝑦|𝑋) = ∫ 𝑝(𝑦|𝑓)𝑝(𝑓|𝑋)𝑑𝑓 = ∫ 𝑒𝑥𝑝(𝛹(𝑓))𝑑𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' (3) Using a Taylor expansion of 𝛹(𝑓) the approximation 𝑞(𝑦|𝑋) to the marginal likelihood is derived as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 𝑝(𝑦|𝑋) ≃ 𝑞(𝑦|𝑋) = 𝑒𝑥𝑝 (𝛹(𝑓̂)) ∫ 𝑒𝑥𝑝 (− 1 2 (𝑓 − 𝑓̂) 𝑇𝐴(𝑓 − 𝑓̂)) 𝑑𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' (4) An approximation to the log marginal likelihood is derived by analyzing this Gaussian integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 𝑙𝑜𝑔𝑞(𝑦|𝑋, 𝜃) = − 1 2 𝑓̂𝑇𝐾−1𝑓̂ + 𝑙𝑜𝑔𝑝(𝑦|𝑓̂) − 1 2 𝑙𝑜𝑔│𝐵│, (5) where │𝐵│ = │𝐾│.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' │𝐾−1 + 𝑊│ = │𝐼𝑛 + 𝑊 1 2𝐾𝑊 1 2│, (6) and 𝜃 is a vector of hyperparameters of the covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We use a stationary covariance function, Radial Basis Function (RBF), as the Gaussian process kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' With 𝑟 = ‖𝑥 − 𝑥𝑖‖ and a specified shape parameter 𝜀, the Gaussian RBF is given as follows while the schematic working procedure of GPC is illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 𝜑(𝑟) = 𝑒𝑥𝑝(−(𝜀𝑟)2) (7) Dataset Tree 1 Tree 2 Tree 3 Final Prediction Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' GPC architecture for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='7 Support Vector Machine (SVM) Support vector machines (SVMs) are widely used for classification because they build a linear decision surface from a very large feature space to which input vectors are mapped non-linearly [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Based on the properties of the optimal hyperplane (feature map), the SVM algorithm can be classified into linearly separable, linearly inseparable, and non-linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' For non- linear feature mapping, a kernel function is used to map the inputs implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Similar to the GPC, we use the Gaussian RBF as the kernel function for our SVM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' For Gaussian RBF, 𝜑 the kernel function can be written as 𝐾(𝑥𝑖, 𝑥𝑗) = 𝜑(𝑥𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 𝜑(𝑥𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' (8) Then the vector to the hyperplane (weight) is given by 𝑤 = ∑ α𝑖𝑦𝑖φ(𝑥𝑖) 𝑖 (9) The SVM classifier minimizes the following expression to separate the input feature vectors with the parameter 𝜆 > 0, which denotes the tradeoff between the size and flexibility of the margin for classification while the basic architecture for non-linear SVM is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' [ 1 𝑛 ∑ 𝑚𝑎𝑥(0,1 − 𝑦𝑖(𝑤𝑇𝑥𝑖 − 𝑏)) 𝑛 𝑖=1 ] + λ|𝑤|2 (10) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Non-linear SVM with Gaussian RBF kernel for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' y1 y2 y3 yn Prediction Sigmoid f3 Kernel X1 X2 X3 Xn InputsWeights Input Vector (X) Kernel K(Xi, xj) Output (y) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Result and Discussion The experiments were done using MATLAB R2022b and Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='10 in Microsoft Windows 11 (22H2) platform on an AMD Ryzen 7 3750H computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The performance of each model is evaluated using 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The final confusion matrix for each model is derived by taking the average of all confusion matrices, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Average confusion matrices for test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We use precision, recall, and F1-score to evaluate the classification accuracy for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The mathematical expressions for precision, recall, and F1-score are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = 𝑇𝑃 𝑇𝑃+𝐹𝑃, (11) 𝑅𝑒𝑐𝑎𝑙𝑙 = 𝑇𝑃 𝑇𝑃+𝐹𝑁, (12) 𝐹1 = 2×𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛×𝑅𝑒𝑐𝑎𝑙𝑙 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛+𝑅𝑒𝑐𝑎𝑙𝑙 , (13) where 𝑇𝑃, 𝐹𝑃, and 𝐹𝑁 denote true-positive, false-positive, and false-negative predictions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Specificity or true negative rate is defined as the recall of the negative class (control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The accuracy score, precision, recall, and F1 scores for the random forest, GPC, and SVM models are discussed in Table 2, Table 3, and Table 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Classification report for the random forest model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Group Precision (SD) Recall (SD) F1-score (SD) Overall Accuracy (SD) Control 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='93 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='9) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='34 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='84 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='72) FEP 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='89 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='9) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='9) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='27 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Classification report for the GPC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Group Precision (SD) Recall (SD) F1-score (SD) Overall Accuracy (SD) Control 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='93 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='78 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='72 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='7) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='51 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='74) FEP 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='56 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='1) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='26 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8) 11 1 - 0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8 1 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='8 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='0 Random forest GPC SVM Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Classification report for the SVM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Group Precision (SD) Recall (SD) F1-score (SD) Overall Accuracy (SD) Control 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='49 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='6) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='69 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='45 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='2) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='25 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='18) FEP 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='47 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='4) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='45 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='75 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='5) With an accuracy of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='51 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='74)% and specificity of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='78 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='3)%, the GPC model has outperformed the other models (↑9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='67% accuracy over random forest and ↑13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='26% accuracy over SVM) and thus, decided as the best model for PSD-based classification of FEP vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The proposed GPC model has a comparatively small number of parameters and can be considered a ‘shallow’ learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The high accuracy of GPC can be attributed to selecting a suitable covariance function for the input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Other RBF kernels should also be considered for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Deep recurrent neural network (RNN) models trained with time-frequency features, much like the recently proposed models for epilepsy classification, age prediction, and concussion classification [33, 34, 35], can hypothetically outperform this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Another aspect that requires further analysis is the method for computing PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=" Future studies should also consider Welch's method for computing PSD to compare with the results of the DPSS method." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Combining the CSD features with the PSD features can also provide insight into which electrode signals have the most significant impact on classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' This work can also be extended further for a spectrum-wide analysis of the schizophrenia spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Conclusion In this study, we have evaluated the use of machine learning methods for the classification of patients with first-episode psychosis (FEP) and healthy controls based on the Power Spectral Density (PSD) of resting-state EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We have reviewed various feature engineering techniques and machine learning models to demonstrate that FEP patients can be accurately detected utilizing resting-state EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' In addition, we have demonstrated that low-to-medium frequency (delta-to-sigma band) waves are pathological in FEP patients and can differentiate patients from healthy persons with the same degree of accuracy as task/event-related high-frequency waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' PSD is shown to be a reliable characteristic for the effective classification of FEP using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' We conclude that resting-state EEG studies can lead to an accurate diagnosis of FEP/FESz and other psychiatric disorders and should be regarded as equally essential as stimulus-based EEG studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Data Availability The denoised and preprocessed data used in this work is available at https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='org/record/7315010 while the original EEG: First Episode Psychosis vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Control Resting Task 1 dataset is available at doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='18112/openneuro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='ds003944.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' & Etkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Identification of psychiatric disorder subtypes from functional connectivity patterns in resting-state electroencephalography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Nature biomedical engineering, 5(4), 309-323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Dean Salisbury and Dylan Seebold and Brian Coffman (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' EEG: First Episode Psychosis vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Control Resting Task 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' OpenNeuro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' [Dataset] doi: doi:10.' metadata={'source': 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+page_content=' EEG: First Episode Psychosis vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Control Resting Task 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' OpenNeuro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' [Dataset] doi: doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='18112/openneuro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='ds003947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='v1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Resting state alpha oscillatory activity is a valid and reliable marker of schizotypy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Scientific Reports, 11(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='1-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Chedid, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=', Tabbal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=', Kabbara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=', Allouch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' and Hassan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' The development of an automated machine learning pipeline for the detection of Alzheimer’s Disease.' metadata={'source': 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Spectral Power Density analysis of the resting-state as a marker of the central effects of opioid use in fibromyalgia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content=' Scientific reports, 11(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} +page_content='1-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfn_0r/content/2301.01588v1.pdf'} diff --git a/h9AzT4oBgHgl3EQfbPwA/content/tmp_files/2301.01380v1.pdf.txt b/h9AzT4oBgHgl3EQfbPwA/content/tmp_files/2301.01380v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..082572ab81b76b1cd351782a146b77ab09a7ec3e --- /dev/null +++ b/h9AzT4oBgHgl3EQfbPwA/content/tmp_files/2301.01380v1.pdf.txt @@ -0,0 +1,1865 @@ +Ego-Only: Egocentric Action Detection without Exocentric Pretraining +Huiyu Wang1 +Mitesh Kumar Singh1 +Lorenzo Torresani1 +1Meta AI +Abstract +We present Ego-Only, the first training pipeline that en- +ables state-of-the-art action detection on egocentric (first- +person) videos without any form of exocentric (third- +person) pretraining. Previous approaches found that ego- +centric models cannot be trained effectively from scratch +and that exocentric representations transfer well to first- +person videos. In this paper we revisit these two obser- +vations. Motivated by the large content and appearance +gap separating the two domains, we propose a strategy that +enables effective training of egocentric models without ex- +ocentric pretraining. Our Ego-Only pipeline is simple. It +trains the video representation with a masked autoencoder +finetuned for temporal segmentation. The learned features +are then fed to an off-the-shelf temporal action localization +method to detect actions. We evaluate our approach on two +established egocentric video datasets: Ego4D and EPIC- +Kitchens-100. On Ego4D, our Ego-Only is on-par with exo- +centric pretraining methods that use an order of magnitude +more labels. On EPIC-Kitchens-100, our Ego-Only even +outperforms exocentric pretraining (by 2.1% on verbs and +by 1.8% on nouns), setting a new state-of-the-art. +1. Introduction +In this paper we consider the problem of action detec- +tion from egocentric videos [18, 21, 29] captured by head- +mounted devices. While action detection in third-person +videos [6,36] has been the topic of extended and active re- +search by the computer vision community, the formulation +of this task in the first-person setting is underexplored. +Despite the dramatically different viewpoint of first- +person videos, most action detection methods pretrain video +models on large-scale exocentric (third-person) videos [37] +or images [22], under the assumption that the learned rep- +resentations transfer well to egocentric videos, with [21,72] +or without finetuning [29, 44, 75]. This expectation is rein- +forced by the observation that deep neural networks exhibit +invariance to object viewpoints [56], as evidenced by the +transfer results from ImageNet pretraining to various still- +Data +Exocentric pretraining +Ego-Only +Model +(previous) +(ours) +Ego4D +ViT-B +240K labels +(exocentric) ++ +14K labels +(egocentric) +14K labels +(egocentric) +mAP: +14.4% +14.4% +EPIC +ViT-L +240K labels +(exocentric) ++ +67K labels +(egocentric) +67K labels +(egocentric) +Verb mAP: +25.6% +27.7% +Noun mAP: +26.3% +28.1% +Table 1. Our Ego-Only pipeline achieves state-of-the-art results on +Ego4D [29] and EPIC-Kitchens-100 [21] without any extra data or +label. Compared with the previous exocentric pretraining pipeline, +our Ego-Only approach uses an order of magnitude fewer human +annotated labels (Ego4D Moments) and scales better when more +egocentric labels are available (EPIC-Kitchens-100). Ego-Only +also simplifies the full training pipeline by using one dataset only. +Exocentric Videos +Egocentric Videos +(length: 10 seconds) +(length: 480 seconds) +Figure 1. Exocentric videos (Kinetics-400 [37]) vs. egocentric +videos (Ego4D [29]). Exocentric videos are typically in the form +of short trimmed clips, which show the actors as well as the con- +textual scene. Egocentric videos are dramatically longer, capture +close-up object interactions but only the hands of the actor. These +differences make it challenging to transfer models from exocentric +action classification to egocentric action detection. +arXiv:2301.01380v1 [cs.CV] 3 Jan 2023 + +3627image [34,48,63] and video understanding tasks [2,5,37]. +However, in addition to viewpoint changes, egocentric +action detection poses a set of new challenges, as illustrated +in Figure 1: (1) No actor in view. In egocentric videos +the subject is behind the camera and is never visible, ex- +cept for their hands. Conversely, third-person videos usu- +ally capture the actors as well as informative spatial con- +text around them. +(2) Domain shift. +Egocentric videos +entail daily life activities such as cooking, playing, per- +forming household chores, which are poorly represented in +third-person datasets. (3) Class granularity. First-person +vision requires fine-grained recognition of actions within +the same daily life category, such as “wipe oil metallic +item”, “wipe kitchen counter”, “wipe kitchen appliance”, +and “wipe other surface or object” [29]. (4) Object interac- +tion. Egocentric videos capture a lot of human-object inter- +actions as a result of the first-person viewpoint. The scales +and views of the objects are dramatically different than in +exocentric videos. (5) Long-form. Egocentric videos are +typically much longer than exocentric videos and thus re- +quire long-term reasoning of the human-object interactions +rather than single frame classification. (6) Long-tail. Real- +world long-tail distribution is often observed in egocentric +datasets, as they are uncurated and thus reflect the in-the- +wild true distribution of activities, which is far from uni- +form. (7) Localization. Egocentric action detection requires +temporally sensitive representations which are difficult to +obtain from third-person video classification on short and +trimmed clips. We argue that these challenges impede effec- +tive transfer from the exocentric to the egocentric domain +and may actually cause detrimental biases when adapting +third-person models to the first-person setting. +While prior approaches [41, 57] have demonstrated the +performance benefits of transferring from exocentric rep- +resentations over learning egocentric representations from +scratch, we argue that these empirical observations need +to be revisited in light of the scale growth of egocen- +tric data collections (e.g., the recently introduced Ego4D +dataset [29]) as well as the development of data-efficient +training methods, such as masked autoencoders [27,31,60]. +In this paper, we study the possibility of training +with only egocentric video data by proposing a simple +“Ego-Only” training approach. Specifically, the Ego-Only +pipeline consists of three training stages: (1) a masked au- +toencoder stage that bootstraps the backbone representa- +tion, (2) a simple finetuning stage that performs temporal +semantic segmentation of egocentric actions, and (3) a fi- +nal detection stage using an off-the-shelf temporal action +detector, such as ActionFormer [72], without any modifica- +tion. This pipeline enables us to train an egocentric action +detector from random initialization without any exocentric +data or image pretraining. +Empirically, we evaluate Ego-Only on the two largest +egocentric datasets, +Ego4D [29] and EPIC-Kitchens- +100 [21] where our method outperforms all previous results +based on exocentric video pretraining, setting a new state- +of-the-art, obtained for the first time without additional data. +Specifically, Ego-Only achieves 15.7% average mAP on +Ego4D Moments Queries, outperforming the best published +results by 4.3% absolute points. +On EPIC-Kitchens-100 +Action Detection, Ego-Only improves over the state-of-the- +art by 4.2% and 6.2% on verbs and nouns, reaching the per- +formance of 27.7% and 28.1%, respectively. +In summary, we propose the first training pipeline that +enables temporal action detection on egocentric videos +without any form of exocentric pretraining but with results +outperforming the previous state-of-the-art. +2. Related Work +Action recognition methods learn to classify actions in +videos. Recent action recognition models include convo- +lutional neural networks [10, 26, 28, 43, 61, 62, 65, 66] and +vision transformers [2,5,24,25,42,50]. The learned action +representations can serve as features for downstream tasks. +Temporal action localization aims to detect action in- +stances from long videos. Most methods [45,46,70,75] de- +tect actions using frozen video features from action recog- +nition models. Recently, ActionFormer [72] models long- +sequence features with transformers. SegTAD [74] detects +actions via temporal segmentation. TALLFormer [17] trains +the feature backbone end-to-end with the detector. +Self-supervised learning aims to learn visual representa- +tion without human annotation. +Traditional methods in- +clude hand-crafted pretext tasks [23, 38, 55, 68] and con- +trastive learning [7–9, 15, 16, 30, 32, 64, 69]. +Recently, +masked autoencoders [4,27,31,67,76] demonstrate training +efficiency [31], model scalability [31], data efficiency [60], +and effectiveness on videos [27,60,67]. +Egocentric video datasets [20, 21, 29, 58] increase in size +by orders of magnitude over the past few years, presenting +new challenges [21] and opportunities [29], such as egocen- +tric action recognition [21,41] and detection [21]. Existing +egocentric action detection methods [21, 29, 44, 72] follow +temporal action localization practices [45, 70, 72, 75] and +adopt exocentric pretrained checkpoints [2,3,5,10,28]. +In this paper, we study the possibility of detecting ego- +centric actions without any form of exocentric pretraining. +3. Method +In Section 3.1, we provide an overview of our Ego-Only +approach which enables egocentric action detection without +relying on exocentric pretraining. The proposed Ego-Only +pipeline consists of three training stages: a standard masked + +Exocentric +Videos +(240K labels) +Images +(>14M labels) +Egocentric +Videos +(14K labels) +Exocentric +MAE +Pretraining +Exocentric +Classification +Image +Classification +Egocentric +MAE +Pretraining +Egocentric +Finetuning +Egocentric +Action +Detection +Image MAE +Pretraining +Our +Ego-Only +Pipeline +Previous +Exocentric +Pretraining +Previous +Image +Pretraining +ViViT, +VideoSwin, +TimeSformer, +SlowFast, +ViT, MViT, +Figure 2. Our Ego-Only approach simplifies the previous pipeline +by removing the dependence on pretrained exocentric checkpoints +obtained with extra data, extra labels, and extra pretraining stages. +autoencoder (MAE) pretraining stage, an egocentric fine- +tuning stage, which we present in Section 3.2, and finally a +standard temporal action detection stage. +3.1. Ego-Only Pipeline +There is an extensive literature for training object detec- +tors [34, 48] on images end-to-end from random initializa- +tion [33]. However, these approaches are difficult to adapt +to egocentric action detection where both the videos and the +actions are long-form. For example, Ego4D [29] Moments +clips are 8 minutes long, and around half of the actions are +longer than 10 seconds which is the typical length of an +exocentric video. In this case, end-to-end training of an ac- +tion detector is impossible due to GPU memory limitations +unless one reduces aggressively the model size, the spatial +resolution, or the temporal sampling density, which would +lead to degradation in performance. +This empirical challenge calls for a “proxy” objective +that enables learning visual representations with a large +model size, a high spatial resolution, and a high temporal +sampling density. This surrogate objective is usually real- +ized by pretraining on short exocentric videos. However, as +discussed in Section 1, the learned representation may not +transfer effectively. Instead, in our Ego-Only pipeline, we +approximate the temporal action detection task by perform- +ing temporal semantic segmentation that predicts action la- +bels at each frame. Note that this approximation is not exact +because we truncate long-form videos into clips, throwing +away the action context outside the sampled clip. Such ap- +proximation leads to a trade-off between the action context +and the temporal sampling density, ablated in Section 4.2. +This simple surrogate objective allows us to train visual +representations from random initialization towards tempo- +ral action detection. However, we empirically find that the +learned representation generalizes poorly even with strong +augmentation and regularization. In order to further im- +prove generalization, we introduce an additional MAE pre- +training stage which have demonstrated strong generaliza- +tion ability in low data regime [60]. This additional pre- +training improves generalization as shown in Table 2. +Putting these pieces together, Figure 2 summarizes our +complete Ego-Only pipeline that includes the initial MAE +pretraining, the egocentric finetuning task as an approxi- +mation of action detection, and the final temporal action +detector that incorporates full context of the whole long- +form video. This pipeline differs from existing methods in +the absence of an exocentric pretraining stage that requires +large-scale annotated exocentric videos or images. For ex- +ample, most prior approaches pretrain egocentric models on +Kinetics-400 (K400) with 240K annotated videos, while our +Ego-Only pipeline uses merely 14K annotated action seg- +ments on Ego4D and achieves on-par results (Table 2). +Next, we describe in more detail the initial MAE pre- +training stage and the final action detection stage that are +both adopted from existing literature without any modifica- +tion. Note that this paper aims to revisit the value of ex- +ocentric pretraining and does so by proposing an ego-only +pipeline that is intentionally kept as simple as possible. +Masked Autoencoder. Our pipeline applies the original +MAE [31] and video MAE [27] algorithms. Specifically, +we consider the vanilla vision transformers [24, 27], ViT- +B and ViT-L, as our architectures, due to the native sup- +port by MAE. We do not consider convolutional architec- +tures [28] or hierarchical transformers [25, 42, 49, 50] that +require adaptation of the MAE algorithm. Since videos are +highly redundant, we use a very high masking ratio (90%) +with a random masking strategy and all of the pretraining +recipes as suggested in video MAE [27]. The only adapta- +tion we make is to sample each video with a probability pro- +portional to its temporal length, because of the long-form +property of egocentric videos. This ensures equal sampling +probability for any possible clip in the dataset. +Action Detector. After the egocentric finetuning stage +(Section 3.2) that trains the backbone representation to- +wards action detection, we apply an existing temporal ac- +tion localization algorithm to detect the actions. Specif- +ically, given the finetuned video backbone, features are +extracted from the frozen model with sliding windows, +following standard practice in temporal action localiza- +tion [72, 75]. Then, the action detector is trained on top of +a long sequence of frozen video features to produce tempo- +ral segments as outputs. There is a potential risk of overfit- +ting since our finetuning stage and action detection stage are +trained on the same training set, but empirically we do not +find this to be a significant issue in practice, probably be- +cause the detector takes as input a long-form video instead +of a clip and the detector loss differs from simply segmenta- +tion. For better performance, we choose ActionFormer [72] +as our default detector as it has demonstrated good accu- +racy on temporal action localization benchmarks. As we +work on egocentric videos, we follow exactly the previous +ActionFormer architecture for EPIC-Kitchens-100 [21]. + +1 +1 +1 +1 +Vision Transformer +Frame 1 +Frame 2 +Frame 3 +Frame 4 +2 3 4 +2 3 4 +2 3 4 +2 3 4 +1 +1 +1 +1 +2 3 4 +2 3 4 +2 3 4 +2 3 4 +Pool +Pool +Pool +Pool +0 +CLS +0 +Feature +Feature +Feature +Finetuning +Feature +Feature +Feature +Feature +Action Detection +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Feature +Pool +Pool +Pool +Pool +Pool +Temporal Action Detector +Classes +Classes +Classes +Feature +Classes +T1 +T2 +T3 +T4 +T5 +T +T1 +T2 +T3 +T4 +T +Figure 3. Ego-Only finetuning stage (left) and action detection stage (right). In the finetuning stage, the vision transformer is finetuned to +predict action classes at each frame from spatially-pooled features (colors represent frame indices within a clip). In the detection stage, +finetuned backbone features are frozen and extracted using a sliding window. Features at the same timestamp (e.g. T1) but from different +windows are average-pooled. On top of the long sequence of frozen features, a detector is then trained to temporally localize the actions. +3.2. Finetuning via Temporal Segmentation +Inspired by TSN [65] and SegTAD [74] that detect ac- +tions via temporal semantic segmentation, we finetune our +backbone features from MAE pretraining by predicting +class labels for each frame, as illustrated in Figure 3 (left). +This is akin to the task of image semantic segmenta- +tion [11–14] which predicts class labels for each pixel. For- +mally, given an input video clip with a certain temporal +span, a temporal segmentation model predicts output log- +its L ∈ RT ×C where T denotes the temporal dimension of +the logits and C is the total number of action classes. +We follow a few principles in defining this simple fine- +tuning objective: (1) A video clip of a certain temporal span +is taken as the input instead of the full long-form video. +This temporal approximation enables us to train large-scale +models within the given GPU memory limit. (2) We em- +ploy a fixed temporal span which is consistent with both +MAE pretraining and detection feature extraction. This re- +moves potential domain gaps when models are trained and +inferred with different temporal spans. (3) The temporal +segmentation objective trains models to distinguish frames +of different classes within one video clip, especially when +a long temporal span is adopted. (4) We train with clips +uniformly sampled over the dataset, making full use of all +positive and negative samples in the dataset. +Next, we discuss the loss function that we choose to fine- +tune the backbone, and how backbone features are extracted +for the subsequent action detection stage. +Loss function. Egocentric videos usually contain overlap- +ping actions of different classes. For example, a person +could be taking a photo while speaking on the phone. This +makes the finetuning stage a multi-label classification task. +Therefore, we employ a loss function independent of action +classes so that the activation of one class does not suppress +another. +Specifically, we adopt per-frame binary cross- +entropy (BCE) as the loss function on the logits, instead +of cross-entropy which suppresses non-maximum classes. +Imbalance challenges. The long-tail imbalance in egocen- +tric videos (Section 1) poses a major challenge to our fine- +tuning stage, due to the less curated nature and the long- +form property of egocentric videos. Specifically, there are +usually (1) imbalanced numbers of videos across action +classes, (2) imbalanced action lengths within one class, +and (3) imbalanced numbers of foreground frames vs back- +ground frames within one class. Inspired by the literature +of one-stage object detection, we mitigate the imbalance is- +sue by adopting focal loss [47] in the BCE objective and +biasing the logits towards background at initialization. We +also reweight the loss of each class by the inverse-square- +root [53,54] of the total number of foreground frames. +Feature extraction. Once our video backbone is finetuned +on sampled clips, features are extracted using a sliding win- +dow on both the training set and validation set for train- +ing the detector on long-form videos and validating the ap- +proach. According to temporal action localization litera- +ture [72, 75], clip features are average-pooled spatiotem- +porally following the exocentric classification practice [10, +28]. However, in our temporal segmentation case on long- +form videos, our spatially-pooled features are trained to be +temporally different within a video clip, encoding their own +local context. Therefore, as illustrated in Figure 3 (right), +given the sliding windows of features, we average-pool fea- +tures at the same wall-clock timestamp from all sliding win- +dows. This enables the usage of a long temporal span, such +as 64 seconds (Figure 4), by extracting temporally variable +features from a window. +4. Experiments +We report our results on the two largest egocentric video +datasets, Ego4D [29] and EPIC-Kitchens-100 [21], mea- +sured by average mAP at tIoU {0.1, 0.2, 0.3, 0.4, 0.5} on +the validation set. Technical details are discussed in Sec- +tion 4.1. We validate our Ego-Only pipeline by comparing +with egocentric pretraining and by ablating the importance + +of each component in Section 4.2. We then compare our re- +sults with state-of-the-art methods on Ego4D (Section 4.3) +and EPIC-Kitchens-100 (Section 4.4). We also study the +transfer ability of our Ego-Only pipeline in Section 4.5. +Ego4D +[29] offers 3,670 hours of daily life egocentric +videos from hundreds of scenarios, providing massive-scale +data for self-supervised pretraining. The Ego4D Moments +Queries (MQ) task in the Episodic Memory benchmark con- +tains 110 moments classes, 326.4 hours of videos (194.9h in +train, 68.5h in val, 62.9h in test), 2522 clips (1486 in train, +521 in val, 481 in test), and 22.2K annotated temporal ac- +tion segments (13.6K in train, 4.3K in val, 4.3K in test). +EPIC-Kitchens-100 [21] offers 100 hours (74.7h in train, +13.2h in val, 12.1h in test) of egocentric videos from 700 +sessions (495 in train, 138 in val, 67 in test) in 45 kitchens. +The Action Detection challenge contains 97 verb classes +(97 in train, 78 in val, 84 in test), 300 noun classes (289 +in train, 211 in val, 207 in test), and 90.0K temporal action +segments (67.2K in train, 9.7K in val, 13.1K in test). +4.1. Technical Details +MAE pretraining. As discussed in Section 3.1, we follow +the technical details in video MAE [27] unless noted other- +wise. However, as egocentric datasets contain long videos +with hundreds or thousands of hours, it is hard to define a +meaningful epoch. In this paper, we define one epoch as +245,760 clips sampled from data, so that the compute bud- +get is comparable to one Kinetics-400 [37] epoch. With this +definition, we pretrain egocentric MAE for 800 epochs with +120-epoch warm-up, batch size 512, without repeated sam- +pling for simplicity, learning rate 8e-4, by default. We sam- +ple clips of 16 frames with a temporal span of 2 seconds, +equivalent to a sampling rate of 4 in 30-fps videos. +Finetuning. We finetune for 20 epochs with 2-epoch +warm-up, batch size 256, RandAugment [19], stochastic +depth [35] 0.2, dropout [59] 0.5, label smoothing 0.0001 for +BCE, no mixup [73] or cutmix [71] as they are not common +for segmentation. We use SGD with learning rate 2.0 weight +decay 0.0 on Ego4D, while we use AdamW [51] with learn- +ing rate 8e-4 weight decay 0.05 on EPIC-Kitchens-100. For +finetuning on EPIC-Kitchens-100, we concatenate all verb +and noun classes so that we finetune only once. +Action detection. As discussed in Section 3.1, we follow +the details of ActionFormer [72] for EPIC-Kitchens-100 un- +less noted otherwise. Our Ego4D features are extracted at +stride 8 which equals the transformer output stride, with +frame sampling rate 4 and temporal patch stride 2. The slid- +ing windows use stride 8 as well. We train for 10 epochs +with 8-epoch warm-up, learning rate 2e-4. EPIC-Kitchens- +100 features use stride 16 [72] for fair comparison. We train +for 20 epochs with 16-epoch warm-up, learning rate 2e-4. +We report an average of 3 runs. +Self-supervised +Supervised +Supervised +Ego4D +MAE pretrain +exo finetune +ego finetune +mAP +K400 +K400 (240K) +Ego4D (14K) +14.4 +K400 +- +Ego4D (14K) +12.4 +- +- +Ego4D (14K) +4.4 +Ego4D +- +Ego4D (14K) +14.4 +Table 2. Varying the pretraining stage. Ego-Only matches exocen- +tric pretraining but with much fewer labels (14K vs. 240K+14K). +4.2. Ablation Study +In order to validate our Ego-Only pipeline, especially to +compare with exocentric pretraining solutions, we first set +up a strong exocentric baseline. Then, we compare with +Ego-Only and ablate the importance of each stage in Ego- +Only. +We also scale the amount of data consumed, the +model sizes, as well as the number of pretraining epochs. +We perform all ablation studies on Ego4D MQ. +A strong baseline. For fair comparison, we first build a +strong exocentric pretraining baseline that achieves top per- +formance [27] on Kinetics-400 classification [37]. Specif- +ically, this baseline is pretrained on Kinetics-400 with +MAE [27] and finetuned on Kinetics-400 with 240K ex- +ocentric classification labels [27]. +Then, this exocentric +pretrained checkpoint is further finetuned on Ego4D with +the same temporal segmentation objective and recipe as +Ego-Only. Finally, features are extracted and an Action- +Former [72] is trained on the features with again exactly +the same setting as Ego-Only. This strong exocentric pre- +training baseline achieves 14.4% average mAP with ViT-B +and 15.8% with ViT-L. Next, we compare variants of our +Ego-Only pipeline with the strong baseline. We ablate with +ViT-B unless noted otherwise. +Varying the pretraining stage. Table 2 reports our results +with different pretraining stages. Compared with the strong +exocentric pretraining baseline, our Ego-Only with exactly +the same backbone, the same finetuning, and the same de- +tector, matches the performance of 14.4% mAP by using +egocentric data only and with merely 14K labels, instead of +the 240K labels of the exocentric pretraining. +Next, we consider skipping the MAE pretraining and +training from scratch the model via temporal segmenta- +tion on egocentric data. +Our best attempt for this strat- +egy uses AdamW [51] with weight decay 0.05, layer decay +1.0, learning rate 2e-5, and 100 epochs. However, our best +model learned from scratch only reaches the mAP of 4.4% +(vs. 14.4% with MAE pretraining in Ego-Only), due to the +limited number of labels available on Ego4D, only 14K. +This is smaller than the number of labels in MNIST [40] +or CIFAR [39] but the task of egocentric action detection is +significantly more challenging. + +Self-supervised +Supervised +Supervised +Ego4D +MAE pretrain +exo finetune +ego finetune +mAP +K400 +K400 (240K) +- +13.5 +K400 +- +- +6.7 +Ego4D +- +- +7.8 +Ego4D +- +Ego4D (14K) +14.4 +Table 3. Varying the finetuning stage. +0.5 +1.0 +2.0 +4.0 +8.0 +16.0 +32.0 +64.0 +Temporal Span (seconds) +2 +4 +6 +8 +10 +12 +14 +Average mAP (%) +ActionFormer (finetuned features) +ActionFormer (frozen MAE features) +VSGN (finetuned features) +Blob detector (finetuned features) +Figure 4. Varying detectors and temporal spans. The blob detec- +tor performs surprisingly well and perfers a long temporal span, +while ActionFormer and VSGN prefer short spans due to their +transformer or graph neural network based architectures. +In addition to the model trained from scratch, we also +compare with self-supervised MAE pretraining on Kinetics- +400. When this checkpoint is finetuned, it achieves 12.4% +mAP which is 2.0% worse than the counterpart pretrained +on Ego4D. This gap is reasonable since the model is pre- +trained on out-of-domain data but does not benefit from the +large-scale exocentric labels. Once the extra labels are used, +Kinetics supervised pretraining yields performance on-par +with our much simpler Ego-Only pipeline. +Varying the finetuning stage. After varying the pretrain- +ing stage, we study the importance of finetuning. For this +purpose, we extract features from pretrained models, with- +out any form of finetuning on egocentric data. Contrary +to the strong linear probing results of MAE on ImageNet- +1K [22], we observe that frozen MAE features perform +poorly on egocentric action detection, leading to an absolute +drop of 6.6% points in average mAP. Kinetics-400 MAE +features perform even worse (as expected), but finetuning +on Kinetics with 240K labels is helpful, achieving a 13.5% +mAP which is 0.9% worse than the same model finetuned +on Ego4D and 0.9% worse than our Ego-Only pipeline. We +also try concatenating frozen MAE features from multiple +blocks, inspired by DINO [9], but only observe a marginal +gain (Section A.1). +200 +400 +800 +Ego-Only pretraining epochs +13.0 +13.5 +14.0 +14.5 +15.0 +15.5 +16.0 +Average mAP (%) +ViT-L Ego-Only +ViT-L Exocentric +ViT-B Ego-Only +ViT-B Exocentric +Figure 5. Scaling models and pretraining epochs. At around 800 +epochs, our Ego-Only starts to match exocentric pretraining. +Detectors and temporal spans. Next, we compare tempo- +ral action detector choices in Ego-Only and vary the tem- +poral span at the same time. As we use a consistent tem- +poral span for the whole pipeline, including MAE, finetun- +ing, and feature extraction (Section 3.2), we pretrain MAE +with each temporal span for 200 epochs only. Then, we de- +fine a simple baseline of a 1D blob detector [52] using the +Laplacian of Gaussian kernel. To our surprise, as shown in +Figure 4, this simple blob detection baseline achieves 8.2% +mAP which is already better than the Ego4D [29] paper +baseline of 6.0% mAP with pretrained SlowFast [28] fea- +tures and VSGN [75], thanks to the effectiveness of Ego- +Only features. We also notice that the blob detector and +the frozen MAE feature prefer a longer temporal span of 16 +or 32 seconds, demonstrating the importance of long-term +context in egocentric videos. On the other hand, VSGN [75] +and ActionFormer [72] prefer short feature spans probably +because the graph neural network or the transformer cap- +tures long-term relations internally, benefiting more from +local features that represent dense temporal motion. Finally, +ActionFormer with finetuned features achieves the best re- +sult of 12.9%, outperforming VSGN by 4.0% consistently. +Scaling models and pretraining epochs. In addition to +ablating the three stages in our Ego-Only pipeline, we also +scale the model size from ViT-B to ViT-L and benchmark +results under different computation budgets. We keep the +relatively cheap finetuning of 20 epochs unchanged, but +vary the MAE pretraining epochs. As shown in Figure 5, +both ViT-B and ViT-L results improve consistently when +they are pretrained longer. At around the budget of 800 +epochs, our Ego-Only models start to match Kinetics-400 +pretrained models with both ViT-B and ViT-L. The Kinet- +ics baselines, before finetuned on egocentric data, are pre- +trained with 800-epoch MAE and 100/150 epoch exocentric +finetuning that consumes not only more data and labels but +also more computation resources than Ego-Only. + +Ego MAE pretrain +Ego finetune +mAP +Random initialization (0h) +195h +4.4 +Ego4D MQ clips (195h) +195h +12.9 +Ego4D MQ videos (487h) +195h +13.2 +Ego4D EM videos (838h) +195h +13.5 +Ego4D ALL videos (3560h) +195h +13.9 +Table 4. Scaling the amount of pretraining data. MQ clips: all MQ +training clips [29]. MQ videos: all videos in the MQ task training +set. EM videos: all videos in the Episodic Memory benchmark +training set. ALL videos: all Ego4D videos except MQ val and +test videos. Our Ego-Only results improve consistently with re- +spect to the amount of data consumed in the pretraining stage. +MAE +Exo +Ego +Rebalancing +Ego4D +pretrain +finetune +finetune +method +mAP +K400 +K400 +Ego4D +Reweighting +14.4 +K400 +K400 +Ego4D +Resampling +16.2 +Ego4D +- +Ego4D +Reweighting +14.4 +Ego4D +- +Ego4D +Resampling +16.3 +Table 5. Varying rebalancing techniques. Ego-Only still matches +exocentric pretraining when a stronger rebalancing technique (ac- +tion resampling in this case) is employed. +Scaling the amount of pretraining data. Beyond standard +ablations on pretraining epochs, an intriguing dimension for +study offered by the massive scale of Ego4D is the different +amounts of large-scale unsupervised video data. Specifi- +cally, given the fixed amount of finetuning data, we select +four subsets and amounts of unsupervised data in Ego4D +to study the data scaling property of the Ego-Only pretrain- +ing stage. Note that in all cases, we exclude val and test +videos of the MQ task from the pretraining set. All models +are pretrained for 200 epochs instead of 800 epochs to save +computation resources. From the results in Table 4, we see +that the performance of Ego-Only consistently improves as +more unsupervised data is provided for MAE pretraining. +Varying rebalancing techniques. As discussed in Sec- +tion 3.2, we are currently mitigating the imbalance chal- +lenges by simply reweighting the loss according to the num- +ber of positive frames in each action class, ignoring the vari- +able action lengths within one class. Beyond this current +technique, we also study a simple action resampling option +as a natural alternative. Specifically, instead of uniformly +sampling all the clips within the train data, we sample only +the center 2 seconds of each action regardless of the action +length, similar to an action classification task. As shown +in Table 5, this resampling option outperforms the default +reweighting by around 1.8% mAP in both settings with and +without exocentric pretraining. In this case, our Ego-Only +pipeline still matches Kinetics pretraining, without any exo- +centric data or label, and regardless of the rebalancing tech- +niques employed. We consider further exploration of better +rebalancing methods as an open research problem and leave +it to future work beyond the scope of this paper. +4.3. Comparison on Ego4D +We compare our results on the Ego4D [29] MQ val set +with state-of-the-art methods in Table 6, using ViT-B and +ViT-L, with and without exocentric pretraining. +We no- +tice that our strong baseline performs better than the re- +sults reported in the Ego4D paper with VSGN [75] and +SlowFast [28] features, because of the stronger Action- +Former [72] detector and the stronger ViT [24] pretrain- +ing [27]. However, Ego-Only is able to match this strong +performance in a fair comparison but without any extra ex- +ocentric data or label, in both ViT-B and ViT-L cases. Ego- +Only achieves the average mAP of 15.7%, producing a rel- +ative improvement of 160% over the Ego4D paper base- +line [29], and setting a new state-of-the-art on this bench- +mark without any extra data or label. +4.4. Comparison on EPIC-Kitchens-100 +Following the exploration on Ego4D, we validate our +Ego-Only approach on the EPIC-Kitchens-100 [21] Action +Detection benchmark. We can see from Table 7 that, on +EPIC-Kitchens-100, Ego-Only achieves much stronger re- +sults compared with exocentric pretraining. +Specifically, only when we evaluate on nouns and only +when a small ViT-B is used, exocentric pretraining matches +Ego-Only (25.1% vs. 25.3%), possibly due to the bias of +Kinetics towards scene classification. Once we evaluate on +verbs or the model is scaled to ViT-L, Ego-Only shows a +significant absolute gain of 2% over exocentric pretraining. +This stronger scaling is probably due to more labels (67K) +available than in Ego4D (14K) which allow the large trans- +formers to generalize better. +Compared with previous methods that adopt Kinet- +ics [37] SlowFast [28] features finetuned on EPIC- +Kitchens-100 Action Recognition, our baseline models with +exocentric pretraining already performs better, regardless of +the model size. Our Ego-Only approach improves further +on top of the strong baselines and sets a new state-of-the-art +result of 27.7% mAP on verbs and 28.1% mAP on nouns. +Furthermore, we analyze our ViT-L results and baselines +using DETAD [1] (Section A.2), and notice that Ego-Only +significantly reduces false positives on backgrounds, com- +pared with exocentric pretraining baselines, probably be- +cause Kinetics contains mostly trimmed videos with fore- +ground actions only. This analysis validates the benefit of +our Ego-Only approach. + +Detector +Backbone +Extra data +Extra labels # labels +0.1 +0.2 +0.3 +0.4 +0.5 +Average +VSGN [75] +SlowFast [28] +Kinetics-400 [37] +240K +14K +9.10 +7.16 +5.76 +4.62 3.41 +6.03 +VSGN [75] +Frozen [3] +IN-21K [22] + EgoClip [44] +14M + 4M +14K +16.63 +- +11.45 +- +6.57 +11.39 +ActionFormer† +ViT-B +Kinetics-400 +240K +14K +20.5 +17.1 +14.0 +11.4 +9.0 +14.4 +ActionFormer +ViT-B +- +- +14K +20.5 +17.3 +14.3 +11.3 +8.8 +14.4 +ActionFormer† +ViT-L +Kinetics-400 +240K +14K +22.5 +18.8 +15.6 +12.5 +9.6 +15.8 +ActionFormer +ViT-L +- +- +14K +22.0 +18.7 +15.3 +12.6 10.2 +15.7 +Table 6. Comparing with the state-of-the-art on Ego4D MQ val set. †our strong exocentric pretraining baselines. +Task +Detector +Backbone +Extra data +Extra labels +# labels +0.1 +0.2 +0.3 +0.4 +0.5 +Average +Verb +BMN [21,45] +SlowFast [28] +K400 [37] +240K +67K +10.8 +9.8 +8.4 +7.1 +5.6 +8.4 +G-TAD [70] +SlowFast [28] +K400 [37] +240K +67K +12.1 +11.0 +9.4 +8.1 +6.5 +9.4 +ActionFormer [72] +SlowFast [28] +K400 [37] +240K +67K +26.6 +25.4 +24.2 +22.3 +19.1 +23.5 +ActionFormer† +ViT-B +K400 +240K +67K +27.1 +26.2 +25.0 +22.2 +18.9 +23.9 +ActionFormer +ViT-B +- +- +67K +29.6 +28.6 +27.0 +24.2 +21.3 +26.1 +ActionFormer† +ViT-L +K400 +240K +67K +29.3 +28.3 +26.8 +23.5 +20.0 +25.6 +ActionFormer +ViT-L +- +- +67K +30.8 +30.1 +28.7 +26.3 +22.5 +27.7 +Noun +BMN [21,45] +SlowFast [28] +K400 [37] +240K +67K +10.3 +8.3 +6.2 +4.5 +3.4 +6.5 +G-TAD [70] +SlowFast [28] +K400 [37] +240K +67K +11.0 +10.0 +8.6 +7.0 +5.4 +8.4 +ActionFormer [72] +SlowFast [28] +K400 [37] +240K +67K +25.2 +24.1 +22.7 +20.5 +17.0 +21.9 +ActionFormer† +ViT-B +K400 +240K +67K +28.9 +27.7 +26.0 +23.6 +19.2 +25.1 +ActionFormer +ViT-B +- +- +67K +29.0 +27.8 +26.1 +23.7 +19.9 +25.3 +ActionFormer† +ViT-L +K400 +240K +67K +30.4 +29.1 +27.3 +24.5 +20.1 +26.3 +ActionFormer +ViT-L +- +- +67K +31.8 +30.8 +29.3 +26.6 +22.0 +28.1 +Table 7. Comparing with the state-of-the-art on EPIC-Kitchens-100 Action Detection val set. †our strong exocentric pretraining baselines. +4.5. Transfer Learning +Beyond comparing with state-of-the-art methods, we +study the possibility of transferring representations learned +from the massive-scale diverse Ego4D to a different ego- +centric dataset. Specifically, we take EPIC-Kitchens-100 as +the target data and compare ViT-L results with various pre- +training setups. As shown in Table 8, we observe that none +of the transferring model performs on par with MAE pre- +trained on the target data, which is expected. However, we +notice that Ego4D pretraining transfers significantly better +than Kinetics-400 pretraining when supervised exocentric +finetuning is not available, suggesting a large domain gap +between egocentric and exocentric datasets. In addition, +our Ego4D pretraining even outperforms exocentric models +finetuned using large-scale exocentric labels, further con- +firming the effectiveness of Ego-Only. +5. Conclusion +In this work, we have shown for the first time that the cur- +rent egocentric videos collected by the community are suf- +Self-supervised +Supervised +Supervised +Verbs Nouns +MAE pretrain +exo finetune ego finetune mAP +mAP +EPIC +- +EPIC +27.7 +28.1 +K400 +K400 +- +17.9 +14.6 +K400 +- +EPIC +24.5 +25.8 +K400 +K400 +EPIC +25.6 +26.3 +Ego4D +- +EPIC +25.9 +26.4 +Table 8. Transfer learning from Ego4D outperforms Kinetics-400. +ficient to train a state-of-the-art egocentric action detector +without any exocentric pretraining. Our proposed Ego-Only +simplifies the current learning pipeline by removing the pre- +vious need for supervised pretraining on large-scale exocen- +tric video or image datasets before transferring to egocen- +tric videos. Instead, we hope our attempt inspires the com- +munity to rethink the trade-off between training in-domain +with ego-only data and transferring from out-of-domain ex- +ocentric learning. We also hope that our Ego-Only results +provide a strong baseline for future research that aims to +improve egocentric learning by leveraging exocentric data. + +A. Appendix +A.1. Ablation on Concatenated Features +In Figure A.1, we present the ablation of concatenat- +ing features from the last few (2, 3, 6, or 12) transformer +blocks, instead of our default choice of the last block only +(Section 4.2). +This is inspired by the linear protocol in +DINO [9] that was aimed to improve results with frozen +self-supervised learning features (in our case frozen MAE +features) but we ablate this choice for all models, with and +without finetuning. However, we see a marginal gain for +frozen MAE features, which confirms the necessity of the +egocentric finetuning stage in Ego-Only. +A.2. Error Analyses +False positive analysis. In Figure A.2, we analyze false +positive errors on EPIC-Kitchens-100 [21] with ViT-L [24] +models using the DETAD [1] error diagnosing tool (Sec- +tion 4.4). We notice that Ego-Only reduces false positive er- +rors on backgrounds, compared with exocentric pretraining +baselines, probably because Kinetics [37] contains mostly +trimmed videos with foreground actions only. +Sensitivity analysis. In Figure A.3, we analyze the model +sensitivity according to DETAD characteristics [1] on +EPIC-Kitchens-100 [21] with ViT-L [24] models. We ob- +serve that our Ego-Only improves significantly when there +are multiple verb instances of the same category in a video. +A.3. Visualization of MAE Reconstructions +In Figure A.4, we visualize the MAE [27, 31] recon- +struction results on a few Ego4D [29] examples with +a ViT-B [24] trained for 200 epochs without per-patch +normalization. +We notice that egocentric MAE learns +human-object interactions (d,f,g,h,i,k) and temporal corre- +spondence across frames (c,j), even in cases with strong +head/camera motion (a,b,e,l). +1 +2 +3 +6 +12 +Number of concatenated blocks +0 +2 +4 +6 +8 +10 +12 +14 +16 +Average mAP (%) +Ego-Only (finetuned features) +Exocentric (finetuned features) +Ego-Only (frozen MAE features) +Exocentric (frozen MAE features) +Figure A.1. Ego4D Moments Queries results with concatenated +features from the last few (2, 3, 6, 12) transformer blocks (12 +blocks in total for the ViT-B [24] architecture), instead of our de- +fault choice of the last block only. The detection results are almost +not affected in any of the four models studied. This stable gap +between finetuned features and frozen MAE features verifies the +necessity of the egocentric finetuning stage in Ego-Only. + +EPIC +Exocentric pretraining (previous) +Ego-Only (ours) +25.6% mAP +27.7% mAP +Verb +1G +2G +3G +4G +5G +6G +7G +8G +9G +10G +Top Predictions +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Error Breakdown (%) +False Positive Profile +Background Err +Confusion Err +Localization Err +Wrong Label Err +Double Detection Err +True Positive +Error Type +0 +1 +2 +3 +4 +5 +6 +7 +8 +Average-mAPN +Improvment (%) +0.4 +6.8 +0.9 1.0 +2.7 +Removing Error Impact +1G +2G +3G +4G +5G +6G +7G +8G +9G +10G +Top Predictions +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Error Breakdown (%) +False Positive Profile +Background Err +Confusion Err +Localization Err +Wrong Label Err +Double Detection Err +True Positive +Error Type +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Average-mAPN +Improvment (%) +0.5 +7.6 +0.8 0.9 +2.3 +Removing Error Impact +26.3% mAP +28.1% mAP +Noun +1G +2G +3G +4G +5G +6G +7G +8G +9G +10G +Top Predictions +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Error Breakdown (%) +False Positive Profile +Background Err +Confusion Err +Localization Err +Wrong Label Err +Double Detection Err +True Positive +Error Type +0 +1 +2 +3 +4 +5 +6 +7 +8 +Average-mAPN +Improvment (%) +0.4 +7.1 +1.0 1.0 +2.9 +Removing Error Impact +1G +2G +3G +4G +5G +6G +7G +8G +9G +10G +Top Predictions +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Error Breakdown (%) +False Positive Profile +Background Err +Confusion Err +Localization Err +Wrong Label Err +Double Detection Err +True Positive +Error Type +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Average-mAPN +Improvment (%) +0.3 +7.6 +0.9 1.1 +2.4 +Removing Error Impact +Figure A.2. False positive analysis on EPIC-Kitchens-100 [21] with DETAD [1]. The error types are determined by the tIoU between +ground-truth and predicted segments, as well as the correctness of the predicted labels. Background error: tIoU < 1e-5; confusion error: +1e-5 < tIoU < α and label is wrong; localization error: label is correct but 1e-5 < tIoU < α; wrong label error: tIoU >= α but label is +wrong, where α refers to the tIoU thresholds {0.1, 0.2, 0.3, 0.4, 0.5}. ‘G’ refers to the number of ground-truth instances. According to +the error breakdown, although the large-scale exocentric pretraining helps reducing wrong label errors, our Ego-Only predicts more true +positives correctly and reduces background errors, probably because Kinetics [37] contains mostly trimmed videos with foreground actions +only. + +EPIC +Exocentric pretraining (previous) +Ego-Only (ours) +25.6% mAP +27.7% mAP +Verb +XS +S +M +L XL +XS +S +M +L XL +0 +20 +40 +60 +80 +100 +Average-mAPN (%) +20.8 +26.6 +34.835.337.6 +Length +35.540.7 +24.125.8 +14.7 +# Instances +XS +S +M +L XL +XS +S +M +L XL +0 +20 +40 +60 +80 +100 +Average-mAPN (%) +20.8 +30.935.240.536.1 +Length +36.5 +46.4 +29.231.8 +19.8 +# Instances +26.3% mAP +28.1% mAP +Noun +XS +S +M +L XL +XS +S +M +L XL +0 +20 +40 +60 +80 +100 +Average-mAPN (%) +22.7 +30.735.8 +29.4 +35.0 +Length +32.532.027.129.225.9 +# Instances +XS +S +M +L XL +XS +S +M +L XL +0 +20 +40 +60 +80 +100 +Average-mAPN (%) +25.931.336.5 +30.433.7 +Length +32.031.928.928.231.1 +# Instances +Figure A.3. Sensitivity analysis on EPIC-Kitchens-100 [21] with DETAD [1]. Ground-truth segments are divided into 5 equal buckets +according to their characteristic [1] percentiles. Then, average mAPN [1] metrics are computed for each characteristic bucket. The ‘length’ +characteristic measures the length of the ground-truth action segment in seconds. The ‘# instances’ characteristic measures the number of +action instances belonging to the same category as the ground-truth segment in the same video. According to the average mAPN in each +bucket, we observe that our Ego-Only improves significantly when there are multiple verb instances of the same category in a video. + +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +(j) +(k) +(l) +Figure A.4. MAE [27, 31] reconstruction results on Ego4D [29] MQ val set. For each sample, we show the original video (top), the +randomly masked video (middle), and the MAE reconstruction (bottom). We visualize 8 frames [27] out of 16 with a temporal stride +of 2. 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In ICLR, 2022. 2 + diff --git a/h9AzT4oBgHgl3EQfbPwA/content/tmp_files/load_file.txt b/h9AzT4oBgHgl3EQfbPwA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1be34fa0432877dbed1fcda72c91e1999a140eac --- /dev/null +++ b/h9AzT4oBgHgl3EQfbPwA/content/tmp_files/load_file.txt @@ -0,0 +1,1007 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf,len=1006 +page_content='Ego-Only: Egocentric Action Detection without Exocentric Pretraining Huiyu Wang1 Mitesh Kumar Singh1 Lorenzo Torresani1 1Meta AI Abstract We present Ego-Only, the first training pipeline that en- ables state-of-the-art action detection on egocentric (first- person) videos without any form of exocentric (third- person) pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Previous approaches found that ego- centric models cannot be trained effectively from scratch and that exocentric representations transfer well to first- person videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In this paper we revisit these two obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Motivated by the large content and appearance gap separating the two domains, we propose a strategy that enables effective training of egocentric models without ex- ocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our Ego-Only pipeline is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' It trains the video representation with a masked autoencoder finetuned for temporal segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The learned features are then fed to an off-the-shelf temporal action localization method to detect actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We evaluate our approach on two established egocentric video datasets: Ego4D and EPIC- Kitchens-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' On Ego4D, our Ego-Only is on-par with exo- centric pretraining methods that use an order of magnitude more labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' On EPIC-Kitchens-100, our Ego-Only even outperforms exocentric pretraining (by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1% on verbs and by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8% on nouns), setting a new state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Introduction In this paper we consider the problem of action detec- tion from egocentric videos [18, 21, 29] captured by head- mounted devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' While action detection in third-person videos [6,36] has been the topic of extended and active re- search by the computer vision community, the formulation of this task in the first-person setting is underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Despite the dramatically different viewpoint of first- person videos, most action detection methods pretrain video models on large-scale exocentric (third-person) videos [37] or images [22], under the assumption that the learned rep- resentations transfer well to egocentric videos, with [21,72] or without finetuning [29, 44, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This expectation is rein- forced by the observation that deep neural networks exhibit invariance to object viewpoints [56], as evidenced by the transfer results from ImageNet pretraining to various still- Data Exocentric pretraining Ego-Only Model (previous) (ours) Ego4D ViT-B 240K labels (exocentric) + 14K labels (egocentric) 14K labels (egocentric) mAP: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% EPIC ViT-L 240K labels (exocentric) + 67K labels (egocentric) 67K labels (egocentric) Verb mAP: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7% Noun mAP: 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1% Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our Ego-Only pipeline achieves state-of-the-art results on Ego4D [29] and EPIC-Kitchens-100 [21] without any extra data or label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Compared with the previous exocentric pretraining pipeline, our Ego-Only approach uses an order of magnitude fewer human annotated labels (Ego4D Moments) and scales better when more egocentric labels are available (EPIC-Kitchens-100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego-Only also simplifies the full training pipeline by using one dataset only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Exocentric Videos Egocentric Videos (length: 10 seconds) (length: 480 seconds) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Exocentric videos (Kinetics-400 [37]) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' egocentric videos (Ego4D [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Exocentric videos are typically in the form of short trimmed clips, which show the actors as well as the con- textual scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric videos are dramatically longer, capture close-up object interactions but only the hands of the actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' These differences make it challenging to transfer models from exocentric action classification to egocentric action detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='01380v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='CV] 3 Jan 2023 3627image [34,48,63] and video understanding tasks [2,5,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, in addition to viewpoint changes, egocentric action detection poses a set of new challenges, as illustrated in Figure 1: (1) No actor in view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In egocentric videos the subject is behind the camera and is never visible, ex- cept for their hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Conversely, third-person videos usu- ally capture the actors as well as informative spatial con- text around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (2) Domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric videos entail daily life activities such as cooking, playing, per- forming household chores, which are poorly represented in third-person datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (3) Class granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' First-person vision requires fine-grained recognition of actions within the same daily life category, such as “wipe oil metallic item”, “wipe kitchen counter”, “wipe kitchen appliance”, and “wipe other surface or object” [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (4) Object interac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric videos capture a lot of human-object inter- actions as a result of the first-person viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The scales and views of the objects are dramatically different than in exocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (5) Long-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric videos are typically much longer than exocentric videos and thus re- quire long-term reasoning of the human-object interactions rather than single frame classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (6) Long-tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Real- world long-tail distribution is often observed in egocentric datasets, as they are uncurated and thus reflect the in-the- wild true distribution of activities, which is far from uni- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (7) Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric action detection requires temporally sensitive representations which are difficult to obtain from third-person video classification on short and trimmed clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We argue that these challenges impede effec- tive transfer from the exocentric to the egocentric domain and may actually cause detrimental biases when adapting third-person models to the first-person setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' While prior approaches [41, 57] have demonstrated the performance benefits of transferring from exocentric rep- resentations over learning egocentric representations from scratch, we argue that these empirical observations need to be revisited in light of the scale growth of egocen- tric data collections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=', the recently introduced Ego4D dataset [29]) as well as the development of data-efficient training methods, such as masked autoencoders [27,31,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In this paper, we study the possibility of training with only egocentric video data by proposing a simple “Ego-Only” training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, the Ego-Only pipeline consists of three training stages: (1) a masked au- toencoder stage that bootstraps the backbone representa- tion, (2) a simple finetuning stage that performs temporal semantic segmentation of egocentric actions, and (3) a fi- nal detection stage using an off-the-shelf temporal action detector, such as ActionFormer [72], without any modifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This pipeline enables us to train an egocentric action detector from random initialization without any exocentric data or image pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Empirically, we evaluate Ego-Only on the two largest egocentric datasets, Ego4D [29] and EPIC-Kitchens- 100 [21] where our method outperforms all previous results based on exocentric video pretraining, setting a new state- of-the-art, obtained for the first time without additional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, Ego-Only achieves 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7% average mAP on Ego4D Moments Queries, outperforming the best published results by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3% absolute points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' On EPIC-Kitchens-100 Action Detection, Ego-Only improves over the state-of-the- art by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2% on verbs and nouns, reaching the per- formance of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7% and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In summary, we propose the first training pipeline that enables temporal action detection on egocentric videos without any form of exocentric pretraining but with results outperforming the previous state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Related Work Action recognition methods learn to classify actions in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Recent action recognition models include convo- lutional neural networks [10, 26, 28, 43, 61, 62, 65, 66] and vision transformers [2,5,24,25,42,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The learned action representations can serve as features for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Temporal action localization aims to detect action in- stances from long videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Most methods [45,46,70,75] de- tect actions using frozen video features from action recog- nition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Recently, ActionFormer [72] models long- sequence features with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' SegTAD [74] detects actions via temporal segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' TALLFormer [17] trains the feature backbone end-to-end with the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Self-supervised learning aims to learn visual representa- tion without human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Traditional methods in- clude hand-crafted pretext tasks [23, 38, 55, 68] and con- trastive learning [7–9, 15, 16, 30, 32, 64, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Recently, masked autoencoders [4,27,31,67,76] demonstrate training efficiency [31], model scalability [31], data efficiency [60], and effectiveness on videos [27,60,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric video datasets [20, 21, 29, 58] increase in size by orders of magnitude over the past few years, presenting new challenges [21] and opportunities [29], such as egocen- tric action recognition [21,41] and detection [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Existing egocentric action detection methods [21, 29, 44, 72] follow temporal action localization practices [45, 70, 72, 75] and adopt exocentric pretrained checkpoints [2,3,5,10,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In this paper, we study the possibility of detecting ego- centric actions without any form of exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Method In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1, we provide an overview of our Ego-Only approach which enables egocentric action detection without relying on exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The proposed Ego-Only pipeline consists of three training stages: a standard masked Exocentric Videos (240K labels) Images (>14M labels) Egocentric Videos (14K labels) Exocentric MAE Pretraining Exocentric Classification Image Classification Egocentric MAE Pretraining Egocentric Finetuning Egocentric Action Detection Image MAE Pretraining Our Ego-Only Pipeline Previous Exocentric Pretraining Previous Image Pretraining ViViT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' VideoSwin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' TimeSformer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' SlowFast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' ViT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' MViT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our Ego-Only approach simplifies the previous pipeline by removing the dependence on pretrained exocentric checkpoints obtained with extra data, extra labels, and extra pretraining stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' autoencoder (MAE) pretraining stage, an egocentric fine- tuning stage, which we present in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2, and finally a standard temporal action detection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego-Only Pipeline There is an extensive literature for training object detec- tors [34, 48] on images end-to-end from random initializa- tion [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, these approaches are difficult to adapt to egocentric action detection where both the videos and the actions are long-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For example, Ego4D [29] Moments clips are 8 minutes long, and around half of the actions are longer than 10 seconds which is the typical length of an exocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In this case, end-to-end training of an ac- tion detector is impossible due to GPU memory limitations unless one reduces aggressively the model size, the spatial resolution, or the temporal sampling density, which would lead to degradation in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This empirical challenge calls for a “proxy” objective that enables learning visual representations with a large model size, a high spatial resolution, and a high temporal sampling density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This surrogate objective is usually real- ized by pretraining on short exocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, as discussed in Section 1, the learned representation may not transfer effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Instead, in our Ego-Only pipeline, we approximate the temporal action detection task by perform- ing temporal semantic segmentation that predicts action la- bels at each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Note that this approximation is not exact because we truncate long-form videos into clips, throwing away the action context outside the sampled clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Such ap- proximation leads to a trade-off between the action context and the temporal sampling density, ablated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This simple surrogate objective allows us to train visual representations from random initialization towards tempo- ral action detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, we empirically find that the learned representation generalizes poorly even with strong augmentation and regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In order to further im- prove generalization, we introduce an additional MAE pre- training stage which have demonstrated strong generaliza- tion ability in low data regime [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This additional pre- training improves generalization as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Putting these pieces together, Figure 2 summarizes our complete Ego-Only pipeline that includes the initial MAE pretraining, the egocentric finetuning task as an approxi- mation of action detection, and the final temporal action detector that incorporates full context of the whole long- form video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This pipeline differs from existing methods in the absence of an exocentric pretraining stage that requires large-scale annotated exocentric videos or images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For ex- ample, most prior approaches pretrain egocentric models on Kinetics-400 (K400) with 240K annotated videos, while our Ego-Only pipeline uses merely 14K annotated action seg- ments on Ego4D and achieves on-par results (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Next, we describe in more detail the initial MAE pre- training stage and the final action detection stage that are both adopted from existing literature without any modifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Note that this paper aims to revisit the value of ex- ocentric pretraining and does so by proposing an ego-only pipeline that is intentionally kept as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Masked Autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our pipeline applies the original MAE [31] and video MAE [27] algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, we consider the vanilla vision transformers [24, 27], ViT- B and ViT-L, as our architectures, due to the native sup- port by MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We do not consider convolutional architec- tures [28] or hierarchical transformers [25, 42, 49, 50] that require adaptation of the MAE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Since videos are highly redundant, we use a very high masking ratio (90%) with a random masking strategy and all of the pretraining recipes as suggested in video MAE [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The only adapta- tion we make is to sample each video with a probability pro- portional to its temporal length, because of the long-form property of egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This ensures equal sampling probability for any possible clip in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Action Detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' After the egocentric finetuning stage (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2) that trains the backbone representation to- wards action detection, we apply an existing temporal ac- tion localization algorithm to detect the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specif- ically, given the finetuned video backbone, features are extracted from the frozen model with sliding windows, following standard practice in temporal action localiza- tion [72, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Then, the action detector is trained on top of a long sequence of frozen video features to produce tempo- ral segments as outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' There is a potential risk of overfit- ting since our finetuning stage and action detection stage are trained on the same training set, but empirically we do not find this to be a significant issue in practice, probably be- cause the detector takes as input a long-form video instead of a clip and the detector loss differs from simply segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For better performance, we choose ActionFormer [72] as our default detector as it has demonstrated good accu- racy on temporal action localization benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As we work on egocentric videos, we follow exactly the previous ActionFormer architecture for EPIC-Kitchens-100 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Vision Transformer ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Temporal Action Detector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Classes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Classes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Classes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Classes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego-Only finetuning stage (left) and action detection stage (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In the finetuning stage, the vision transformer is finetuned to predict action classes at each frame from spatially-pooled features (colors represent frame indices within a clip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In the detection stage, finetuned backbone features are frozen and extracted using a sliding window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Features at the same timestamp (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' T1) but from different windows are average-pooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' On top of the long sequence of frozen features, a detector is then trained to temporally localize the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Finetuning via Temporal Segmentation Inspired by TSN [65] and SegTAD [74] that detect ac- tions via temporal semantic segmentation, we finetune our backbone features from MAE pretraining by predicting class labels for each frame, as illustrated in Figure 3 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This is akin to the task of image semantic segmenta- tion [11–14] which predicts class labels for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For- mally, given an input video clip with a certain temporal span, a temporal segmentation model predicts output log- its L ∈ RT ×C where T denotes the temporal dimension of the logits and C is the total number of action classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We follow a few principles in defining this simple fine- tuning objective: (1) A video clip of a certain temporal span is taken as the input instead of the full long-form video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This temporal approximation enables us to train large-scale models within the given GPU memory limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (2) We em- ploy a fixed temporal span which is consistent with both MAE pretraining and detection feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This re- moves potential domain gaps when models are trained and inferred with different temporal spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (3) The temporal segmentation objective trains models to distinguish frames of different classes within one video clip, especially when a long temporal span is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (4) We train with clips uniformly sampled over the dataset, making full use of all positive and negative samples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Next, we discuss the loss function that we choose to fine- tune the backbone, and how backbone features are extracted for the subsequent action detection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Egocentric videos usually contain overlap- ping actions of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For example, a person could be taking a photo while speaking on the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This makes the finetuning stage a multi-label classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Therefore, we employ a loss function independent of action classes so that the activation of one class does not suppress another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, we adopt per-frame binary cross- entropy (BCE) as the loss function on the logits, instead of cross-entropy which suppresses non-maximum classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Imbalance challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The long-tail imbalance in egocen- tric videos (Section 1) poses a major challenge to our fine- tuning stage, due to the less curated nature and the long- form property of egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, there are usually (1) imbalanced numbers of videos across action classes, (2) imbalanced action lengths within one class, and (3) imbalanced numbers of foreground frames vs back- ground frames within one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Inspired by the literature of one-stage object detection, we mitigate the imbalance is- sue by adopting focal loss [47] in the BCE objective and biasing the logits towards background at initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We also reweight the loss of each class by the inverse-square- root [53,54] of the total number of foreground frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Once our video backbone is finetuned on sampled clips, features are extracted using a sliding win- dow on both the training set and validation set for train- ing the detector on long-form videos and validating the ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' According to temporal action localization litera- ture [72, 75], clip features are average-pooled spatiotem- porally following the exocentric classification practice [10, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, in our temporal segmentation case on long- form videos, our spatially-pooled features are trained to be temporally different within a video clip, encoding their own local context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Therefore, as illustrated in Figure 3 (right), given the sliding windows of features, we average-pool fea- tures at the same wall-clock timestamp from all sliding win- dows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This enables the usage of a long temporal span, such as 64 seconds (Figure 4), by extracting temporally variable features from a window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Experiments We report our results on the two largest egocentric video datasets, Ego4D [29] and EPIC-Kitchens-100 [21], mea- sured by average mAP at tIoU {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5} on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Technical details are discussed in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We validate our Ego-Only pipeline by comparing with egocentric pretraining and by ablating the importance of each component in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We then compare our re- sults with state-of-the-art methods on Ego4D (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3) and EPIC-Kitchens-100 (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We also study the transfer ability of our Ego-Only pipeline in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego4D [29] offers 3,670 hours of daily life egocentric videos from hundreds of scenarios, providing massive-scale data for self-supervised pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The Ego4D Moments Queries (MQ) task in the Episodic Memory benchmark con- tains 110 moments classes, 326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 hours of videos (194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9h in train, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5h in val, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9h in test), 2522 clips (1486 in train, 521 in val, 481 in test), and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2K annotated temporal ac- tion segments (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6K in train, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3K in val, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3K in test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' EPIC-Kitchens-100 [21] offers 100 hours (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7h in train, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2h in val, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1h in test) of egocentric videos from 700 sessions (495 in train, 138 in val, 67 in test) in 45 kitchens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The Action Detection challenge contains 97 verb classes (97 in train, 78 in val, 84 in test), 300 noun classes (289 in train, 211 in val, 207 in test), and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0K temporal action segments (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2K in train, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7K in val, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1K in test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Technical Details MAE pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1, we follow the technical details in video MAE [27] unless noted other- wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, as egocentric datasets contain long videos with hundreds or thousands of hours, it is hard to define a meaningful epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In this paper, we define one epoch as 245,760 clips sampled from data, so that the compute bud- get is comparable to one Kinetics-400 [37] epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' With this definition, we pretrain egocentric MAE for 800 epochs with 120-epoch warm-up, batch size 512, without repeated sam- pling for simplicity, learning rate 8e-4, by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We sam- ple clips of 16 frames with a temporal span of 2 seconds, equivalent to a sampling rate of 4 in 30-fps videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We finetune for 20 epochs with 2-epoch warm-up, batch size 256, RandAugment [19], stochastic depth [35] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2, dropout [59] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5, label smoothing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0001 for BCE, no mixup [73] or cutmix [71] as they are not common for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We use SGD with learning rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 on Ego4D, while we use AdamW [51] with learn- ing rate 8e-4 weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='05 on EPIC-Kitchens-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For finetuning on EPIC-Kitchens-100, we concatenate all verb and noun classes so that we finetune only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Action detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1, we follow the details of ActionFormer [72] for EPIC-Kitchens-100 un- less noted otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our Ego4D features are extracted at stride 8 which equals the transformer output stride, with frame sampling rate 4 and temporal patch stride 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The slid- ing windows use stride 8 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We train for 10 epochs with 8-epoch warm-up, learning rate 2e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' EPIC-Kitchens- 100 features use stride 16 [72] for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We train for 20 epochs with 16-epoch warm-up, learning rate 2e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We report an average of 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Self-supervised Supervised Supervised Ego4D MAE pretrain exo finetune ego finetune mAP K400 K400 (240K) Ego4D (14K) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 K400 Ego4D (14K) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Ego4D (14K) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Ego4D Ego4D (14K) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying the pretraining stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego-Only matches exocen- tric pretraining but with much fewer labels (14K vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 240K+14K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ablation Study In order to validate our Ego-Only pipeline, especially to compare with exocentric pretraining solutions, we first set up a strong exocentric baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Then, we compare with Ego-Only and ablate the importance of each stage in Ego- Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We also scale the amount of data consumed, the model sizes, as well as the number of pretraining epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We perform all ablation studies on Ego4D MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' A strong baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For fair comparison, we first build a strong exocentric pretraining baseline that achieves top per- formance [27] on Kinetics-400 classification [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specif- ically, this baseline is pretrained on Kinetics-400 with MAE [27] and finetuned on Kinetics-400 with 240K ex- ocentric classification labels [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Then, this exocentric pretrained checkpoint is further finetuned on Ego4D with the same temporal segmentation objective and recipe as Ego-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Finally, features are extracted and an Action- Former [72] is trained on the features with again exactly the same setting as Ego-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This strong exocentric pre- training baseline achieves 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% average mAP with ViT-B and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8% with ViT-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Next, we compare variants of our Ego-Only pipeline with the strong baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We ablate with ViT-B unless noted otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying the pretraining stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Table 2 reports our results with different pretraining stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Compared with the strong exocentric pretraining baseline, our Ego-Only with exactly the same backbone, the same finetuning, and the same de- tector, matches the performance of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% mAP by using egocentric data only and with merely 14K labels, instead of the 240K labels of the exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Next, we consider skipping the MAE pretraining and training from scratch the model via temporal segmenta- tion on egocentric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our best attempt for this strat- egy uses AdamW [51] with weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='05, layer decay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0, learning rate 2e-5, and 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, our best model learned from scratch only reaches the mAP of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% with MAE pretraining in Ego-Only), due to the limited number of labels available on Ego4D, only 14K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This is smaller than the number of labels in MNIST [40] or CIFAR [39] but the task of egocentric action detection is significantly more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Self-supervised Supervised Supervised Ego4D MAE pretrain exo finetune ego finetune mAP K400 K400 (240K) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 K400 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 Ego4D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 Ego4D Ego4D (14K) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying the finetuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 Temporal Span (seconds) 2 4 6 8 10 12 14 Average mAP (%) ActionFormer (finetuned features) ActionFormer (frozen MAE features) VSGN (finetuned features) Blob detector (finetuned features) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying detectors and temporal spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The blob detec- tor performs surprisingly well and perfers a long temporal span, while ActionFormer and VSGN prefer short spans due to their transformer or graph neural network based architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In addition to the model trained from scratch, we also compare with self-supervised MAE pretraining on Kinetics- 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' When this checkpoint is finetuned, it achieves 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4% mAP which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0% worse than the counterpart pretrained on Ego4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This gap is reasonable since the model is pre- trained on out-of-domain data but does not benefit from the large-scale exocentric labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Once the extra labels are used, Kinetics supervised pretraining yields performance on-par with our much simpler Ego-Only pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying the finetuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' After varying the pretrain- ing stage, we study the importance of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For this purpose, we extract features from pretrained models, with- out any form of finetuning on egocentric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Contrary to the strong linear probing results of MAE on ImageNet- 1K [22], we observe that frozen MAE features perform poorly on egocentric action detection, leading to an absolute drop of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6% points in average mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Kinetics-400 MAE features perform even worse (as expected), but finetuning on Kinetics with 240K labels is helpful, achieving a 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5% mAP which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9% worse than the same model finetuned on Ego4D and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9% worse than our Ego-Only pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We also try concatenating frozen MAE features from multiple blocks, inspired by DINO [9], but only observe a marginal gain (Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 200 400 800 Ego-Only pretraining epochs 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 Average mAP (%) ViT-L Ego-Only ViT-L Exocentric ViT-B Ego-Only ViT-B Exocentric Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Scaling models and pretraining epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' At around 800 epochs, our Ego-Only starts to match exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Detectors and temporal spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Next, we compare tempo- ral action detector choices in Ego-Only and vary the tem- poral span at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As we use a consistent tem- poral span for the whole pipeline, including MAE, finetun- ing, and feature extraction (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2), we pretrain MAE with each temporal span for 200 epochs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Then, we de- fine a simple baseline of a 1D blob detector [52] using the Laplacian of Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' To our surprise, as shown in Figure 4, this simple blob detection baseline achieves 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2% mAP which is already better than the Ego4D [29] paper baseline of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0% mAP with pretrained SlowFast [28] fea- tures and VSGN [75], thanks to the effectiveness of Ego- Only features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We also notice that the blob detector and the frozen MAE feature prefer a longer temporal span of 16 or 32 seconds, demonstrating the importance of long-term context in egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' On the other hand, VSGN [75] and ActionFormer [72] prefer short feature spans probably because the graph neural network or the transformer cap- tures long-term relations internally, benefiting more from local features that represent dense temporal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Finally, ActionFormer with finetuned features achieves the best re- sult of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9%, outperforming VSGN by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0% consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Scaling models and pretraining epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In addition to ablating the three stages in our Ego-Only pipeline, we also scale the model size from ViT-B to ViT-L and benchmark results under different computation budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We keep the relatively cheap finetuning of 20 epochs unchanged, but vary the MAE pretraining epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As shown in Figure 5, both ViT-B and ViT-L results improve consistently when they are pretrained longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' At around the budget of 800 epochs, our Ego-Only models start to match Kinetics-400 pretrained models with both ViT-B and ViT-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The Kinet- ics baselines, before finetuned on egocentric data, are pre- trained with 800-epoch MAE and 100/150 epoch exocentric finetuning that consumes not only more data and labels but also more computation resources than Ego-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego MAE pretrain Ego finetune mAP Random initialization (0h) 195h 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Ego4D MQ clips (195h) 195h 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 Ego4D MQ videos (487h) 195h 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 Ego4D EM videos (838h) 195h 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 Ego4D ALL videos (3560h) 195h 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Scaling the amount of pretraining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' MQ clips: all MQ training clips [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' MQ videos: all videos in the MQ task training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' EM videos: all videos in the Episodic Memory benchmark training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' ALL videos: all Ego4D videos except MQ val and test videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our Ego-Only results improve consistently with re- spect to the amount of data consumed in the pretraining stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' MAE Exo Ego Rebalancing Ego4D pretrain finetune finetune method mAP K400 K400 Ego4D Reweighting 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 K400 K400 Ego4D Resampling 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 Ego4D Ego4D Reweighting 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Ego4D Ego4D Resampling 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying rebalancing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego-Only still matches exocentric pretraining when a stronger rebalancing technique (ac- tion resampling in this case) is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Scaling the amount of pretraining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Beyond standard ablations on pretraining epochs, an intriguing dimension for study offered by the massive scale of Ego4D is the different amounts of large-scale unsupervised video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifi- cally, given the fixed amount of finetuning data, we select four subsets and amounts of unsupervised data in Ego4D to study the data scaling property of the Ego-Only pretrain- ing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Note that in all cases, we exclude val and test videos of the MQ task from the pretraining set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' All models are pretrained for 200 epochs instead of 800 epochs to save computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' From the results in Table 4, we see that the performance of Ego-Only consistently improves as more unsupervised data is provided for MAE pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Varying rebalancing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As discussed in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2, we are currently mitigating the imbalance chal- lenges by simply reweighting the loss according to the num- ber of positive frames in each action class, ignoring the vari- able action lengths within one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Beyond this current technique, we also study a simple action resampling option as a natural alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, instead of uniformly sampling all the clips within the train data, we sample only the center 2 seconds of each action regardless of the action length, similar to an action classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As shown in Table 5, this resampling option outperforms the default reweighting by around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8% mAP in both settings with and without exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In this case, our Ego-Only pipeline still matches Kinetics pretraining, without any exo- centric data or label, and regardless of the rebalancing tech- niques employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We consider further exploration of better rebalancing methods as an open research problem and leave it to future work beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Comparison on Ego4D We compare our results on the Ego4D [29] MQ val set with state-of-the-art methods in Table 6, using ViT-B and ViT-L, with and without exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We no- tice that our strong baseline performs better than the re- sults reported in the Ego4D paper with VSGN [75] and SlowFast [28] features, because of the stronger Action- Former [72] detector and the stronger ViT [24] pretrain- ing [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, Ego-Only is able to match this strong performance in a fair comparison but without any extra ex- ocentric data or label, in both ViT-B and ViT-L cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego- Only achieves the average mAP of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7%, producing a rel- ative improvement of 160% over the Ego4D paper base- line [29], and setting a new state-of-the-art on this bench- mark without any extra data or label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Comparison on EPIC-Kitchens-100 Following the exploration on Ego4D, we validate our Ego-Only approach on the EPIC-Kitchens-100 [21] Action Detection benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We can see from Table 7 that, on EPIC-Kitchens-100, Ego-Only achieves much stronger re- sults compared with exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, only when we evaluate on nouns and only when a small ViT-B is used, exocentric pretraining matches Ego-Only (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3%), possibly due to the bias of Kinetics towards scene classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Once we evaluate on verbs or the model is scaled to ViT-L, Ego-Only shows a significant absolute gain of 2% over exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This stronger scaling is probably due to more labels (67K) available than in Ego4D (14K) which allow the large trans- formers to generalize better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Compared with previous methods that adopt Kinet- ics [37] SlowFast [28] features finetuned on EPIC- Kitchens-100 Action Recognition, our baseline models with exocentric pretraining already performs better, regardless of the model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our Ego-Only approach improves further on top of the strong baselines and sets a new state-of-the-art result of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7% mAP on verbs and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1% mAP on nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Furthermore, we analyze our ViT-L results and baselines using DETAD [1] (Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2), and notice that Ego-Only significantly reduces false positives on backgrounds, com- pared with exocentric pretraining baselines, probably be- cause Kinetics contains mostly trimmed videos with fore- ground actions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This analysis validates the benefit of our Ego-Only approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Detector Backbone Extra data Extra labels # labels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 Average VSGN [75] SlowFast [28] Kinetics-400 [37] 240K 14K 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='03 VSGN [75] Frozen [3] IN-21K [22] + EgoClip [44] 14M + 4M 14K 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='63 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='57 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='39 ActionFormer† ViT-B Kinetics-400 240K 14K 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 ActionFormer ViT-B 14K 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 ActionFormer† ViT-L Kinetics-400 240K 14K 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 ActionFormer ViT-L 14K 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Comparing with the state-of-the-art on Ego4D MQ val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' †our strong exocentric pretraining baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Task Detector Backbone Extra data Extra labels # labels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 Average Verb BMN [21,45] SlowFast [28] K400 [37] 240K 67K 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 G-TAD [70] SlowFast [28] K400 [37] 240K 67K 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 ActionFormer [72] SlowFast [28] K400 [37] 240K 67K 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 ActionFormer† ViT-B K400 240K 67K 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 ActionFormer ViT-B 67K 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 ActionFormer† ViT-L K400 240K 67K 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 ActionFormer ViT-L 67K 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 Noun BMN [21,45] SlowFast [28] K400 [37] 240K 67K 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 G-TAD [70] SlowFast [28] K400 [37] 240K 67K 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 ActionFormer [72] SlowFast [28] K400 [37] 240K 67K 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 ActionFormer† ViT-B K400 240K 67K 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 ActionFormer ViT-B 67K 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 ActionFormer† ViT-L K400 240K 67K 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 ActionFormer ViT-L 67K 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Comparing with the state-of-the-art on EPIC-Kitchens-100 Action Detection val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' †our strong exocentric pretraining baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Transfer Learning Beyond comparing with state-of-the-art methods, we study the possibility of transferring representations learned from the massive-scale diverse Ego4D to a different ego- centric dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Specifically, we take EPIC-Kitchens-100 as the target data and compare ViT-L results with various pre- training setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' As shown in Table 8, we observe that none of the transferring model performs on par with MAE pre- trained on the target data, which is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, we notice that Ego4D pretraining transfers significantly better than Kinetics-400 pretraining when supervised exocentric finetuning is not available, suggesting a large domain gap between egocentric and exocentric datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In addition, our Ego4D pretraining even outperforms exocentric models finetuned using large-scale exocentric labels, further con- firming the effectiveness of Ego-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Conclusion In this work, we have shown for the first time that the cur- rent egocentric videos collected by the community are suf- Self-supervised Supervised Supervised Verbs Nouns MAE pretrain exo finetune ego finetune mAP mAP EPIC EPIC 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 K400 K400 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 K400 EPIC 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 K400 K400 EPIC 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 Ego4D EPIC 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Transfer learning from Ego4D outperforms Kinetics-400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' ficient to train a state-of-the-art egocentric action detector without any exocentric pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Our proposed Ego-Only simplifies the current learning pipeline by removing the pre- vious need for supervised pretraining on large-scale exocen- tric video or image datasets before transferring to egocen- tric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Instead, we hope our attempt inspires the com- munity to rethink the trade-off between training in-domain with ego-only data and transferring from out-of-domain ex- ocentric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We also hope that our Ego-Only results provide a strong baseline for future research that aims to improve egocentric learning by leveraging exocentric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ablation on Concatenated Features In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1, we present the ablation of concatenat- ing features from the last few (2, 3, 6, or 12) transformer blocks, instead of our default choice of the last block only (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This is inspired by the linear protocol in DINO [9] that was aimed to improve results with frozen self-supervised learning features (in our case frozen MAE features) but we ablate this choice for all models, with and without finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' However, we see a marginal gain for frozen MAE features, which confirms the necessity of the egocentric finetuning stage in Ego-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Error Analyses False positive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2, we analyze false positive errors on EPIC-Kitchens-100 [21] with ViT-L [24] models using the DETAD [1] error diagnosing tool (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We notice that Ego-Only reduces false positive er- rors on backgrounds, compared with exocentric pretraining baselines, probably because Kinetics [37] contains mostly trimmed videos with foreground actions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3, we analyze the model sensitivity according to DETAD characteristics [1] on EPIC-Kitchens-100 [21] with ViT-L [24] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We ob- serve that our Ego-Only improves significantly when there are multiple verb instances of the same category in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Visualization of MAE Reconstructions In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4, we visualize the MAE [27, 31] recon- struction results on a few Ego4D [29] examples with a ViT-B [24] trained for 200 epochs without per-patch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We notice that egocentric MAE learns human-object interactions (d,f,g,h,i,k) and temporal corre- spondence across frames (c,j), even in cases with strong head/camera motion (a,b,e,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 1 2 3 6 12 Number of concatenated blocks 0 2 4 6 8 10 12 14 16 Average mAP (%) Ego-Only (finetuned features) Exocentric (finetuned features) Ego-Only (frozen MAE features) Exocentric (frozen MAE features) Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ego4D Moments Queries results with concatenated features from the last few (2, 3, 6, 12) transformer blocks (12 blocks in total for the ViT-B [24] architecture), instead of our de- fault choice of the last block only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The detection results are almost not affected in any of the four models studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' This stable gap between finetuned features and frozen MAE features verifies the necessity of the egocentric finetuning stage in Ego-Only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' EPIC Exocentric pretraining (previous) Ego-Only (ours) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6% mAP 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7% mAP Verb 1G 2G 3G 4G 5G 6G 7G 8G 9G 10G Top Predictions 0 10 20 30 40 50 60 70 80 90 100 Error Breakdown (%) False Positive Profile Background Err Confusion Err Localization Err Wrong Label Err Double Detection Err True Positive Error Type 0 1 2 3 4 5 6 7 8 Average-mAPN Improvment (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 Removing Error Impact 1G 2G 3G 4G 5G 6G 7G 8G 9G 10G Top Predictions 0 10 20 30 40 50 60 70 80 90 100 Error Breakdown (%) False Positive Profile Background Err Confusion Err Localization Err Wrong Label Err Double Detection Err True Positive Error Type 0 1 2 3 4 5 6 7 8 9 Average-mAPN Improvment (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 Removing Error Impact 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3% mAP 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1% mAP Noun 1G 2G 3G 4G 5G 6G 7G 8G 9G 10G Top Predictions 0 10 20 30 40 50 60 70 80 90 100 Error Breakdown (%) False Positive Profile Background Err Confusion Err Localization Err Wrong Label Err Double Detection Err True Positive Error Type 0 1 2 3 4 5 6 7 8 Average-mAPN Improvment (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 Removing Error Impact 1G 2G 3G 4G 5G 6G 7G 8G 9G 10G Top Predictions 0 10 20 30 40 50 60 70 80 90 100 Error Breakdown (%) False Positive Profile Background Err Confusion Err Localization Err Wrong Label Err Double Detection Err True Positive Error Type 0 1 2 3 4 5 6 7 8 9 Average-mAPN Improvment (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 Removing Error Impact Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' False positive analysis on EPIC-Kitchens-100 [21] with DETAD [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The error types are determined by the tIoU between ground-truth and predicted segments, as well as the correctness of the predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Background error: tIoU < 1e-5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' confusion error: 1e-5 < tIoU < α and label is wrong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' localization error: label is correct but 1e-5 < tIoU < α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' wrong label error: tIoU >= α but label is wrong, where α refers to the tIoU thresholds {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' ‘G’ refers to the number of ground-truth instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' According to the error breakdown, although the large-scale exocentric pretraining helps reducing wrong label errors, our Ego-Only predicts more true positives correctly and reduces background errors, probably because Kinetics [37] contains mostly trimmed videos with foreground actions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' EPIC Exocentric pretraining (previous) Ego-Only (ours) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6% mAP 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7% mAP Verb XS S M L XL XS S M L XL 0 20 40 60 80 100 Average-mAPN (%) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='6 Length 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 # Instances XS S M L XL XS S M L XL 0 20 40 60 80 100 Average-mAPN (%) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 Length 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 # Instances 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3% mAP 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1% mAP Noun XS S M L XL XS S M L XL 0 20 40 60 80 100 Average-mAPN (%) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='0 Length 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='9 # Instances XS S M L XL XS S M L XL 0 20 40 60 80 100 Average-mAPN (%) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='7 Length 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='1 # Instances Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Sensitivity analysis on EPIC-Kitchens-100 [21] with DETAD [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Ground-truth segments are divided into 5 equal buckets according to their characteristic [1] percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Then, average mAPN [1] metrics are computed for each characteristic bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The ‘length’ characteristic measures the length of the ground-truth action segment in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The ‘# instances’ characteristic measures the number of action instances belonging to the same category as the ground-truth segment in the same video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' According to the average mAPN in each bucket, we observe that our Ego-Only improves significantly when there are multiple verb instances of the same category in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' MAE [27, 31] reconstruction results on Ego4D [29] MQ val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' For each sample, we show the original video (top), the randomly masked video (middle), and the MAE reconstruction (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We visualize 8 frames [27] out of 16 with a temporal stride of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' The model predicts RGB pixels without patch normalization with a masking ratio of 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' We notice that egocentric MAE learns human-object interactions (d,f,g,h,i,k) and temporal correspondence across frames (c,j), even in cases with strong head/camera motion (a,b,e,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' References [1] Humam Alwassel, Fabian Caba Heilbron, Victor Escorcia, and Bernard Ghanem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Diagnosing error in temporal action detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ECCV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 7, 9, 10, 11 [2] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luˇci´c, and Cordelia Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Vivit: A video vi- sion transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2 [3] Max Bain, Arsha Nagrani, G¨ul Varol, and Andrew Zisser- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Frozen in time: A joint video and image encoder for end-to-end retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2, 8 [4] Hangbo Bao, Li Dong, and Furu Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' BEiT: BERT pre- training of image transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ICLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2 [5] Gedas Bertasius, Heng Wang, and Lorenzo Torresani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Is space-time attention all you need for video understanding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ICML, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2 [6] Fabian Caba Heilbron, Victor Escorcia, Bernard Ghanem, and Juan Carlos Niebles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Activitynet: A large-scale video benchmark for human activity understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In CVPR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 1 [7] Mathilde Caron, Piotr Bojanowski, Armand Joulin, and Matthijs Douze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Deep clustering for unsupervised learning of visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ECCV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2 [8] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Pi- otr Bojanowski, and Armand Joulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Unsupervised learn- ing of visual features by contrasting cluster assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2 [9] Mathilde Caron, Hugo Touvron, Ishan Misra, Herv´e J´egou, Julien Mairal, Piotr Bojanowski, and Armand Joulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Emerg- ing properties in self-supervised vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2, 6, 9 [10] Joao Carreira and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Quo vadis, action recognition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' a new model and the kinetics dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In CVPR, 2017.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' In ICLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 5 [74] Chen Zhao, Merey Ramazanova, Mengmeng Xu, and Bernard Ghanem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' Segtad: Precise temporal action detection via semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' ECCVW, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} +page_content=' 2, 4 [75] Chen Zhao, Ali K Thabet, and Bernard Ghanem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf'} diff --git a/i9E1T4oBgHgl3EQfzwXg/content/tmp_files/2301.03449v1.pdf.txt b/i9E1T4oBgHgl3EQfzwXg/content/tmp_files/2301.03449v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d37931c5e90ad04ad4bf56cc672b7f756759a1c3 --- /dev/null +++ b/i9E1T4oBgHgl3EQfzwXg/content/tmp_files/2301.03449v1.pdf.txt @@ -0,0 +1,427 @@ +1 + +Edges fractal approach in graphene – defects density gain +I. Janowska,a* M. Lafjah,a V. Papaefthymiou,a S. Pronkin,a C. Ulhaq-Bouilletb +aInstitut de Chimie et Procédés pour l’Énergie, l'Environnement et la Santé (ICPEES), CNRS +UMR 7515-University of Strasbourg, France +bInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS UMR 7504- +University of Strasbourg, France +*Corresponding author: janowskai@unistra.fr, tel: 33 (0)368852633 + To optimize the technological development together with energy gain, the crucial +materials needs to be tailored in a smart way to benefit a maximum from the property- +structure relation. Herein we propose the way to increase a defect density in graphene +moving from 1D linear edges to fractal edges, also highlight the fractal nature of +graphene structures them-self. The tentative oxidation and hydrogenation catalytic +etching of few layer graphene results in structures with edges fractals, jaggy edges at +graphene periphery or in newly formed etched holes. + + The sustainable development emerging from accelerating societal requirements is a driving +force for the recent progress in green technologies and crucial (nano)materials. To face up the +economical and environmental issues a smart choice of crucial materials including their +abundance, price, production and appropriate tailoring is necessary. + Due to diver, often baronial and combined properties, the important materials of choice are +graphitic materials and especially 2D graphene or few layer graphene and their composites. +Although their intrinsic features are mostly related to the C=C conjugated honeycomb lattice, + +2 + +their +efficient +use +calls +for +application +dependent +tailoring +of +their +chemical/electronic/geometry and are often related to the presence of defects. Different +structural defects (Stone-Wales, vacancy, point, linear, edges), oxygen, hydrogen +functionalities or doped lattice heteroatoms induce a positive or penalizing effect and are +intended to be removed or introduced, consequently. The intentional introduction of defects +aims often to improve or even provide the reactivity (i.e. metal-free catalyst), hydrophylicity, +(opto) electronic, magnetic or sensing properties among others [1]. The important defects are +graphene edges, the linear defects usually introduced by a catalytic etching (cutting) of +graphene lattice. Since the etching of graphite is known for dozen of years, its recent revival +concerns mostly the hydrogenation etching with transition metal NPs (Ni, Co, Fe) to reach the +nanoribbons or dots possessing well defined “arm-chair” or “zig-zag” edges conformation +[2,3]. The preferential etching direction is determined by graphene lattice crystallography and +no matter the direction is, the most reported work focus on straight line channels formation if +not bearing in mind the arm-chair” or “zig-zag conformation at atomic level related to sp2 +hybridized C atom geometry. +The synthesized and tailored graphitic materials belong typically to Euclidean geometry (1D +tubes or ribbons, 0D fullerens or dots, 2D graphenes, 3D graphites), meanwhile an expanded +in 2D space the hexagonal structure in perfect graphene can already be considered as a +hierarchical multifractal [4]. Only one group reported at presence the catalytic in-situ fractal +etching of CVD-graphene [5], and it is much worthy to urgently take care for non-integer +dimension of graphenes. A defined by Mandelbrot “fractals” are indeed omnipresent in +natural matter geometry, physicochemical phenomena, mathematical analysis methods, and +fractal design is more and more explored. The scarce reports link to graphene. Very recently, +a fractal quantum Hall effect in graphene-BN heterostructure and structure-due tenfold +enhancement of photocurrent generation in fractal metasurface graphene designed by e-beam + +3 + +lithography, were observed [6, 7]. The fractal approach was also applied for the formation of +conductive self-assembled fractal branched patterns, with reduced percolation threshold for a +given substrate surface [8]. + Here, we highlight the fractal feature of graphene and exemplary possibilities offered by +graphene structure in term of its fractal geometry pattering but primary we focus on the +patterning of the graphene edges. Essentially, we propose a new look on the graphene defects +(edges), which can have space expanding fractal dimension (and can have mixed zig-zag and +arm-chair conformations at atomic level). We claim that the density of active sites in graphene +can be strongly enhanced not only by increasing the edges number but increasing it in +nonlinear manner to reach, “jaggy”, fractal edges. The concept reflects a known fractal +expansion of 1D line in 2D plane, with exemplary illustrated in fig. 1a. Such successive +expansion (from I to III) increases the length of the line at the distance from point A to B. +Let’s imagine now these lines are the edges of graphene, in other words: active sites. The +number of active sites does not increase by multiplication (or elongation) of straight edges, as +it is a case of most of the etched channels, but by increasing their density in 2D plane. To +achieve the length of edge III (or the same active sites number) by simple elongation of edge I +the latter needs to be elongated almost twice. The examples of such pattering in nature are +common including the plants’ leaves (fig. 1b). + +4 + + +Fig. 1 a) Schema of exemplary fractal expansion of a 1D line, b) optical photo of trees’ leaves +with eye visible jaggy edges (left, right) and fractal edges/structure (right). +While the reported etching of graphene mostly looks for straight line channeling, any +asymmetry of the etched lines, rarely discussed, is seen by us as benefic for active site density +enhancement. We have already observed earlier some partially asymmetric, “step-like”, edges +in a case of FeNx NPs assisted etching, were the “creep” movement of the faceted active FeNx +NP occurred [3]. The intended and controlled expansion of the edges through the right +experimental approach would be however highly suitable. + At present we show the results of tentative graphene etching, which proceeds in a way to +provide the graphene structures without the pronounced channels but with asymmetric, +strongly “jagged” edges. For this purpose, few layer graphene (FLG) flakes obtained +A +B +I +II +III +a +b + +5 + +previously by liquid exfoliation method, of lateral size ~3 µm and decorated with Fe3-xO4 (SI) +was submitted to the oxidative treatment with air flow under atmosphere pressure at low +temperature (300°C) for 0.5h. Next to this, the temperature was increased to 800°C under Ar +and hydrogenation treatment with H2/Ar flow was applied. Fig. 2 a shows TEM micrographs +of the Fe-based/FLG structures after the performed treatment. For the reason of comparison, +the results from the etching running under “standard” hydrogenation conditions at 800°C with +the same catalytic system are also presented in fig. 2 b. The difference between the samples +submitted to oxygen assisted and purely hydrogen assisted treatments is clear. While in the +case of “standard” hydrogenation the straight line channels with the active NPs at their ends +are formed in the FLG flakes under well defined angles directions to periphery edges (often +perpendicularly) (fig. 2b), the addition of pre-oxidation process results in the flakes sheets +with “jaggy” edges and the channels are quasi absent (fig. 2a). + +6 + + +Fig. 2 TEM micrographs of a) FLG with jagged edges obtained after oxidation, hydrogenation +treatment, b) FLG with straight edges etched by hydrogenation +1 µm +1 µm +1 µm +0.5 µm +b +a + +2.μm7 + +At the moment it is difficult to make a final statement about the impact of the atomic oxygen +potentially produced from the dissociative chemisorptions of molecular oxygen on NPs +surface and of the NPs themselves. It can be found indeed in the literature that in the case of +iron oxide, a dissociation and recombination of oxygen can take place at the temperature even +lower then oxidation of NPs. The recent work concerning the investigation of oxidation of +graphene supported on Ir(111) also claimed the role of atomic oxygen, which production was +facilitated by Ir(111) at the graphene edges [9]. Those oxidized edges were however only +slightly “jagged”, rather wavy, at lower scale and degree that the ones presented here. Other +thing is that we did not observe any oxidative channeling of FLG Fe3-xO4 NPs after oxygen +treatment at 300°C. We suppose then that the strongly asymmetric “jaggy” edges obtained +with O2 + H2/Ar treatments are the effect of both the oxidation through atomic oxygen and +hydrogenation process. The lack of numerous visible typical long straight channels after the +subsequent hydrogenation can be explained by strongly oxidized edges, excessive coalescence +of NPs and rapid etching. All together result in fast etching and sometimes in total +gasification of the etched path. If then the NPs are larger than the thickness of FLG, the flakes +cut in small pieces can be observed from time to time (TEM, SI). This confirms the general +statement of the etching proceeding easier at more defected/oxygen rich carbon, the reason +also why the channeling in general initiates at graphene edges. The primary importance is +however the fact that most of the herein etching directions are not well oriented by +crystallographic lattice of graphene, which is breached, and NPs change often their trajectory +adopting the creep movement due to modified interactions with oxidized edges. Contrary then +to several works investigating the oxidation of graphene (graphite) as a thermal/air stability +drawback; we see the controlled oxidation as support for defect density rich tailoring of +graphene planar structures. + +8 + +Since the edges are “jugged” at various degrees we can consider them as fractal edges, +according to description of schema 1, in which the equilateral triangular extension may be +replaced by less symmetric or others geometry analogue such as for example trapeze. + +Fig. 3 a) TEM micrographs of FLG with periphery jagged edges and formed holes, also with +jagged edges obtained after oxidation, hydrogenation treatment b) representative TEM +micrograph of jagged FLG edge, c) schema of benzenoid with rotary symmetry of n = 4. + +Apart from the periphery jaggy edges, the etched large holes, also possessing jagged edges +have been observed (fig. 3a). They formation is benefic for additional enhancement of the +edges-to-plane ratio and so increasing of defects density, as well as for introducing the +porosity in the bulk material. Fig. 3b shows representative TEM micrographs of the edges. +1 µm +1 µm +a +b +c + +10nm9 + +The fractal edges form part of “surface fractal”, where the surface corresponds to “surface of +edges” as analog to other surfaces, for instance, the chemically active surface of metal NPs. In +this context and from a chemistry point of view, the well crystallized graphene planar sheet +are rather a platform for defects active sites, especially in few layer graphene, similarly to +bulk of NPs (obviously the mutual interactions across C=C conjugated planar lattice and +defects exist). Due to however the much higher accessibility of the graphene plane +(monolayer) towards reactants compared to the NPs bulk and the fact that the word “surface +“is firmly linked to 2D planar graphene geometry, the formulation “surface fractals” for the +“edges” would be misunderstood. We call it then “edge fractal” or more generally “defect +fractal”. +The significant amount of this “defected” carbon was confirmed by XPS analysis (fig. 4a, b) +while the improved activity nature of the “jaggy edges” rich FLG was shortly investigated by +the electrochemical approach in aqueous electrolyte (fig. 4c), both referred to the FLG with +straight etched channels. The enhanced contribution of the “non-graphitic” carbon related to +the high density of edge defects is reflected by enormous broadening of the C1s XPS peak +towards higher binding energies (4a vs. 4b). The full width at half maximum (FWHM) is +almost twice (2.8 eV /1.5 eV) with significantly increased contributions from sp3carbon and +oxygen – bound carbon, especially of C-OR and COO type groups (SI). +The activity of these defects/groups is revealed by enhanced electrochemical surface area +(ECSA), times fourteen (3.88 vs 0.28 cm2 / 1 cm2 substrate), determined by cyclic +voltammetry measurements in aqueous 1M NaOH electrolytes (fig. 4c, SI). For the same mass +of deposited material, the double-layer capacitance of “jaggy edges” FLG is thirty times +higher (3.3 F/g vs. 0.14 F/g). The obtained values of specific ESCA and capacitance are small +due to the π-π stacking of the FLG flakes (and presence of Nafion). The interlayer stacking is +a common and well known drawback and might be overpassed by an appropriate formation of + +10 + +3D structures in the final applications, which is not however the goal of the present work. In +turn, the enhancement of ESCA and capacitance in “edges fractal” FLG clearly reveals the +presence of active and more hydrophilic carbon, and possibly of the increased porosity. These +provide larger interface between FLG and electrolyte, still keeping the charge transport ability +through FLG sp2C lattice. + +294 +292 +290 +288 +286 +284 +282 + + +C 1s +Binding Energy (eV) +XPS Intensity (a.u.) + + +sp2C +sp3C +C-OR +O-C=O +C= O +π-π* +def.C +a +b +c +-1,0 +-0,8 +-0,6 +-0,4 +-0,2 +0,0 +-0,02 +0,00 +0,02 +0,04 +I /mA +E/V Hg/HgO + +11 + +Fig. 4 a,b) Experimental and deconvoluted C1s XPS spectra of (a) FLG with straight etched +channels and (b) “jaggy” edges FLG; c) their cyclic voltammetry curves (jagged FLG-black +curve) +More than “defect fractal” approach, we would like to herein refer to the fractal nature of +graphene it-self and notice that obtained here graphene structures with fractal edges are not +far from the model of graphene macro-molecule fractal, which is a benzenoid with rotary +symmetry of higher fractal order. The fig. 3c presents such benzenoid with fractal order 4 (n = +4). Let’s remind that the fractal configuration will depend on the way the self-similar +“circular” units of benzene rings expend in 2D plane. Two different circularly arranged +benzenoids structures with a fractal order of 1, 2 and 3 are presented in fig.5 a, b [4, 10]. +Clearly, the symmetric structures depicted in fig. 5a reflect Koch snow flake, where triangle +corner is replaced by trapeze configuration corresponding to the “arm-chair” conformation of +C=C. + +12 + + +Fig. 5 a, b) schemes of benzenoids structures with increasing fractal order with (a) symmetry +(coronene) and (b) rotational symmetry, c) exponential increase of the structures diameter and +their periphery edges with fractal order. +The arrangement of the units in the structure b cause the rotational symmetry with the +additional counterclockwise rotating angle of 19.1° each time the fractal order increases [4]. +Focalizing however on the edges we underline the fact that they increase exponentially and +together with the fractal order similarly to diameter of the structures (faster for symmetric +structures), and the edges are mixed. Funnily, if we consider the fractal approach for graphene +planar lattice and its related properties, the reasoning of dimension reduction from 2D is right; +in the case of edge fractal, the 1D dimension increases. The “defect fractal” includes also the +n = 1 +n = 2 +n = 3 +Rotational symmetry (19.1 ) +Symmetry +a +b +R² = 1 +0 +1000 +2000 +3000 +4000 +5000 +6000 +2 +4 +6 +8 +10 +lateral diameter (nm) +fractal order (n) +symm. +rot-symm. +Expon. (symm.) +Expon. (rot- +symm.) +R² = 1 +0 +40 +80 +120 +160 +1 +2 +3 +4 +peripheryedges length (nm) +fractal order (n) +symm. (full) +rot.-symm. (full) +Expon. (rot.-symm. +(full)) +c + +13 + +holes in the model macromolecules, which are formed by “the luck” of benzenoids units in +the plane having as well nonlinearly increasing edges as illustrated in the in fig. 5 and in SI. +The illustrated here schema of benzenoids, coroenenes including (fig.5), are up to now the +theoretical “idealistic” relatively small structures and their preparation through a total organic +synthesis would be difficult but easily controlled. Other way to achieve such structures at high +scale (omitting a laser or lithography patterning) would be a well controlled catalytic etching. +The latter route seems also be more realistic for the structures with higher size (higher fractal +order). + In order to explain the exact process of the edges etching and to well control the final +structures, the additional efforts needs to be undertaken and the supporting in-situ TEM and +XPS observations are considered for this purpose in future. The process to get “jagged” but +still sharp and well defined edges is complex and both, oxygen pretreatment and hydrogen are +necessary. At this moment it is difficult to determine the impact of the enhanced oxygenation +degree of FLG once the temperature is increased and hydrogen is supplied. Regardless the +oxidation with atomic oxygen and assuming that C-metal interactions are important factor, +different phenomena can take place during the etching. These are NPs coalescence, splitting, +melting, faceting and structure reorganization [3, 11, 12], which, as chemistry of the NPs- +FLG hybrid changes under etching process (iron, iron oxides and carbide phase) [13], will +depend not only on the etching conditions but also on the preparation and pretreatments of the +hybrid. We suggest that the pre-oxidized edges of FLG have increased affinity towards the +NPs inducing a possible splitting of the latter and allowing for fine-grained jaggy etching due +to the small size NPs (or their quasi-melt phase), while the etching of more defected (active) +carbon it-self happens easier. From the other hand, the faceting is expected especially for +larger NPs as observed in “post-mortem” catalyst and has an impact on the edges structure in +general. + +14 + + The presented concept of etching to edge fractal structures is different from the one reported +for the in-situ CVD synthesized graphene via Cu catalyst, where either hexagonal structures +or sharp thin highly etched branched structures were obtained [14, 15]. In herein approach, the +significant area of honeycomb structure remains robust while the edges are highly etched. +This leads to the bulk materials with still significant planar surface, related high mechanical +resistance and platform for charge transport, but also great density of nonlinearly etched edges +active sites. The more controlled etching would entail not only controlled defect density but +also surface area and porosity (size, volume, distribution). Such tailoring and activation of +large and bulky graphene structures are benefic for many applications including the electrodes +(supercapacitors, fuel cells), where enhanced (electro)catalytic activity, appropriate porosity, +surface area and electrode-electrolyte interface interactions, but also sufficient conductive +platform for efficient charge transport, are reached. Number of other fields would gain from +controlled fractal edges tailoring, not to mention the electronic, optical, sensing or magnetic +properties, especially in the case of well defined smaller fractal order structures. The efforts +for their preparation and theoretical calculations should be then made. The in-situ CVD- +fractal etching of the individual graphene sheets would seem to be easier way for the etching +control that the etching of “bulk” graphene, however, the in-situ CVD-fractal etching over +catalytic substrate has similar advantages and drawbacks as CVD synthesis of graphene. +Despite the easier control of the final patterns, the use it for “bulk” applications is strongly +scale limited along with the substrate transfer issue, which would be much harder in the case +of etched structures. +Autor contributions: I. Janowska: principal author, project investigator and coordinator, +writing; M. Lafjah: initial experiment, V. Papaefthymiou: XPS analysis, S. 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Mater. 2015; 27: 4195– +4199. + + + + diff --git a/i9E1T4oBgHgl3EQfzwXg/content/tmp_files/load_file.txt b/i9E1T4oBgHgl3EQfzwXg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5061a40e7774c9fe4c7ab10e0efbb5abd24b4494 --- /dev/null +++ b/i9E1T4oBgHgl3EQfzwXg/content/tmp_files/load_file.txt @@ -0,0 +1,249 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf,len=248 +page_content='1 Edges fractal approach in graphene – defects density gain I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Janowska,a* M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Lafjah,a V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Papaefthymiou,a S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Pronkin,a C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=" Ulhaq-Bouilletb aInstitut de Chimie et Procédés pour l’Énergie, l'Environnement et la Santé (ICPEES), CNRS UMR 7515-University of Strasbourg, France bInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS UMR 7504- University of Strasbourg, France *Corresponding author: janowskai@unistra." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='fr, tel: 33 (0)368852633 To optimize the technological development together with energy gain, the crucial materials needs to be tailored in a smart way to benefit a maximum from the property- structure relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Herein we propose the way to increase a defect density in graphene moving from 1D linear edges to fractal edges, also highlight the fractal nature of graphene structures them-self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The tentative oxidation and hydrogenation catalytic etching of few layer graphene results in structures with edges fractals, jaggy edges at graphene periphery or in newly formed etched holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The sustainable development emerging from accelerating societal requirements is a driving force for the recent progress in green technologies and crucial (nano)materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' To face up the economical and environmental issues a smart choice of crucial materials including their abundance, price, production and appropriate tailoring is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Due to diver, often baronial and combined properties, the important materials of choice are graphitic materials and especially 2D graphene or few layer graphene and their composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Although their intrinsic features are mostly related to the C=C conjugated honeycomb lattice, 2 their efficient use calls for application dependent tailoring of their chemical/electronic/geometry and are often related to the presence of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Different structural defects (Stone-Wales, vacancy, point, linear, edges), oxygen, hydrogen functionalities or doped lattice heteroatoms induce a positive or penalizing effect and are intended to be removed or introduced, consequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The intentional introduction of defects aims often to improve or even provide the reactivity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' metal-free catalyst), hydrophylicity, (opto) electronic, magnetic or sensing properties among others [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The important defects are graphene edges, the linear defects usually introduced by a catalytic etching (cutting) of graphene lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Since the etching of graphite is known for dozen of years, its recent revival concerns mostly the hydrogenation etching with transition metal NPs (Ni, Co, Fe) to reach the nanoribbons or dots possessing well defined “arm-chair” or “zig-zag” edges conformation [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The preferential etching direction is determined by graphene lattice crystallography and no matter the direction is, the most reported work focus on straight line channels formation if not bearing in mind the arm-chair” or “zig-zag conformation at atomic level related to sp2 hybridized C atom geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The synthesized and tailored graphitic materials belong typically to Euclidean geometry (1D tubes or ribbons, 0D fullerens or dots, 2D graphenes, 3D graphites), meanwhile an expanded in 2D space the hexagonal structure in perfect graphene can already be considered as a hierarchical multifractal [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Only one group reported at presence the catalytic in-situ fractal etching of CVD-graphene [5], and it is much worthy to urgently take care for non-integer dimension of graphenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' A defined by Mandelbrot “fractals” are indeed omnipresent in natural matter geometry, physicochemical phenomena, mathematical analysis methods, and fractal design is more and more explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The scarce reports link to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Very recently, a fractal quantum Hall effect in graphene-BN heterostructure and structure-due tenfold enhancement of photocurrent generation in fractal metasurface graphene designed by e-beam 3 lithography, were observed [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The fractal approach was also applied for the formation of conductive self-assembled fractal branched patterns, with reduced percolation threshold for a given substrate surface [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Here, we highlight the fractal feature of graphene and exemplary possibilities offered by graphene structure in term of its fractal geometry pattering but primary we focus on the patterning of the graphene edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Essentially, we propose a new look on the graphene defects (edges), which can have space expanding fractal dimension (and can have mixed zig-zag and arm-chair conformations at atomic level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' We claim that the density of active sites in graphene can be strongly enhanced not only by increasing the edges number but increasing it in nonlinear manner to reach, “jaggy”, fractal edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The concept reflects a known fractal expansion of 1D line in 2D plane, with exemplary illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Such successive expansion (from I to III) increases the length of the line at the distance from point A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Let’s imagine now these lines are the edges of graphene, in other words: active sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The number of active sites does not increase by multiplication (or elongation) of straight edges, as it is a case of most of the etched channels, but by increasing their density in 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' To achieve the length of edge III (or the same active sites number) by simple elongation of edge I the latter needs to be elongated almost twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The examples of such pattering in nature are common including the plants’ leaves (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 1 a) Schema of exemplary fractal expansion of a 1D line, b) optical photo of trees’ leaves with eye visible jaggy edges (left, right) and fractal edges/structure (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' While the reported etching of graphene mostly looks for straight line channeling, any asymmetry of the etched lines, rarely discussed, is seen by us as benefic for active site density enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' We have already observed earlier some partially asymmetric, “step-like”, edges in a case of FeNx NPs assisted etching, were the “creep” movement of the faceted active FeNx NP occurred [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The intended and controlled expansion of the edges through the right experimental approach would be however highly suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' At present we show the results of tentative graphene etching, which proceeds in a way to provide the graphene structures without the pronounced channels but with asymmetric, strongly “jagged” edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' For this purpose, few layer graphene (FLG) flakes obtained A B I II III a b 5 previously by liquid exfoliation method, of lateral size ~3 µm and decorated with Fe3-xO4 (SI) was submitted to the oxidative treatment with air flow under atmosphere pressure at low temperature (300°C) for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='5h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Next to this, the temperature was increased to 800°C under Ar and hydrogenation treatment with H2/Ar flow was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 2 a shows TEM micrographs of the Fe-based/FLG structures after the performed treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' For the reason of comparison, the results from the etching running under “standard” hydrogenation conditions at 800°C with the same catalytic system are also presented in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 2 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The difference between the samples submitted to oxygen assisted and purely hydrogen assisted treatments is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' While in the case of “standard” hydrogenation the straight line channels with the active NPs at their ends are formed in the FLG flakes under well defined angles directions to periphery edges (often perpendicularly) (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 2b), the addition of pre-oxidation process results in the flakes sheets with “jaggy” edges and the channels are quasi absent (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 2 TEM micrographs of a) FLG with jagged edges obtained after oxidation, hydrogenation treatment, b) FLG with straight edges etched by hydrogenation 1 µm 1 µm 1 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='5 µm b a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='μm7 At the moment it is difficult to make a final statement about the impact of the atomic oxygen potentially produced from the dissociative chemisorptions of molecular oxygen on NPs surface and of the NPs themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' It can be found indeed in the literature that in the case of iron oxide, a dissociation and recombination of oxygen can take place at the temperature even lower then oxidation of NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The recent work concerning the investigation of oxidation of graphene supported on Ir(111) also claimed the role of atomic oxygen, which production was facilitated by Ir(111) at the graphene edges [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Those oxidized edges were however only slightly “jagged”, rather wavy, at lower scale and degree that the ones presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Other thing is that we did not observe any oxidative channeling of FLG Fe3-xO4 NPs after oxygen treatment at 300°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' We suppose then that the strongly asymmetric “jaggy” edges obtained with O2 + H2/Ar treatments are the effect of both the oxidation through atomic oxygen and hydrogenation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The lack of numerous visible typical long straight channels after the subsequent hydrogenation can be explained by strongly oxidized edges, excessive coalescence of NPs and rapid etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' All together result in fast etching and sometimes in total gasification of the etched path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' If then the NPs are larger than the thickness of FLG, the flakes cut in small pieces can be observed from time to time (TEM, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' This confirms the general statement of the etching proceeding easier at more defected/oxygen rich carbon, the reason also why the channeling in general initiates at graphene edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The primary importance is however the fact that most of the herein etching directions are not well oriented by crystallographic lattice of graphene, which is breached, and NPs change often their trajectory adopting the creep movement due to modified interactions with oxidized edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Contrary then to several works investigating the oxidation of graphene (graphite) as a thermal/air stability drawback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' we see the controlled oxidation as support for defect density rich tailoring of graphene planar structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 8 Since the edges are “jugged” at various degrees we can consider them as fractal edges, according to description of schema 1, in which the equilateral triangular extension may be replaced by less symmetric or others geometry analogue such as for example trapeze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 3 a) TEM micrographs of FLG with periphery jagged edges and formed holes, also with jagged edges obtained after oxidation, hydrogenation treatment b) representative TEM micrograph of jagged FLG edge, c) schema of benzenoid with rotary symmetry of n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Apart from the periphery jaggy edges, the etched large holes, also possessing jagged edges have been observed (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' They formation is benefic for additional enhancement of the edges-to-plane ratio and so increasing of defects density, as well as for introducing the porosity in the bulk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 3b shows representative TEM micrographs of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 1 µm 1 µm a b c 10nm9 The fractal edges form part of “surface fractal”, where the surface corresponds to “surface of edges” as analog to other surfaces, for instance, the chemically active surface of metal NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' In this context and from a chemistry point of view, the well crystallized graphene planar sheet are rather a platform for defects active sites, especially in few layer graphene, similarly to bulk of NPs (obviously the mutual interactions across C=C conjugated planar lattice and defects exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Due to however the much higher accessibility of the graphene plane (monolayer) towards reactants compared to the NPs bulk and the fact that the word “surface “is firmly linked to 2D planar graphene geometry, the formulation “surface fractals” for the “edges” would be misunderstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' We call it then “edge fractal” or more generally “defect fractal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The significant amount of this “defected” carbon was confirmed by XPS analysis (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 4a, b) while the improved activity nature of the “jaggy edges” rich FLG was shortly investigated by the electrochemical approach in aqueous electrolyte (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 4c), both referred to the FLG with straight etched channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The enhanced contribution of the “non-graphitic” carbon related to the high density of edge defects is reflected by enormous broadening of the C1s XPS peak towards higher binding energies (4a vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The full width at half maximum (FWHM) is almost twice (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='8 eV /1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='5 eV) with significantly increased contributions from sp3carbon and oxygen – bound carbon, especially of C-OR and COO type groups (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The activity of these defects/groups is revealed by enhanced electrochemical surface area (ECSA), times fourteen (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='88 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='28 cm2 / 1 cm2 substrate), determined by cyclic voltammetry measurements in aqueous 1M NaOH electrolytes (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 4c, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' For the same mass of deposited material, the double-layer capacitance of “jaggy edges” FLG is thirty times higher (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='3 F/g vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='14 F/g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The obtained values of specific ESCA and capacitance are small due to the π-π stacking of the FLG flakes (and presence of Nafion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The interlayer stacking is a common and well known drawback and might be overpassed by an appropriate formation of 10 3D structures in the final applications, which is not however the goal of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' In turn, the enhancement of ESCA and capacitance in “edges fractal” FLG clearly reveals the presence of active and more hydrophilic carbon, and possibly of the increased porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' These provide larger interface between FLG and electrolyte, still keeping the charge transport ability through FLG sp2C lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 294 292 290 288 286 284 282 C 1s Binding Energy (eV) XPS Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=') sp2C sp3C C OR O C=O C= O π π def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='C a b c 1,0 0,8 0,6 0,4 0,2 0,0 0,02 0,00 0,02 0,04 I /mA E/V Hg/HgO 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 4 a,b) Experimental and deconvoluted C1s XPS spectra of (a) FLG with straight etched channels and (b) “jaggy” edges FLG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' c) their cyclic voltammetry curves (jagged FLG-black curve) More than “defect fractal” approach, we would like to herein refer to the fractal nature of graphene it-self and notice that obtained here graphene structures with fractal edges are not far from the model of graphene macro-molecule fractal, which is a benzenoid with rotary symmetry of higher fractal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 3c presents such benzenoid with fractal order 4 (n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Let’s remind that the fractal configuration will depend on the way the self-similar “circular” units of benzene rings expend in 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Two different circularly arranged benzenoids structures with a fractal order of 1, 2 and 3 are presented in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='5 a, b [4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Clearly, the symmetric structures depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 5a reflect Koch snow flake, where triangle corner is replaced by trapeze configuration corresponding to the “arm-chair” conformation of C=C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 5 a, b) schemes of benzenoids structures with increasing fractal order with (a) symmetry (coronene) and (b) rotational symmetry, c) exponential increase of the structures diameter and their periphery edges with fractal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The arrangement of the units in the structure b cause the rotational symmetry with the additional counterclockwise rotating angle of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='1° each time the fractal order increases [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Focalizing however on the edges we underline the fact that they increase exponentially and together with the fractal order similarly to diameter of the structures (faster for symmetric structures), and the edges are mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Funnily, if we consider the fractal approach for graphene planar lattice and its related properties, the reasoning of dimension reduction from 2D is right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' in the case of edge fractal, the 1D dimension increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The “defect fractal” includes also the n = 1 n = 2 n = 3 Rotational symmetry (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='1 ) Symmetry a b R² = 1 0 1000 2000 3000 4000 5000 6000 2 4 6 8 10 lateral diameter (nm) fractal order (n) symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' rot-symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Expon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' (symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=') Expon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' (rot- symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=') R² = 1 0 40 80 120 160 1 2 3 4 peripheryedges length (nm) fractal order (n) symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' (full) rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='-symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' (full) Expon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' (rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='-symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' (full)) c 13 holes in the model macromolecules, which are formed by “the luck” of benzenoids units in the plane having as well nonlinearly increasing edges as illustrated in the in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 5 and in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The illustrated here schema of benzenoids, coroenenes including (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content='5), are up to now the theoretical “idealistic” relatively small structures and their preparation through a total organic synthesis would be difficult but easily controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Other way to achieve such structures at high scale (omitting a laser or lithography patterning) would be a well controlled catalytic etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The latter route seems also be more realistic for the structures with higher size (higher fractal order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' In order to explain the exact process of the edges etching and to well control the final structures, the additional efforts needs to be undertaken and the supporting in-situ TEM and XPS observations are considered for this purpose in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The process to get “jagged” but still sharp and well defined edges is complex and both, oxygen pretreatment and hydrogen are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' At this moment it is difficult to determine the impact of the enhanced oxygenation degree of FLG once the temperature is increased and hydrogen is supplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Regardless the oxidation with atomic oxygen and assuming that C-metal interactions are important factor, different phenomena can take place during the etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' These are NPs coalescence, splitting, melting, faceting and structure reorganization [3, 11, 12], which, as chemistry of the NPs- FLG hybrid changes under etching process (iron, iron oxides and carbide phase) [13], will depend not only on the etching conditions but also on the preparation and pretreatments of the hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' We suggest that the pre-oxidized edges of FLG have increased affinity towards the NPs inducing a possible splitting of the latter and allowing for fine-grained jaggy etching due to the small size NPs (or their quasi-melt phase), while the etching of more defected (active) carbon it-self happens easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' From the other hand, the faceting is expected especially for larger NPs as observed in “post-mortem” catalyst and has an impact on the edges structure in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' 14 The presented concept of etching to edge fractal structures is different from the one reported for the in-situ CVD synthesized graphene via Cu catalyst, where either hexagonal structures or sharp thin highly etched branched structures were obtained [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' In herein approach, the significant area of honeycomb structure remains robust while the edges are highly etched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' This leads to the bulk materials with still significant planar surface, related high mechanical resistance and platform for charge transport, but also great density of nonlinearly etched edges active sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The more controlled etching would entail not only controlled defect density but also surface area and porosity (size, volume, distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Such tailoring and activation of large and bulky graphene structures are benefic for many applications including the electrodes (supercapacitors, fuel cells), where enhanced (electro)catalytic activity, appropriate porosity, surface area and electrode-electrolyte interface interactions, but also sufficient conductive platform for efficient charge transport, are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Number of other fields would gain from controlled fractal edges tailoring, not to mention the electronic, optical, sensing or magnetic properties, especially in the case of well defined smaller fractal order structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The efforts for their preparation and theoretical calculations should be then made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' The in-situ CVD- fractal etching of the individual graphene sheets would seem to be easier way for the etching control that the etching of “bulk” graphene, however, the in-situ CVD-fractal etching over catalytic substrate has similar advantages and drawbacks as CVD synthesis of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Despite the easier control of the final patterns, the use it for “bulk” applications is strongly scale limited along with the substrate transfer issue, which would be much harder in the case of etched structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Autor contributions: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Janowska: principal author, project investigator and coordinator, writing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Lafjah: initial experiment, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Papaefthymiou: XPS analysis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Pronkin: electrochemical analysis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Ulhaq-Bouillet: TEM microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Supplementary Information: The methods are available 15 References [1] Shen A, Zou Y, Wang Q, Dryfe R A W, Huang X, Dou S, Dai L, Wang S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Oxygen Reduction Reaction in a Droplet on Graphite: Direct Evidence that the Edge Is More Active than the Basal Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQfzwXg/content/2301.03449v1.pdf'} +page_content=' Int.' metadata={'source': 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Roth, Member, IEEE, +Dong-Yeop Na, Member, IEEE, and Weng Cho Chew, Life Fellow, IEEE +Abstract—Potential-based formulation with generalized Lorenz +gauge can be used in the quantization of electromagnetic fields +in inhomogeneous media [1]. However, one often faces the +redundancy of modes when finding eigenmodes from potential- +based formulation. In free space, this can be explained by +the connection to the well-known Helmholtz decomposition. In +this work, we generalize the Helmholtz decomposition to its +generalized form, echoing the use of generalized Lorenz gauge +in inhomogeneous media [2]. We formulate electromagnetics +eigenvalue problems using vector potential formulation which +is often used in numerical quantization [1]. The properties of +the differential operators are mathematically analyzed. Orthog- +onality relations between the two classes of modes are proved +in both continuous and discrete space. Completeness of two +sets of modes and the orthogonality relations are numerically +validated in inhomogeneous anisotropic media. This work serves +as a foundation for numerical quantization of electromagnetic +fields in inhomogeneous media with potential-based formulation. +I. INTRODUCTION +The formulation of electromagnetic theory based on E, +H, D, and B offers physical insight that has resulted in the +development of many electromagnetic-related technologies. +However, classical electromagnetic equations formulated in +terms of E-H have low-frequency or long-wavelength break- +down [2]. Many numerical methods based on E-H formulation +are unstable at low frequencies. Therefore, E-H formulation +is not truly multi-scale, but exhibits catastrophic breakdown +when the dimensions of objects become much smaller than +the local wavelength, or when the frequency is low. +Much previous work has been devoted to eliminating low- +frequency breakdown of E-H formulation. For differential +equation solvers, Manges and Cendes [3] proposed a gen- +eralized tree-cotree gauge to eliminate the matrix null space +of the magnetostatic curl-curl equation. In another work [4], +hierarchical basis functions and tree-cotree splitting are used to +enhance the stability of the finite-element procedure. For time +domain solvers, method based on changing curl-curl operator +to Laplacian is proposed to handle the ill-conditioning of the +system matrix [5]. For integral equation solvers, loop-tree +decomposition was first proposed which separates the electro- +static and magnetostatic physics [6]–[11]. Augmented electric +field integral equation was then proposed which eliminated the +need to search for loops [12]. +Electromagnetic formulation based on potential theory has +received increasing attention recently since it eliminates the +low-frequency breakdown issue elegantly. It removes the null +space solution by converting singular matrices to full-rank +Laplacian-like matrices. In a homogeneous medium, it is easy +to find its connection to the Helmholtz decomposition. A +number of previous works have dealt with potential-based +(or A-Φ) formulation with both differential equations [13]– +[16], and integral equations [17]–[24]. Moreover, potential- +based formulation is more compatible with quantum theory. +Aharonov-Bohm effect is an example in quantum theory where +E-H are zero, but vector potential A is not zero, and yet, +the effect of A is felt [25]. Besides, in many quantum optics +studies, both vector potential A and scalar potential Φ are +used when the electromagnetic fields are incorporated into the +Schr¨odinger equation [15]. +To perform numerical quantization of electromagnetic fields +in the mode picture with potential-based formulation, it is +often needed to find the electromagnetic eigenmodes of the +system with inhomogeneous media by numerical methods [1], +[26], [27]. When solving potential-based formulation as elec- +tromagnetics eigenvalue problems, null space solution does +not exist, but one often obtains some “extra solutions” that +feature zero E and H even though A and Φ are nonzero. +In a homogeneous medium, this can be explained by making +connection to the Helmholtz decomposition, as will be shown +in Section II. +Due to the need of numerical quantization in complex media +in modern quantum technology [1], [26], an understanding of +the solution space of vector potential wave equation in general +inhomogeneous media is called for. The connection between +the extra modes and the solution space should be made. A +general scheme is needed to explain it in inhomogeneous +media. +In this work, we investigate how to generalize Helmholtz +decomposition to inhomogeneous media. We formulate elec- +tromagnetics eigenvalue problems using the vector potential +formulation, and investigate the modes from solving the sys- +tem equations. The formulation can be readily used in the +numerical quantization scheme in quantum electromagnetics +[1], [26], [27]. We first review Helmholtz theorem and moti- +vate the need for its generalization. We then begin the proof +by analyzing the differential operator associated with the wave +equation for vector potential in Section II. In Section III, we +define div-ε-free modes in inhomogeneous media, which is +the counterpart of div-free modes in homogeneous case. We +prove analytically that the div-ε-free modes are orthogonal +arXiv:2301.03446v1 [physics.optics] 9 Jan 2023 + +2 +to the curl-free modes in source-less inhomogeneous regions, +by both continuous-space and discrete-space calculus. No such +proof has been found in the literature. The proof can be readily +generalized to anisotropic media. We provide numerical results +in Section IV: We first show numerically that Helmholtz +decomposition can be extended to inhomogeneous case, which +we call generalized Helmholtz decomposition (GHD). We then +show numerically the orthogonality condition. We conclude +the work in Section V. +II. A REVIEW OF HELMHOLTZ THEOREM AND +MOTIVATION OF GENERALIZATION +A. A Traditional Proof +Two important theorems establish the conditions for the +existence and uniqueness of solutions to time-independent +problems in electromagnetic theory. The first one is [28]: +A vector field is uniquely specified by giving its +divergence and its curl within a simply connected +region and its normal component on the boundary. +It can be proved by contradiction. Assume that two different +vector fields A and A′ have the same divergence and curl +∇ · A = ∇ · A′ = s +(1) +∇ × A = ∇ × A′ = c +(2) +and their normal component An and A′ +n on the boundary are +also the same. Let F = A − A′. Then ∇ · F, ∇ × F, and Fn +are all zero. Thus there exists a φ such that F = −∇φ and it +satisfies +∇2φ = 0. +(3) +Due to Earnshaw’s theorem, the solution to (3) must have its +maximum and minimum on the boundary. Together with the +condition Fn = 0, or ∂φ +∂n = 0, it must have that φ is a constant +everywhere. Hence F = 0 everywhere. Thus A and A′ are +identical. +Given the proof of uniqueness, we then review the +Helmholtz theorem. The Helmholtz theorem states [28]–[30]: +A vector A with both source and circulation densi- +ties vanishing at infinity may be written as the sum +of two parts, one of which is irrotational, curl-free, +or longitudinal, the other of which is solenoidal, div- +free, or transverse. +That is, +A = A∥ + A⊥ = −∇φ + ∇ × P +(4) +where A∥ = −∇φ and A⊥ = ∇ × P. Here we use A∥ +and A⊥ to denote longitudinal fields and transverse fields, +respectively.1 Apparently ∇×A∥ = 0 and ∇·A⊥ = 0. Since +a vector can be uniquely defined by its divergence ∇ · A = s +1It should be noted that the term longitudinal and transverse are first used +in the context of plane wave, where ∇ × A = 0 → k × A = 0 and +∇ · A = 0 → k · A = 0, suggesting that the longitudinal (transverse) field +is parallel (perpendicular) to the propagation direction in the Fourier space. +Here we port the use of these terms to curl-free and divergence free in general +cases, with a caveat that they do not have the same physical meaning as in +plane wave. +and curl ∇ × A = c, taking the divergence and curl of the +above, we have +∇ · A = −∇2φ = s +(5) +∇ × A = ∇ × ∇ × P = c +(6) +where s acts as the source for Poisson equation, and c the +source for P. We proceed the proof of Helmholtz theorem by +showing the expression for φ and A that recover s and c. Since +the solution of P in (6) is not unique, we set ∇ · P = 0, and +(6) becomes vector Poisson equation with a unique solution. +The solution to (5) and (6) are then well known as [30] +φ(r) = 1 +4π +� +s(r′) +|r − r′|dr′ +(7) +P(r) = 1 +4π +� +c(r′) +|r − r′|dr′. +(8) +The above can be proved by direct substitution of (7) and (8) +into (5) and (6). +B. Solution Space of Wave Equations +Helmholtz theorem can be used to explain the null-space +solution of electric field wave equation. We consider the elec- +tric field wave equations in a homogeneous medium without +sources +∇ × ∇ × E = ω2εµE. +(9) +It can be formulated as an eigenvalue problem. According to +the Helmholtz theorem, the solution to (9) must be either div- +free, or curl-free. Apparently curl-free solutions are the null- +space solutions to (9), where the eigenvalues are zero. On the +other hand, div-free solutions are the non-null-space solutions. +This is because when ∇ · E = 0, we can add −∇∇ · E to the +left-hand side of (9) without changing the equation. Then the +left-hand side becomes ∇×∇×E−∇∇·E = −∇2E and the +Laplacian ∇2 is a negative definite operator. Thus the div-free +solutions must have nonzero eigenvalues. +Helmholtz theorem also connects the solution to the vector +potential wave equation with the solution to the electric +field wave equation. We consider the vector potential wave +equations with Lorenz gauge ∇ · A = −εµ∂tΦ in the same +homogeneous medium without sources +∇ × ∇ × A − ∇∇ · A = ω2εµA. +(10) +Both (9) and (10) have two families of solutions following the +Helmholtz theorem. The div-free solution to (9) must also be +the div-free solution to (10), and vice versa. This is because +the second term of (10) can be dropped in the case of div-free +solution. However, since null-space solution does not exist in +(10) due to the negative-definiteness of Laplacian, the space +spanned by the null-space solutions to (9) must be filled by +the same number of non-null-space solutions to (10). Again +due to the Helmholtz theorem, these solutions are curl-free +solutions in (10) as well. The mapping between the solution +space of (9) and (10) is illustrated in Fig. 1. Apparently curl- +free vector potential in electrodynamics leads to zero E and H +even though A and Φ are nonzero. This also justifies the use of +Φ = 0 gauge due to Lorenz gauge to remove the redundancy of + +3 +Fig. 1. Mapping between solution space of the electric field wave equation +and vector potential wave equation. The double-headed arrow represents the +same space spanned by the eigenvectors of both equations. +the curl-free solutions in homogeneous case for many practical +problems. +C. A New Look of Helmholtz Theorem +We can also prove Helmholtz theorem from a different +perspective by looking at (9). We first prove that the curl +operator is a self-adjoint operator [31] using inner product. +We begin with +⟨A∗, CB⟩ = +� +V +dr A∗ · (∇ × B) += +� +V +dr B · (∇ × A∗) − +� +S +dr ˆn · (A∗ × B) +(11) +where C denotes the curl operator. With the boundary condi- +tion that ˆn × A = 0 and ˆn × B = 0,2 and the surface integral +term vanishes, and hence the curl operator is self-adjoint for +fields of the prescribed boundary conditions. With the same +approach, it can also be proved that ∇ × ∇× is a self-adjoint +operator [16]. Thus the solution vectors to (9) form a complete +basis. +The proceeding proof also suggests that the curl operator +has the same null-space as its power. In general when an +operator maps a domain space function (or a row space vector, +if the operator is a finite-sized matrix) into its range space +(or column space), the new function may contain null-space +component. However, the self-adjointness and symmetry of the +curl operators ensure that its domain space and range space are +the same, and its (right-)null-space and left-null-space are the +same. Thus the curl operator maps a domain space function +into the same space. Hence it will not generate a null-space +component. +Now we are ready to prove the Helmholtz theorem by +analyzing (9). We only consider the non-trivial solution to (9) +where E ̸= 0 almost everywhere. Taking the divergence of +(9), we have ω2∇ · E = 0. It has three possible classes of +solutions: (i) ω = 0 and ∇ · E ̸= 0; (ii) ω ̸= 0 and ∇ · E = 0; +and (iii) ω = 0 and ∇·E = 0. We can rule out (iii) because it +would make (9) the same as (10), and ω = 0 solution cannot +exist in (10). This is because ∇ × ∇ × A − ∇∇ · A = −∇2A +and the Laplacian ∇2 is a negative definite operator. For (i), +apparently ∇ × ∇ × E = 0, and in turn ∇ × E = 0. For (ii), +2The rationale of this boundary condition is explained in the next section. +we have ∇ × E ̸= 0. Concluding the above, we can clearly +see two classes of solution to (9) similar to the Helmholtz +theorem, where +1) ∇ × E = 0, ∇ · E ̸= 0; +2) ∇ · E = 0, ∇ × E ̸= 0. +Due to the self-adjointness of ∇ × ∇× operator, the solution +to (9) form a complete basis to the entire space. Hence +any function satisfying the proposed boundary condition can +be separated into two classes. This completes the proof of +Helmholtz theorem. Note that analyzing (10) also leads to the +same two families of solutions described above. This will be +shown in greater details similar to the inhomogeneous media +case next. +D. Motivation of Generalization +As shown above, the Helmholtz theorem works well in +explaining the solution space of the electric field wave equa- +tion and vector potential wave equation with Lorenz gauge +in homogeneous medium. However, in inhomogeneous dis- +persionless media, it is suggested that generalized Lorenz +gauge should be used instead [2]. Thus, to better explain the +eigenmodes found by potential-based formulation, it would +be ideal if Helmholtz theorem can be generalized in a way +that ∇ · ε(r)Aε⊥(r) = 0,3 where Aε⊥ denotes the div-ε-free +(or generalized transverse) fields. Considering this, we seek to +prove the following generalized Helmholtz decomposition: +The vector field A given by the Helmholtz wave +equation in source-free inhomogeneous media can +be decomposed into two components Aε⊥ and A∥, +where the div-ε-free (or generalized transverse) +fields satisfy ∇ · εAε⊥ = 0 and the curl-free (or +longitudinal) fields satisfy ∇ × A∥ = 0. +In the remainder of this work, we use solution of the wave +equation for vector potential A to arrive at the generalized +Helmholtz decomposition. A modified generalized Lorenz +gauge ∇ · εA = −ε2 +0µ0∂tΦ is used. We follow matrix theory +generally in the proof. It will be shown that the vector potential +wave equation has two classes of solutions. They degenerate +into the Helmholtz decomposition in free space. Thus the +vector potential wave equation for inhomogeneous media can +be used to motivate the generalized Helmholtz decomposition. +We also note that there is no null space in the resulting +generalized eigenvalue problem. +III. PROOF OF ORTHOGONALITY +We start from proving the orthogonality between the two +families of fields proposed in the generalized Helmholtz +decomposition, assuming the decomposition does exist. We +leave it to the next section to demonstrate the existence of the +decomposition through numerical examples. +We consider the space terminated by continuous perfect +electric conductor (PEC) boundary, filled with inhomogeneous +media (Fig. 2). Because of the gauge ∇ · εA = −ε2 +0µ0∂tΦ, +the boundary condition for ∇ · εA is actually that for Φ on a +3Position-dependent ε(r) is assumed from now on. For simplicity, we only +write ε. + +Electric Field Wave Equation +Vector Potential Wave +Curl-free (null- +Curl-free solution +Equation +space) solution +(dynamic) +Divergence-free +Divergence-free +solution (dynamic) +solution (dynamic)4 +Fig. 2. Schematic of the region of interest. +PEC, which is Φ equals a constant voltage [16], [18]. If there +is only one PEC, we can choose this as our reference. Thus +ˆn × A = 0 +(12) +which is consistent with ˆn × E = 0, and +∇ · εA = 0. +(13) +The latter is equivalent to Φ = 0 on the boundary when the +PEC has a constant voltage of zero, i.e. “grounded”. +If there are additional PECs inside the domain, they may +not be grounded. In this case, ∇ · εA = Φ is equal to +a constant voltage on each conductor. Additionally, charge +neutrality constraint can be enforced on each conductor, which +requires [16], [18] +� +S +dr ˆn · εA = 0. +(14) +where the surface integral is over the surface of each PEC. +Next, we proceed to show the proof of orthogonality first in +continuous space, followed by a similar procedure in discrete +space. Different definitions of orthogonality will be discussed. +It will also be shown that discrete approximation of Maxwell’s +equations is homomorphic to the continuum case if done +properly. +A. Continuous Space +Wave equation is usually the starting point of many A- +Φ based solver. We start the mathematical proof of mode +orthogonality by defining proper operator and vector space. +We consider the wave equation for vector potential A in lossy +dispersionless inhomogeneous isotropic media4 with modified +generalized Lorenz gauge [2] +∇ × 1 +µ∇ × A − +1 +ε2 +0µ0 +ε∇∇ · εA = ω2εA. +(15) +Let +W1 = ∇ × 1 +µ∇× +(16) +W2 = − +1 +ε2 +0µ0 +ε∇∇ · ε +(17) +L = W1 + W2 +(18) +U = ε +(19) +λ = ω2. +(20) +4To satisfy the Kramers–Kronig relations, this assumption is only valid over +a narrow bandwidth. +Then (15) can be written abstractly and concisely as +(W1 + W2)A = λUA +(21) +where the operator notation is defined as a differential operator +acting on a function A. Note that we use boldface calligraphic +font with an overbar to denote a Hilbert space operator. The +function A in the abovementioned space is in turn square +integrable [32]. +It has been proved that the L operator is self-adjoint in +lossless media given the boundary conditions defined in the +beginning of this section [16]. Here we investigate the general +case with lossy media. Since the L operator consists of two +terms W1 and W2, it is clearer to look at the two terms +separately [16]. +We first define Aε⊥ to be the solution to the eigenvalue +problem W1Aε⊥ = λε⊥UAε⊥ with eigenvalue λε⊥. Here +we only study the solutions with nonzero eigenvalues. This +gives us, from the definition above, +∇ × 1 +µ∇ × Aε⊥ = λε⊥εAε⊥. +(22) +Taking divergence of the above equation gives +0 = λε⊥∇ · εAε⊥. +(23) +Thus Aε⊥ is in the null space of the W2 operator defined +in (17), and it is also a solution to the original eigenvalue +problem (21) with eigenvalue λε⊥. +Similarly, we define A∥ to be the solution to W2A∥ = +λ∥UA∥ with nonzero eigenvalue λ∥, or more explicitly, +− +1 +ε2 +0µ0 +ε∇∇ · εA∥ = λ∥εA∥. +(24) +Dividing the above by ε and taking curl of the resulting +equation yields +0 = λ∥∇ × A∥. +(25) +Thus A∥ is in the null space of the W1 operator defined in +(16), and it is also a solution to the original eigenvalue problem +(21) with eigenvalue λ∥. +Next, we will manipulate the above equations, and use (23) +and (25) to prove orthogonality of two sets of modes. +1) Proof 1: To begin, we take the left dot product of (24) +by Aε⊥, and integrate it over the space filled with dielectrics +to yield +− +1 +ε2 +0µ0 +� +V +dr +� +Aε⊥ · ε∇∇ · εA∥ +� += +� +V +dr +� +Aε⊥ · λ∥εA∥ +� +. +(26) +Using integration by parts, the left-hand side becomes +1 +ε2 +0µ0 +� +V +dr (∇ · εAε⊥)(∇ · εA∥) +− +1 +ε2 +0µ0 +� +S +dr ˆn· +� +(∇ · εA∥)εAε⊥ +� +(27) +where the surface integral is over the PEC surfaces. From +(23), the first term vanishes. At the grounded PEC boundary, + +E1 +E2 +PEC5 +∇ · εA∥ = 0, and then the second term also vanishes. At the +PEC not connected to the ground, the second term becomes +− +1 +ε2 +0µ0 +(∇ · εA∥) +� +S +dr ˆn · εAε⊥ = 0 +(28) +where the divergence term is taken outside the integral because +Φ is equal to a constant voltage on the conductor, and the +integral vanishes because of the charge neutrality constraint +(14). Thus +⟨Aε⊥, εA∥⟩ = 0 +(29) +as long as λ∥ ̸= 0, where ⟨F, G⟩ = +� +F · G dr is defined as +the reaction inner product by Rumsey [33]. In other words, +the two sets of modes Aε⊥ and A∥ are ε-orthogonal to each +other. +2) Proof 2: Now, taking the left dot product of (22) with +A∥, and integrating it over space yields +� +V +dr +� +A∥ · ∇ × 1 +µ∇ × Aε⊥ +� += +� +V +dr +� +A∥ · λε⊥εAε⊥ +� +. +(30) +The right-hand side is denoted λε⊥⟨A∥, εAε⊥⟩. Using inte- +gration by parts, the left-hand side becomes +� +V +dr +� 1 +µ∇ × Aε⊥ +� +· +� +∇ × A∥ +� ++ +� +S +dr ˆn· +�� 1 +µ∇ × Aε⊥ +� +× A∥ +� +. +(31) +From (25), the first term vanishes. Implementing the boundary +condition that ˆn×A∥ = 0, the second term also vanishes. Thus +⟨A∥, εAε⊥⟩ = 0 +(32) +as long as λε⊥ ̸= 0. In other words, the two sets of modes +A∥ and Aε⊥ are ε-orthogonal to each other. +In the above derivation, if we replace A∥ by A∗ +∥, the +derivation is still valid. That is +⟨A∗ +∥, εAε⊥⟩ = 0. +(33) +However, the same token does not apply to Proof 1 in III-A. +This may motivate new definition of inner product in the +problems concerning potential-based formulation. +B. Discrete Space +The problem in the previous subsection is homomorphic +to numerical linear algebra if we find the matrix represen- +tation of the operators using subspace projection methods +[34], including finite difference method [35], finite element +method [36], or discrete exterior calculus [37]. These methods +should be chosen properly to preserve certain properties of the +continuum calculus, i.e. ∇·(∇×A) = 0 and ∇×∇f = 0. In +this work we use finite difference method. After discretization, +we have +∇ × A ⇒ C1 · A1 +(34) +∇ × ∇ × A ⇒ C2 · C1 · A1 +(35) +∇ · εA ⇒ D1 · A1 +(36) +ε∇f ⇒ E1 · f1 +(37) +ε ⇒ U +(38) +where the operators on the LHS are in the continuous Hilbert +space, while the RHS are the linear algebra approximation of +the continuum space. Here we use boldface font with overbar +to denote finite size matrices, and sans serif typestyle to denote +finite length vectors stored as 1D arrays in computers. +Due to the discretization, each operator may have different +representations depending on the space it operates on. To use +finite difference method, Yee grid is constructed with both +primal and dual grids [35], [38]. On each grid, we place a 3D +vector on the face of the cube, and a scalar on the center of +the cube. Details of the discretization scheme can be found in +the Appendix A. +In (34) to (38), the discrete operators with subscript 1 +denotes operations on the primal mesh, while subscript 2 +denotes operations on the dual mesh. The discrete curl operator +C1 acts on a vector on the primal mesh, resulting in a vector +on the dual mesh, while C2 acts on a vector on the dual mesh +and results in a vector on the primal mesh. It can be shown +that these two discrete curl operators are transpose to each +other C1 = C +T +2 , where T denotes matrix transpose. Also, +it is easy to prove that the discrete divergence operator and +the negative of discrete gradient operator are transpose to each +other D +T +1,2 = −E1,2. The U operator is a square matrix whose +dimension depends on the vector it operates on. The following +vector identities are preserved after discretization using finite +difference method [39]–[41] +∇ · (∇ × A) = 0 ⇒ D2 · U +−1 +2 +· C1 · A1 = 0 +and D1 · U +−1 +1 +· C2 · A2 = 0 +(39) +∇ × ∇f = 0 ⇒ C1 · U +−1 +1 +· E1 · f1 += −C1 · U +−1 +1 +· D +T +1 · f1 = 0. +(40) +Without loss of generality, we let µ = 1.5 Also we denote +α = 1/ε2 +0µ0. We assume the solution vector is on the primal +mesh, and thus the subscript 1 is omitted. Then (21) becomes +C2 · C1 · A − αE1 · D1 · A = λU1 · A. +(41) +We break the above into two eigenvalue problems corre- +sponding to the two operators on the left-hand side +C2 · C1 · Aε⊥ = λε⊥U1 · Aε⊥ +(42) +−αE1 · D1 · A∥ = λ∥U1 · A∥. +(43) +Taking the left dot product of (42) with D1 · U +−1 +1 , we get +D1 · U +−1 +1 +· C2 · C1 · Aε⊥ = λε⊥D1 · U +−1 +1 +· U1 · Aε⊥. (44) +5It can be seen that the proof is still valid when µ is inhomogeneous, +anisotropic, or complex. + +6 +Due to (39), LHS of the above vanishes. Thus +D1 · Aε⊥ = 0. +(45) +Taking the left dot product of (43) with C1 · U +−1 +1 , we get +−αC1 · U +−1 +1 +· E1 · D1 · A∥ = λ∥C1 · U +−1 +1 +· U1 · A∥. +(46) +Due to (40), the LHS of the above vanishes. Thus +C1 · A∥ = 0. +(47) +1) Proof 1: Taking the left dot product of (43) with AT +ε⊥ and +using the symmetry property between divergence and gradient +operators, we have +αAT +ε⊥ · D +T +1 · D1 · A∥ = λ∥AT +ε⊥ · U1 · A∥. +(48) +Due to (45) and its transpose, LHS of the above vanishes. +Thus +AT +ε⊥ · U · A∥ = 0. +(49) +which is analogous to (29). +2) Proof 2: Now, taking the left dot product of (42) with +AT +∥ , where +† denotes conjugate transpose, and using the +symmetry property of curl operator, we have +AT +∥ · C +T +1 · C1 · Aε⊥ = λε⊥AT +∥ · U1 · Aε⊥. +(50) +Due to (47) and its transpose, LHS of the above vanishes. +Thus +AT +∥ · U1 · Aε⊥ = 0 +(51) +which is homomorphic to (32). However, we can also left dot +product (42) by A† +∥, and get +A† +∥ · C +T +1 · C1 · Aε⊥ = λε⊥A† +∥ · U1 · Aε⊥. +(52) +Due to (47) and its Hermitian conjugate, and the realness of +the curl operator under finite difference method, LHS of the +above vanishes. Thus +A† +∥ · U · Aε⊥ = 0 +(53) +which is analogous to (33). The above shows that the finite +difference approximation of Maxwell’s equations is homomor- +phic to the continuum case if done properly. +C. Anisotropic Case +The mathematical proof in the previous two subsections +still applies in anisotropic media. Consider the wave equa- +tion for vector potential A in dispersionless inhomogeneous +anisotropic media with modified generalized Lorenz gauge in +frequency domain [2] +∇ × µ−1∇ × A − +1 +ε2 +0µ0 +ε · ∇ (∇ · ε · A) = ω2ε · A. +(54) +The mathematical proof of mode orthogonality follows exactly +as Section III-A for continuous space and as Section III-B for +discrete space, with the requirement that +ε = εT +(55) +which implies that the material is reciprocal. +Fig. 3. Geometries of the numerical examples with dimensions. +IV. NUMERICAL RESULTS +A. Numerical Demonstration of Completeness +In this section, we first demonstrate that the two sets of +modes given in the previous section are complete in the +discrete space. We solve (15) numerically. Finite difference +method is used corresponding to the discrete space proof in +Section III. Uniform discretization is used, and the discretiza- +tion scheme follows [15], [39], as described in Appendix A. +For an empty or homogeneous structure, the proposed gen- +eralized Helmholtz decomposition degenerates into the usual +Helmholtz decomposition. To demonstrate it, we consider a +rectangular 7 × 8 × 6 cavity homogeneously filled with a +εr = 3 material, as shown in Fig. 3(a). Discretization step +is 0.5, resulting in 6938 × 6938 matrices. We use natural +unit so that ε0 = µ0 = 1. After solving the generalized +eigenvalue problem, we take divergence and curl on each mode +to identify the two classes of modes. The results are displayed +in Fig. 4. Note that the indices of curl-free modes and div- +ε-free modes are skipped in (b) and (c), respectively. The +total number of div-ε-free and curl-free modes equals to the +dimension of the eigenvalue problem. Thus it clearly shows +two distinct groups of modes corresponding to the Helmholtz +decomposition, which complete the discrete space, and are +consistent across field-based and potential-based formulations. +We then consider a more general example with anisotropic +material. As shown in 3(b), we construct a cylindrical cavity +with radius r = 2 and height h = 2. It is filled with a block +of anisotropic material centered at (1, 0.4, 0.2) with dimension +0.6, 0.4, 0.8 in x, y, z directions, respectively. Discretization +step is 0.2, resulting in 5822×5822 matrices. The permittivity +tensor of the anisotropic material is +ε = +� +� +2 +0 +1 + 3i +0 +3 + 0.2i +0 +1 + 3i +0 +2 +� +� +(56) +which satisfies reciprocity. The matrix of anisotropic permit- +tivity is treated the same way as [42]. The discretized operators +satisfy the discrete vector identities (39) and (40). +Mode degeneracy is observed after solving the generalized +eigenvalue problem. We apply the analysis described in Ap- +pendix B to separate degenerate modes. The results are then +displayed in Fig. 5. We find that the modes can be categorized +into two distinct groups: +1) div-ε-free (or generalized transverse) modes (Fig. 5(a)): +∇ · εA = 0 for the entire grid; + +(a) +(b) +4 +6 +87 +Fig. 4. Two set of modes found in the geometry described by Fig. 3(a) can be +clearly distinguished. (a) The absolute value of eigenvalues are shown for both +potential-based formulation and E-H formulation. It can be seen clearly that +the potential-based formulation does not have null space. For the potential- +based formulation, only the div-ε-free (or generalized transverse) modes are +shown in (b), and only the curl-free (or longitudinal) modes are shown in (c). +Totally there are 4793 div-ε-free modes in (b) and 2145 curl-free modes in +(c). The latter is consistent with the number of null-space eigenvalues with +E-H formulation in (a). The total number of both families of modes is equal +to the dimension of the eigenvalue problem. The indices of curl-free modes +and div-ε-free modes are skipped in (b) and (c), respectively. It can be seen +that the div-ε-free modes satisfy ∇·εA = 0, while the curl-free modes satisfy +∇ × A = 0. Blue open circles: for each mode, take ∇ · εA, and record the +maximum value of abs(∇ · εA) in the entire grid. Red stars: for each mode, +take ∇ × A, and record the maximum magnitude of all components in the +entire grid. +2) Curl-free (or longitudinal) modes (Fig. 5(b)): ∇×A = 0 +for the entire grid.6 +It can be seen that each mode is either div-ε-free, or curl-free. +The div-ε-free modes are consistent with the modes found by +E-H formulation since (15) returns to the wave equation of E +field [2], [43]. The curl-free modes have zero E and H field +by the fact that B = ∇ × A = 0. +In both numerical examples, the two classes of modes +form a complete basis. Although the completeness of basis +expanded by the eigenvectors in the discrete space from +the numerical examples does not necessarily guarantee the +completeness of the two sets of modes in the continuous +Hilbert space, one could use convergence and existence of the +numerical solution to a partial differential equation with finite +difference approximation [44] to imply completeness of these +modes in the continuous Hilbert space. Thus the numerical +proof could motivate further study in the continuous Hilbert +space as future work. +6The curl-free modes can be eliminated by Φ = 0 gauge. This will be +explained in future work. +Fig. 5. Two set of modes found in the geometry described by Fig. 3(b) can +be clearly distinguished. Totally there are 4049 div-ε-free modes and 1773 +curl-free modes. The description of the figure follows the caption of Fig. 4(b) +and (c). +B. Numerical Validation of Orthogonality +We then proceed to validate the orthogonality conditions. +For the numerical example described by Fig. 3(b), we calculate +the four sets of inner products between the div-ε-free modes +and curl-free modes defined in Section III-B after separating +the degenerate modes. The colormaps of the resulting matrices +are shown in Fig. 6, and the maximum magnitude of the matrix +elements are shown in Table I. For readability, only the first +100×100 elements of each matrix are displayed in Fig. 6, but +these results are representative of the full matrices. It can be +seen from Fig. 6 and Table I that +⟨Aε⊥, εA∥⟩ = 0 +(57) +⟨A∥, εAε⊥⟩ = 0 +(58) +⟨A∗ +ε⊥, εA∥⟩ ̸= 0 +(59) +⟨A∗ +∥, εAε⊥⟩ = 0 +(60) +which are consistent with the mathematical proofs in Sec- +tion III. This result, supplemented by the numerical proof +of completeness in the previous subsection, completes our +demonstration of the generalized Helmholtz decomposition. +TABLE I +ORTHOGONALITY RESULTS FROM THE SECOND NUMERICAL EXAMPLE. +THE MAXIMUM MAGNITUDE OF MATRIX ELEMENTS IN THE +CORRESPONDING MATRICES ARE SHOWN IN THE TABLE. HERE | · | +DENOTES ELEMENT-WISE MAGNITUDE. +max(|V +T +ε⊥ · ε · V∥|) +5.1694e-10 +max(|V +T +∥ · ε · Vε⊥|) +5.1694e-10 +max(|V +† +ε⊥ · ε · V∥|) +0.5779 +max(|V +† +∥ · ε · Vε⊥|) +7.2430e-10 + +(a) +abs(eigenvalue) +100 +- vector-potential-based formulation +-E-Hformulation +Index: 2145 +10-20 +1000 +2000 +3000 +4000 +5000 +6000 +mode index +curl-free modes only +(c) +100 +100 +max(V. EA) +max(V.EA) +max(V × A) +max(V × A) +米 +米 +10-10 +10-10 +米 +米米*米 +2000 +4000 +6000 +2000 +4000 +6000 +mode index +mode indexdiv-e-free modes only +(a) +(b) +curl-free modes only +100 +100 +max(V.·A) +max(V.·A) +max(V × A) +max(V × A) +米 +米 +000 +10-10 +10-10 +* +米米 +888811888 +巫 +变 +1000 +3000 +5000 +1000 +3000 +5000 +modeindex +modeindex8 +Fig. 6. Color map of four sets of inner products from the numerical example +described by Fig. 3(b). (a) V +T +ε⊥ ·ε·V∥; (b) V +T +∥ ·ε·Vε⊥; (c) V +† +ε⊥ ·ε·V∥; +(d) V +† +∥ · ε · Vε⊥. Here Vε⊥(V∥) is a matrix where each column vector +is a div-ε-free (curl-free) mode. Each pixel represents the magnitude of the +corresponding matrix element. The indices are the indices of matrix elements. +Note the different scales of the colorbars. The values of the resulting matrices +are essentially 0 in (a), (b), and (d). +V. CONCLUSION +In this work, we combine mathematical proof and numer- +ical results to demonstrate the generalization of Helmholtz +decomposition to inhomogeneous media. The two families +of fields in the generalized Helmholtz decomposition, div-ε- +free and curl-free fields, are connected to the two classes of +solutions to the vector potential wave equation. The div-ε- +free and curl-free solutions form a complete set of the basis, +as demonstrated in the numerical results and its discussion. +They are also orthogonal to each other. The div-ε-free field +manifests charge-free condition, while the curl-free field is +associated with charges. In the literature, the curl-free fields +are often eliminated by setting Φ = 0. However, when +there are charges present, we need both families of solutions. +It also provides a theoretical background to the numerical +quantization [1] based on the generalized Lorenz gauge. Future +work will be devoted to the application of Φ = 0 gauge +and rigorously determining suitable conditions for eliminating +the curl-free modes in general inhomogeneous media while +preserving full rank of the resulting system matrix for practical +applications. +APPENDIX A +DISCRETE VECTOR CALCULUS ON A GRID +Discrete vector calculus on a Yee lattice [35], [38] is the +basis for the proof of orthogonality in discrete space. We give +a brief overview of the discretization. The details can be found +in [39]. +The staggered grid with both primal (solid line) and dual +(dashed line) grids are shown in Fig. 7. The dual grid is shifted +Fig. 7. Schematic of the grid. The primal and dual grids are shifted by half +grid step. The solution vector A is placed on the face of the primal grid. +half a grid width in all three directions from the primal grid. +On each grid, we place a 3D vector on the face of the cube, +and a scalar in the center of the cube. +We index the grid point at the cube center of the primal grid +by integers (i, j, k). Below we explicitly formulate the discrete +differential operations, where central difference is used. +A discrete gradient g = ∇f on the primal grid is given by +gi+0.5,j,k +x += f i+1,j,k − f i,j,k +∆x +, +(61) +gi,j+0.5,k +y += f i,j+1,k − f i,j,k +∆y +, +(62) +gi,j,k+0.5 +z += f i,j,k+1 − f i,j,k +∆z +. +(63) +A discrete divergence f = ∇ · g on the primal grid is given +by +f i,j,k = gi+0.5,j,k +x +− gi−0.5,j,k +x +∆x ++ gi,j+0.5,k +y +− gi,j−0.5,k +y +∆y ++ gi,j,k+0.5 +z +− gi,j,k−0.5 +z +∆z +. +(64) +A discrete curl h = ∇ × g on a vector on the primal grid +resulting in a vector on the dual grid is given by +hi,j+0.5,k+0.5 +x += gi,j+1,k+0.5 +z +− gi,j,k+0.5 +z +∆y +− gi,j+0.5,k+1 +y +− gi,j+0.5,k +y +∆z +, +(65) +hi+0.5,j,k+0.5 +y += gi+0.5,j,k+1 +x +− gi+0.5,j,k +x +∆z +− gi+1,j,k+0.5 +z +− gi,j,k+0.5 +z +∆x +, +(66) +hi+0.5,j+0.5,k +z += gi+1,j+0.5,k +y +− gi,j+0.5,k +y +∆x +− gi+0.5,j+1,k +x +− gi+0.5,j,k +x +∆y +. +(67) + +.E.Vel +Vel-E.v +×10-11 +×10-11 +(a) +(b) +5 +5 +20 +20 +4 +4 +40 +40 +3 +3 +60 +60 +2 +2 +80 +80 +1 +1 +100 +100 +20 +60 +100 +20 +60 +100 +E.V +× 10-11 +(c) +(d) +0.05 +6 +20 +20 +40 +40 +4 +60 +60 +2 +80 +80 +100 +0 +100 +20 +60 +100 +20 +60 +100Ay +A +29 +A discrete curl g = ∇ × h on a vector on the dual grid +resulting in a vector on the primal grid is given by +gi+0.5,j,k +x += hi+0.5,j+0.5,k +z +− hi+0.5,j−0.5,k +z +∆y +− hi+0.5,j,k+0.5 +y +− hi+0.5,j,k−0.5 +y +∆z +, +(68) +gi,j+0.5,k +y += hi,j+0.5,k+0.5 +x +− hi,j+0.5,k−0.5 +x +∆z +− hi+0.5,j+0.5,k +z +− hi−0.5,j+0.5,k +z +∆x +, +(69) +gi,j,k+0.5 +z += hi+0.5,j,k+0.5 +y +− hi−0.5,j,k+0.5 +y +∆x +− hi,j+0.5,k+0.5 +x +− hi,j−0.5,k+0.5 +x +∆y +. +(70) +APPENDIX B +ANALYSIS OF DEGENERATE MODES +Given a set of eigenmodes A1, A2, . . . , An with corre- +sponding eigenvalues λ1, λ2, . . . , λn obtained from the gen- +eralized eigenvalue problem, the space expanded by +1 +λi +ε−1∇ × 1 +µ∇ × Ai +(71) +has the same dimension as the number of independent div-ε- +free modes. This is because the curl-free modes are eliminated +by the operation due to (25) λ∥∇×A∥ = 0, and the div-ε-free +modes are recovered due to (22) ∇× 1 +µ∇×Aε⊥ = λε⊥εAε⊥. +Besides, the space expanded by +− 1 +λi +1 +ε2 +0µ0 +∇∇ · εAi +(72) +has the same dimension as the number of independent curl-free +modes. This is because the div-ε-free modes are eliminated by +the operation due to (23) λε⊥∇ · εAε⊥ = 0, and the curl-free +modes are recovered due to (24) − +1 +ε2 +0µ0 ε∇∇ · εA∥ = λ∥εA∥. +The dimension of the two spaces from the numerical exam- +ple described by Fig. 3(b) is shown in Fig. 8. The number of +nonzero singular values indicate the dimension of the space +spanned by a set of vectors. It can be seen that the sum of the +dimension of the space of (71) and (72) equals the dimension +of the problem. Thus it proves that the div-ε-free and curl-free +modes as defined in the main text form a complete basis of the +discrete space. Otherwise, if there exists an eigenmode that is +nonzero after both (71) and (72) operations, it must contribute +one “extra dimension”. +Degenerate modes with the same eigenvalue can also be +separated based on this analysis. Given n degenerate modes +A1, A2, . . . , An with the same eigenvalue, we first learn the +number of independent div-ε-free and curl-free modes by +evaluating the dimension of the subspace expanded by (71) and +(72), as shown in Fig. 9. The div-ε-free modes are isolated by +(71) because of (22), and the curl-free modes are isolated by +(72) because of (24). Gram-Schmidt orthogonalization is then +performed within each subspace to extract the independent +div-ε-free and curl-free modes, respectively. +Fig. 8. +The eigenvalues and the resulting solution space of the geometry +described by Fig. 3(b) are shown. (a) Eigenvalues are shown in blue curve; +Orange upward spikes denote a cluster of degenerate modes. (b) The singular +values of the matrix constructed by lining up ε−1 ·∇× 1 +µ∇×Ai are shown. +It can be seen that there are 4049 nonzero singular values. (c) The singular +values of the matrix constructed by lining up −∇∇·ε·Ai are shown. It can +be seen that there are 1773 nonzero singular values. The number of nonzero +singular values indicate the dimension of the space spanned by a set of vectors. +The total number of nonzero singular values from (b) and (c) is 5822, equal +to the dimension of the problem. +Fig. 9. +A closer look at the first cluster of degenerate modes (index 1428 +to 1451) in Fig. 8(a). Similar results are observed for the other clusters of +degenerate modes. Singular values of (a) and (b) are obtained in the same +way as Fig. 8(b) and (c). The total number of nonzero singular values from +(a) and (b) is 24, equal to the dimension of the degenerate subspace. +REFERENCES +[1] D.-Y. Na, J. Zhu, W. C. Chew, and F. L. 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LeVeque, Finite Difference Methods for Ordinary and Partial +Differential Equations: Steady-State and Time-Dependent Problems. +SIAM, 2007. + diff --git a/iNE1T4oBgHgl3EQfzgXI/content/tmp_files/load_file.txt b/iNE1T4oBgHgl3EQfzgXI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f404619dddbd34d9b2a8cfb32434b754e10465a2 --- /dev/null +++ b/iNE1T4oBgHgl3EQfzgXI/content/tmp_files/load_file.txt @@ -0,0 +1,787 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf,len=786 +page_content='1 Generalized Helmholtz Decomposition for Modal Analysis of Electromagnetic Problems in Inhomogeneous Media Jie Zhu, Thomas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Roth, Member, IEEE, Dong-Yeop Na, Member, IEEE, and Weng Cho Chew, Life Fellow, IEEE Abstract—Potential-based formulation with generalized Lorenz gauge can be used in the quantization of electromagnetic fields in inhomogeneous media [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, one often faces the redundancy of modes when finding eigenmodes from potential- based formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In free space, this can be explained by the connection to the well-known Helmholtz decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In this work, we generalize the Helmholtz decomposition to its generalized form, echoing the use of generalized Lorenz gauge in inhomogeneous media [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We formulate electromagnetics eigenvalue problems using vector potential formulation which is often used in numerical quantization [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The properties of the differential operators are mathematically analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Orthog- onality relations between the two classes of modes are proved in both continuous and discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Completeness of two sets of modes and the orthogonality relations are numerically validated in inhomogeneous anisotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This work serves as a foundation for numerical quantization of electromagnetic fields in inhomogeneous media with potential-based formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' INTRODUCTION The formulation of electromagnetic theory based on E, H, D, and B offers physical insight that has resulted in the development of many electromagnetic-related technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, classical electromagnetic equations formulated in terms of E-H have low-frequency or long-wavelength break- down [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Many numerical methods based on E-H formulation are unstable at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Therefore, E-H formulation is not truly multi-scale, but exhibits catastrophic breakdown when the dimensions of objects become much smaller than the local wavelength, or when the frequency is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Much previous work has been devoted to eliminating low- frequency breakdown of E-H formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For differential equation solvers, Manges and Cendes [3] proposed a gen- eralized tree-cotree gauge to eliminate the matrix null space of the magnetostatic curl-curl equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In another work [4], hierarchical basis functions and tree-cotree splitting are used to enhance the stability of the finite-element procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For time domain solvers, method based on changing curl-curl operator to Laplacian is proposed to handle the ill-conditioning of the system matrix [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For integral equation solvers, loop-tree decomposition was first proposed which separates the electro- static and magnetostatic physics [6]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Augmented electric field integral equation was then proposed which eliminated the need to search for loops [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Electromagnetic formulation based on potential theory has received increasing attention recently since it eliminates the low-frequency breakdown issue elegantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It removes the null space solution by converting singular matrices to full-rank Laplacian-like matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In a homogeneous medium, it is easy to find its connection to the Helmholtz decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A number of previous works have dealt with potential-based (or A-Φ) formulation with both differential equations [13]– [16], and integral equations [17]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Moreover, potential- based formulation is more compatible with quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Aharonov-Bohm effect is an example in quantum theory where E-H are zero, but vector potential A is not zero, and yet, the effect of A is felt [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Besides, in many quantum optics studies, both vector potential A and scalar potential Φ are used when the electromagnetic fields are incorporated into the Schr¨odinger equation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' To perform numerical quantization of electromagnetic fields in the mode picture with potential-based formulation, it is often needed to find the electromagnetic eigenmodes of the system with inhomogeneous media by numerical methods [1], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' When solving potential-based formulation as elec- tromagnetics eigenvalue problems, null space solution does not exist, but one often obtains some “extra solutions” that feature zero E and H even though A and Φ are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In a homogeneous medium, this can be explained by making connection to the Helmholtz decomposition, as will be shown in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Due to the need of numerical quantization in complex media in modern quantum technology [1], [26], an understanding of the solution space of vector potential wave equation in general inhomogeneous media is called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The connection between the extra modes and the solution space should be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A general scheme is needed to explain it in inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In this work, we investigate how to generalize Helmholtz decomposition to inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We formulate elec- tromagnetics eigenvalue problems using the vector potential formulation, and investigate the modes from solving the sys- tem equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The formulation can be readily used in the numerical quantization scheme in quantum electromagnetics [1], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We first review Helmholtz theorem and moti- vate the need for its generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We then begin the proof by analyzing the differential operator associated with the wave equation for vector potential in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In Section III, we define div-ε-free modes in inhomogeneous media, which is the counterpart of div-free modes in homogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We prove analytically that the div-ε-free modes are orthogonal arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='03446v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='optics] 9 Jan 2023 2 to the curl-free modes in source-less inhomogeneous regions, by both continuous-space and discrete-space calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' No such proof has been found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The proof can be readily generalized to anisotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We provide numerical results in Section IV: We first show numerically that Helmholtz decomposition can be extended to inhomogeneous case, which we call generalized Helmholtz decomposition (GHD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We then show numerically the orthogonality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We conclude the work in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A REVIEW OF HELMHOLTZ THEOREM AND MOTIVATION OF GENERALIZATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A Traditional Proof Two important theorems establish the conditions for the existence and uniqueness of solutions to time-independent problems in electromagnetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The first one is [28]: A vector field is uniquely specified by giving its divergence and its curl within a simply connected region and its normal component on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be proved by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Assume that two different vector fields A and A′ have the same divergence and curl ∇ · A = ∇ · A′ = s (1) ∇ × A = ∇ × A′ = c (2) and their normal component An and A′ n on the boundary are also the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Let F = A − A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Then ∇ · F, ∇ × F, and Fn are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus there exists a φ such that F = −∇φ and it satisfies ∇2φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (3) Due to Earnshaw’s theorem, the solution to (3) must have its maximum and minimum on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Together with the condition Fn = 0, or ∂φ ∂n = 0, it must have that φ is a constant everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Hence F = 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus A and A′ are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Given the proof of uniqueness, we then review the Helmholtz theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The Helmholtz theorem states [28]–[30]: A vector A with both source and circulation densi- ties vanishing at infinity may be written as the sum of two parts, one of which is irrotational, curl-free, or longitudinal, the other of which is solenoidal, div- free, or transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' That is, A = A∥ + A⊥ = −∇φ + ∇ × P (4) where A∥ = −∇φ and A⊥ = ∇ × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Here we use A∥ and A⊥ to denote longitudinal fields and transverse fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='1 Apparently ∇×A∥ = 0 and ∇·A⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Since a vector can be uniquely defined by its divergence ∇ · A = s 1It should be noted that the term longitudinal and transverse are first used in the context of plane wave, where ∇ × A = 0 → k × A = 0 and ∇ · A = 0 → k · A = 0, suggesting that the longitudinal (transverse) field is parallel (perpendicular) to the propagation direction in the Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Here we port the use of these terms to curl-free and divergence free in general cases, with a caveat that they do not have the same physical meaning as in plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' and curl ∇ × A = c, taking the divergence and curl of the above, we have ∇ · A = −∇2φ = s (5) ∇ × A = ∇ × ∇ × P = c (6) where s acts as the source for Poisson equation, and c the source for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We proceed the proof of Helmholtz theorem by showing the expression for φ and A that recover s and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Since the solution of P in (6) is not unique, we set ∇ · P = 0, and (6) becomes vector Poisson equation with a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The solution to (5) and (6) are then well known as [30] φ(r) = 1 4π � s(r′) |r − r′|dr′ (7) P(r) = 1 4π � c(r′) |r − r′|dr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (8) The above can be proved by direct substitution of (7) and (8) into (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Solution Space of Wave Equations Helmholtz theorem can be used to explain the null-space solution of electric field wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We consider the elec- tric field wave equations in a homogeneous medium without sources ∇ × ∇ × E = ω2εµE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (9) It can be formulated as an eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' According to the Helmholtz theorem, the solution to (9) must be either div- free, or curl-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Apparently curl-free solutions are the null- space solutions to (9), where the eigenvalues are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' On the other hand, div-free solutions are the non-null-space solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This is because when ∇ · E = 0, we can add −∇∇ · E to the left-hand side of (9) without changing the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Then the left-hand side becomes ∇×∇×E−∇∇·E = −∇2E and the Laplacian ∇2 is a negative definite operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus the div-free solutions must have nonzero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Helmholtz theorem also connects the solution to the vector potential wave equation with the solution to the electric field wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We consider the vector potential wave equations with Lorenz gauge ∇ · A = −εµ∂tΦ in the same homogeneous medium without sources ∇ × ∇ × A − ∇∇ · A = ω2εµA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (10) Both (9) and (10) have two families of solutions following the Helmholtz theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The div-free solution to (9) must also be the div-free solution to (10), and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This is because the second term of (10) can be dropped in the case of div-free solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, since null-space solution does not exist in (10) due to the negative-definiteness of Laplacian, the space spanned by the null-space solutions to (9) must be filled by the same number of non-null-space solutions to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Again due to the Helmholtz theorem, these solutions are curl-free solutions in (10) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The mapping between the solution space of (9) and (10) is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Apparently curl- free vector potential in electrodynamics leads to zero E and H even though A and Φ are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This also justifies the use of Φ = 0 gauge due to Lorenz gauge to remove the redundancy of 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Mapping between solution space of the electric field wave equation and vector potential wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The double-headed arrow represents the same space spanned by the eigenvectors of both equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' the curl-free solutions in homogeneous case for many practical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A New Look of Helmholtz Theorem We can also prove Helmholtz theorem from a different perspective by looking at (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We first prove that the curl operator is a self-adjoint operator [31] using inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We begin with ⟨A∗, CB⟩ = � V dr A∗ · (∇ × B) = � V dr B · (∇ × A∗) − � S dr ˆn · (A∗ × B) (11) where C denotes the curl operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' With the boundary condi- tion that ˆn × A = 0 and ˆn × B = 0,2 and the surface integral term vanishes, and hence the curl operator is self-adjoint for fields of the prescribed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' With the same approach, it can also be proved that ∇ × ∇× is a self-adjoint operator [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus the solution vectors to (9) form a complete basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The proceeding proof also suggests that the curl operator has the same null-space as its power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In general when an operator maps a domain space function (or a row space vector, if the operator is a finite-sized matrix) into its range space (or column space), the new function may contain null-space component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, the self-adjointness and symmetry of the curl operators ensure that its domain space and range space are the same, and its (right-)null-space and left-null-space are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus the curl operator maps a domain space function into the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Hence it will not generate a null-space component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Now we are ready to prove the Helmholtz theorem by analyzing (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We only consider the non-trivial solution to (9) where E ̸= 0 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Taking the divergence of (9), we have ω2∇ · E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It has three possible classes of solutions: (i) ω = 0 and ∇ · E ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (ii) ω ̸= 0 and ∇ · E = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' and (iii) ω = 0 and ∇·E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We can rule out (iii) because it would make (9) the same as (10), and ω = 0 solution cannot exist in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This is because ∇ × ∇ × A − ∇∇ · A = −∇2A and the Laplacian ∇2 is a negative definite operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For (i), apparently ∇ × ∇ × E = 0, and in turn ∇ × E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For (ii), 2The rationale of this boundary condition is explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' we have ∇ × E ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Concluding the above, we can clearly see two classes of solution to (9) similar to the Helmholtz theorem, where 1) ∇ × E = 0, ∇ · E ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 2) ∇ · E = 0, ∇ × E ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Due to the self-adjointness of ∇ × ∇× operator, the solution to (9) form a complete basis to the entire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Hence any function satisfying the proposed boundary condition can be separated into two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This completes the proof of Helmholtz theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Note that analyzing (10) also leads to the same two families of solutions described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This will be shown in greater details similar to the inhomogeneous media case next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Motivation of Generalization As shown above, the Helmholtz theorem works well in explaining the solution space of the electric field wave equa- tion and vector potential wave equation with Lorenz gauge in homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, in inhomogeneous dis- persionless media, it is suggested that generalized Lorenz gauge should be used instead [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus, to better explain the eigenmodes found by potential-based formulation, it would be ideal if Helmholtz theorem can be generalized in a way that ∇ · ε(r)Aε⊥(r) = 0,3 where Aε⊥ denotes the div-ε-free (or generalized transverse) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Considering this, we seek to prove the following generalized Helmholtz decomposition: The vector field A given by the Helmholtz wave equation in source-free inhomogeneous media can be decomposed into two components Aε⊥ and A∥, where the div-ε-free (or generalized transverse) fields satisfy ∇ · εAε⊥ = 0 and the curl-free (or longitudinal) fields satisfy ∇ × A∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In the remainder of this work, we use solution of the wave equation for vector potential A to arrive at the generalized Helmholtz decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A modified generalized Lorenz gauge ∇ · εA = −ε2 0µ0∂tΦ is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We follow matrix theory generally in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It will be shown that the vector potential wave equation has two classes of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' They degenerate into the Helmholtz decomposition in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus the vector potential wave equation for inhomogeneous media can be used to motivate the generalized Helmholtz decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We also note that there is no null space in the resulting generalized eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' PROOF OF ORTHOGONALITY We start from proving the orthogonality between the two families of fields proposed in the generalized Helmholtz decomposition, assuming the decomposition does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We leave it to the next section to demonstrate the existence of the decomposition through numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We consider the space terminated by continuous perfect electric conductor (PEC) boundary, filled with inhomogeneous media (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Because of the gauge ∇ · εA = −ε2 0µ0∂tΦ, the boundary condition for ∇ · εA is actually that for Φ on a 3Position-dependent ε(r) is assumed from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For simplicity, we only write ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Electric Field Wave Equation Vector Potential Wave Curl-free (null- Curl-free solution Equation space) solution (dynamic) Divergence-free Divergence-free solution (dynamic) solution (dynamic)4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Schematic of the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' PEC, which is Φ equals a constant voltage [16], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' If there is only one PEC, we can choose this as our reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus ˆn × A = 0 (12) which is consistent with ˆn × E = 0, and ∇ · εA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (13) The latter is equivalent to Φ = 0 on the boundary when the PEC has a constant voltage of zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' “grounded”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' If there are additional PECs inside the domain, they may not be grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In this case, ∇ · εA = Φ is equal to a constant voltage on each conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Additionally, charge neutrality constraint can be enforced on each conductor, which requires [16], [18] � S dr ˆn · εA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (14) where the surface integral is over the surface of each PEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Next, we proceed to show the proof of orthogonality first in continuous space, followed by a similar procedure in discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Different definitions of orthogonality will be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It will also be shown that discrete approximation of Maxwell’s equations is homomorphic to the continuum case if done properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Continuous Space Wave equation is usually the starting point of many A- Φ based solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We start the mathematical proof of mode orthogonality by defining proper operator and vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We consider the wave equation for vector potential A in lossy dispersionless inhomogeneous isotropic media4 with modified generalized Lorenz gauge [2] ∇ × 1 µ∇ × A − 1 ε2 0µ0 ε∇∇ · εA = ω2εA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (15) Let W1 = ∇ × 1 µ∇× (16) W2 = − 1 ε2 0µ0 ε∇∇ · ε (17) L = W1 + W2 (18) U = ε (19) λ = ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (20) 4To satisfy the Kramers–Kronig relations, this assumption is only valid over a narrow bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Then (15) can be written abstractly and concisely as (W1 + W2)A = λUA (21) where the operator notation is defined as a differential operator acting on a function A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Note that we use boldface calligraphic font with an overbar to denote a Hilbert space operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The function A in the abovementioned space is in turn square integrable [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It has been proved that the L operator is self-adjoint in lossless media given the boundary conditions defined in the beginning of this section [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Here we investigate the general case with lossy media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Since the L operator consists of two terms W1 and W2, it is clearer to look at the two terms separately [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We first define Aε⊥ to be the solution to the eigenvalue problem W1Aε⊥ = λε⊥UAε⊥ with eigenvalue λε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Here we only study the solutions with nonzero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This gives us, from the definition above, ∇ × 1 µ∇ × Aε⊥ = λε⊥εAε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (22) Taking divergence of the above equation gives 0 = λε⊥∇ · εAε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (23) Thus Aε⊥ is in the null space of the W2 operator defined in (17), and it is also a solution to the original eigenvalue problem (21) with eigenvalue λε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Similarly, we define A∥ to be the solution to W2A∥ = λ∥UA∥ with nonzero eigenvalue λ∥, or more explicitly, − 1 ε2 0µ0 ε∇∇ · εA∥ = λ∥εA∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (24) Dividing the above by ε and taking curl of the resulting equation yields 0 = λ∥∇ × A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (25) Thus A∥ is in the null space of the W1 operator defined in (16), and it is also a solution to the original eigenvalue problem (21) with eigenvalue λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Next, we will manipulate the above equations, and use (23) and (25) to prove orthogonality of two sets of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 1) Proof 1: To begin, we take the left dot product of (24) by Aε⊥, and integrate it over the space filled with dielectrics to yield − 1 ε2 0µ0 � V dr � Aε⊥ · ε∇∇ · εA∥ � = � V dr � Aε⊥ · λ∥εA∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (26) Using integration by parts, the left-hand side becomes 1 ε2 0µ0 � V dr (∇ · εAε⊥)(∇ · εA∥) − 1 ε2 0µ0 � S dr ˆn· � (∇ · εA∥)εAε⊥ � (27) where the surface integral is over the PEC surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' From (23), the first term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' At the grounded PEC boundary, E1 E2 PEC5 ∇ · εA∥ = 0, and then the second term also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' At the PEC not connected to the ground, the second term becomes − 1 ε2 0µ0 (∇ · εA∥) � S dr ˆn · εAε⊥ = 0 (28) where the divergence term is taken outside the integral because Φ is equal to a constant voltage on the conductor, and the integral vanishes because of the charge neutrality constraint (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus ⟨Aε⊥, εA∥⟩ = 0 (29) as long as λ∥ ̸= 0, where ⟨F, G⟩ = � F · G dr is defined as the reaction inner product by Rumsey [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In other words, the two sets of modes Aε⊥ and A∥ are ε-orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 2) Proof 2: Now, taking the left dot product of (22) with A∥, and integrating it over space yields � V dr � A∥ · ∇ × 1 µ∇ × Aε⊥ � = � V dr � A∥ · λε⊥εAε⊥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (30) The right-hand side is denoted λε⊥⟨A∥, εAε⊥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Using inte- gration by parts, the left-hand side becomes � V dr � 1 µ∇ × Aε⊥ � � ∇ × A∥ � + � S dr ˆn· �� 1 µ∇ × Aε⊥ � × A∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (31) From (25), the first term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Implementing the boundary condition that ˆn×A∥ = 0, the second term also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus ⟨A∥, εAε⊥⟩ = 0 (32) as long as λε⊥ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In other words, the two sets of modes A∥ and Aε⊥ are ε-orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In the above derivation, if we replace A∥ by A∗ ∥, the derivation is still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' That is ⟨A∗ ∥, εAε⊥⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (33) However, the same token does not apply to Proof 1 in III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This may motivate new definition of inner product in the problems concerning potential-based formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Discrete Space The problem in the previous subsection is homomorphic to numerical linear algebra if we find the matrix represen- tation of the operators using subspace projection methods [34], including finite difference method [35], finite element method [36], or discrete exterior calculus [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' These methods should be chosen properly to preserve certain properties of the continuum calculus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' ∇·(∇×A) = 0 and ∇×∇f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In this work we use finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' After discretization, we have ∇ × A ⇒ C1 · A1 (34) ∇ × ∇ × A ⇒ C2 · C1 · A1 (35) ∇ · εA ⇒ D1 · A1 (36) ε∇f ⇒ E1 · f1 (37) ε ⇒ U (38) where the operators on the LHS are in the continuous Hilbert space, while the RHS are the linear algebra approximation of the continuum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Here we use boldface font with overbar to denote finite size matrices, and sans serif typestyle to denote finite length vectors stored as 1D arrays in computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Due to the discretization, each operator may have different representations depending on the space it operates on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' To use finite difference method, Yee grid is constructed with both primal and dual grids [35], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' On each grid, we place a 3D vector on the face of the cube, and a scalar on the center of the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Details of the discretization scheme can be found in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In (34) to (38), the discrete operators with subscript 1 denotes operations on the primal mesh, while subscript 2 denotes operations on the dual mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The discrete curl operator C1 acts on a vector on the primal mesh, resulting in a vector on the dual mesh, while C2 acts on a vector on the dual mesh and results in a vector on the primal mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be shown that these two discrete curl operators are transpose to each other C1 = C T 2 , where T denotes matrix transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Also, it is easy to prove that the discrete divergence operator and the negative of discrete gradient operator are transpose to each other D T 1,2 = −E1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The U operator is a square matrix whose dimension depends on the vector it operates on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The following vector identities are preserved after discretization using finite difference method [39]–[41] ∇ · (∇ × A) = 0 ⇒ D2 · U −1 2 C1 · A1 = 0 and D1 · U −1 1 C2 · A2 = 0 (39) ∇ × ∇f = 0 ⇒ C1 · U −1 1 E1 · f1 = −C1 · U −1 1 D T 1 · f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (40) Without loss of generality, we let µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 Also we denote α = 1/ε2 0µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We assume the solution vector is on the primal mesh, and thus the subscript 1 is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Then (21) becomes C2 · C1 · A − αE1 · D1 · A = λU1 · A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (41) We break the above into two eigenvalue problems corre- sponding to the two operators on the left-hand side C2 · C1 · Aε⊥ = λε⊥U1 · Aε⊥ (42) −αE1 · D1 · A∥ = λ∥U1 · A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (43) Taking the left dot product of (42) with D1 · U −1 1 , we get D1 · U −1 1 C2 · C1 · Aε⊥ = λε⊥D1 · U −1 1 U1 · Aε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (44) 5It can be seen that the proof is still valid when µ is inhomogeneous, anisotropic, or complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 6 Due to (39), LHS of the above vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus D1 · Aε⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (45) Taking the left dot product of (43) with C1 · U −1 1 , we get −αC1 · U −1 1 E1 · D1 · A∥ = λ∥C1 · U −1 1 U1 · A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (46) Due to (40), the LHS of the above vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus C1 · A∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (47) 1) Proof 1: Taking the left dot product of (43) with AT ε⊥ and using the symmetry property between divergence and gradient operators, we have αAT ε⊥ · D T 1 · D1 · A∥ = λ∥AT ε⊥ · U1 · A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (48) Due to (45) and its transpose, LHS of the above vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus AT ε⊥ · U · A∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (49) which is analogous to (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 2) Proof 2: Now, taking the left dot product of (42) with AT ∥ , where † denotes conjugate transpose, and using the symmetry property of curl operator, we have AT ∥ · C T 1 · C1 · Aε⊥ = λε⊥AT ∥ · U1 · Aε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (50) Due to (47) and its transpose, LHS of the above vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus AT ∥ · U1 · Aε⊥ = 0 (51) which is homomorphic to (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, we can also left dot product (42) by A† ∥, and get A† ∥ · C T 1 · C1 · Aε⊥ = λε⊥A† ∥ · U1 · Aε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (52) Due to (47) and its Hermitian conjugate, and the realness of the curl operator under finite difference method, LHS of the above vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus A† ∥ · U · Aε⊥ = 0 (53) which is analogous to (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The above shows that the finite difference approximation of Maxwell’s equations is homomor- phic to the continuum case if done properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Anisotropic Case The mathematical proof in the previous two subsections still applies in anisotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Consider the wave equa- tion for vector potential A in dispersionless inhomogeneous anisotropic media with modified generalized Lorenz gauge in frequency domain [2] ∇ × µ−1∇ × A − 1 ε2 0µ0 ε · ∇ (∇ · ε · A) = ω2ε · A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (54) The mathematical proof of mode orthogonality follows exactly as Section III-A for continuous space and as Section III-B for discrete space, with the requirement that ε = εT (55) which implies that the material is reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Geometries of the numerical examples with dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' NUMERICAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Numerical Demonstration of Completeness In this section, we first demonstrate that the two sets of modes given in the previous section are complete in the discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We solve (15) numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Finite difference method is used corresponding to the discrete space proof in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Uniform discretization is used, and the discretiza- tion scheme follows [15], [39], as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For an empty or homogeneous structure, the proposed gen- eralized Helmholtz decomposition degenerates into the usual Helmholtz decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' To demonstrate it, we consider a rectangular 7 × 8 × 6 cavity homogeneously filled with a εr = 3 material, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Discretization step is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5, resulting in 6938 × 6938 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We use natural unit so that ε0 = µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' After solving the generalized eigenvalue problem, we take divergence and curl on each mode to identify the two classes of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Note that the indices of curl-free modes and div- ε-free modes are skipped in (b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The total number of div-ε-free and curl-free modes equals to the dimension of the eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus it clearly shows two distinct groups of modes corresponding to the Helmholtz decomposition, which complete the discrete space, and are consistent across field-based and potential-based formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We then consider a more general example with anisotropic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' As shown in 3(b), we construct a cylindrical cavity with radius r = 2 and height h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It is filled with a block of anisotropic material centered at (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='2) with dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='8 in x, y, z directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Discretization step is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='2, resulting in 5822×5822 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The permittivity tensor of the anisotropic material is ε = � � 2 0 1 + 3i 0 3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='2i 0 1 + 3i 0 2 � � (56) which satisfies reciprocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The matrix of anisotropic permit- tivity is treated the same way as [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The discretized operators satisfy the discrete vector identities (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Mode degeneracy is observed after solving the generalized eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We apply the analysis described in Ap- pendix B to separate degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The results are then displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We find that the modes can be categorized into two distinct groups: 1) div-ε-free (or generalized transverse) modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 5(a)): ∇ · εA = 0 for the entire grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (a) (b) 4 6 87 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Two set of modes found in the geometry described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(a) can be clearly distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (a) The absolute value of eigenvalues are shown for both potential-based formulation and E-H formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be seen clearly that the potential-based formulation does not have null space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For the potential- based formulation, only the div-ε-free (or generalized transverse) modes are shown in (b), and only the curl-free (or longitudinal) modes are shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Totally there are 4793 div-ε-free modes in (b) and 2145 curl-free modes in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The latter is consistent with the number of null-space eigenvalues with E-H formulation in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The total number of both families of modes is equal to the dimension of the eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The indices of curl-free modes and div-ε-free modes are skipped in (b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be seen that the div-ε-free modes satisfy ∇·εA = 0, while the curl-free modes satisfy ∇ × A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Blue open circles: for each mode, take ∇ · εA, and record the maximum value of abs(∇ · εA) in the entire grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Red stars: for each mode, take ∇ × A, and record the maximum magnitude of all components in the entire grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 2) Curl-free (or longitudinal) modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 5(b)): ∇×A = 0 for the entire grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='6 It can be seen that each mode is either div-ε-free, or curl-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The div-ε-free modes are consistent with the modes found by E-H formulation since (15) returns to the wave equation of E field [2], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The curl-free modes have zero E and H field by the fact that B = ∇ × A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In both numerical examples, the two classes of modes form a complete basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Although the completeness of basis expanded by the eigenvectors in the discrete space from the numerical examples does not necessarily guarantee the completeness of the two sets of modes in the continuous Hilbert space, one could use convergence and existence of the numerical solution to a partial differential equation with finite difference approximation [44] to imply completeness of these modes in the continuous Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus the numerical proof could motivate further study in the continuous Hilbert space as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 6The curl-free modes can be eliminated by Φ = 0 gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This will be explained in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Two set of modes found in the geometry described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(b) can be clearly distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Totally there are 4049 div-ε-free modes and 1773 curl-free modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The description of the figure follows the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 4(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Numerical Validation of Orthogonality We then proceed to validate the orthogonality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For the numerical example described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(b), we calculate the four sets of inner products between the div-ε-free modes and curl-free modes defined in Section III-B after separating the degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The colormaps of the resulting matrices are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 6, and the maximum magnitude of the matrix elements are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' For readability, only the first 100×100 elements of each matrix are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 6, but these results are representative of the full matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 6 and Table I that ⟨Aε⊥, εA∥⟩ = 0 (57) ⟨A∥, εAε⊥⟩ = 0 (58) ⟨A∗ ε⊥, εA∥⟩ ̸= 0 (59) ⟨A∗ ∥, εAε⊥⟩ = 0 (60) which are consistent with the mathematical proofs in Sec- tion III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This result, supplemented by the numerical proof of completeness in the previous subsection, completes our demonstration of the generalized Helmholtz decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' TABLE I ORTHOGONALITY RESULTS FROM THE SECOND NUMERICAL EXAMPLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' THE MAXIMUM MAGNITUDE OF MATRIX ELEMENTS IN THE CORRESPONDING MATRICES ARE SHOWN IN THE TABLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' HERE | · | DENOTES ELEMENT-WISE MAGNITUDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' max(|V T ε⊥ · ε · V∥|) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='1694e-10 max(|V T ∥ · ε · Vε⊥|) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='1694e-10 max(|V † ε⊥ · ε · V∥|) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5779 max(|V † ∥ · ε · Vε⊥|) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='2430e-10 (a) abs(eigenvalue) 100 vector-potential-based formulation E-Hformulation Index: 2145 10-20 1000 2000 3000 4000 5000 6000 mode index curl-free modes only (c) 100 100 max(V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' EA) max(V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='EA) max(V × A) max(V × A) 米 米 10-10 10-10 米 米米*米 2000 4000 6000 2000 4000 6000 mode index mode indexdiv-e-free modes only (a) (b) curl-free modes only 100 100 max(V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='·A) max(V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='·A) max(V × A) max(V × A) 米 米 000 10-10 10-10 米米 888811888 巫 变 1000 3000 5000 1000 3000 5000 modeindex modeindex8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Color map of four sets of inner products from the numerical example described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (a) V T ε⊥ ·ε·V∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (b) V T ∥ ·ε·Vε⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (c) V † ε⊥ ·ε·V∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (d) V † ∥ · ε · Vε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Here Vε⊥(V∥) is a matrix where each column vector is a div-ε-free (curl-free) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Each pixel represents the magnitude of the corresponding matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The indices are the indices of matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Note the different scales of the colorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The values of the resulting matrices are essentially 0 in (a), (b), and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' CONCLUSION In this work, we combine mathematical proof and numer- ical results to demonstrate the generalization of Helmholtz decomposition to inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The two families of fields in the generalized Helmholtz decomposition, div-ε- free and curl-free fields, are connected to the two classes of solutions to the vector potential wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The div-ε- free and curl-free solutions form a complete set of the basis, as demonstrated in the numerical results and its discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' They are also orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The div-ε-free field manifests charge-free condition, while the curl-free field is associated with charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' In the literature, the curl-free fields are often eliminated by setting Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' However, when there are charges present, we need both families of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It also provides a theoretical background to the numerical quantization [1] based on the generalized Lorenz gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Future work will be devoted to the application of Φ = 0 gauge and rigorously determining suitable conditions for eliminating the curl-free modes in general inhomogeneous media while preserving full rank of the resulting system matrix for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' APPENDIX A DISCRETE VECTOR CALCULUS ON A GRID Discrete vector calculus on a Yee lattice [35], [38] is the basis for the proof of orthogonality in discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We give a brief overview of the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The details can be found in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The staggered grid with both primal (solid line) and dual (dashed line) grids are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The dual grid is shifted Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Schematic of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The primal and dual grids are shifted by half grid step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The solution vector A is placed on the face of the primal grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' half a grid width in all three directions from the primal grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' On each grid, we place a 3D vector on the face of the cube, and a scalar in the center of the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' We index the grid point at the cube center of the primal grid by integers (i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Below we explicitly formulate the discrete differential operations, where central difference is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A discrete gradient g = ∇f on the primal grid is given by gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k x = f i+1,j,k − f i,j,k ∆x , (61) gi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y = f i,j+1,k − f i,j,k ∆y , (62) gi,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z = f i,j,k+1 − f i,j,k ∆z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (63) A discrete divergence f = ∇ · g on the primal grid is given by f i,j,k = gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k x − gi−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k x ∆x + gi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y − gi,j−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y ∆y + gi,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z − gi,j,k−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z ∆z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (64) A discrete curl h = ∇ × g on a vector on the primal grid resulting in a vector on the dual grid is given by hi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 x = gi,j+1,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z − gi,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z ∆y − gi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k+1 y − gi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y ∆z , (65) hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 y = gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k+1 x − gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k x ∆z − gi+1,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z − gi,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z ∆x , (66) hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k z = gi+1,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y − gi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y ∆x − gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j+1,k x − gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k x ∆y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (67) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='Vel Vel-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='v ×10-11 ×10-11 (a) (b) 5 5 20 20 4 4 40 40 3 3 60 60 2 2 80 80 1 1 100 100 20 60 100 20 60 100 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='V × 10-11 (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='05 6 20 20 40 40 4 60 60 2 80 80 100 0 100 20 60 100 20 60 100Ay A 29 A discrete curl g = ∇ × h on a vector on the dual grid resulting in a vector on the primal grid is given by gi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k x = hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k z − hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k z ∆y − hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 y − hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 y ∆z , (68) gi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k y = hi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 x − hi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 x ∆z − hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k z − hi−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k z ∆x , (69) gi,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 z = hi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 y − hi−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,j,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 y ∆x − hi,j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 x − hi,j−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5,k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='5 x ∆y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (70) APPENDIX B ANALYSIS OF DEGENERATE MODES Given a set of eigenmodes A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' , An with corre- sponding eigenvalues λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' , λn obtained from the gen- eralized eigenvalue problem, the space expanded by 1 λi ε−1∇ × 1 µ∇ × Ai (71) has the same dimension as the number of independent div-ε- free modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This is because the curl-free modes are eliminated by the operation due to (25) λ∥∇×A∥ = 0, and the div-ε-free modes are recovered due to (22) ∇× 1 µ∇×Aε⊥ = λε⊥εAε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Besides, the space expanded by − 1 λi 1 ε2 0µ0 ∇∇ · εAi (72) has the same dimension as the number of independent curl-free modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' This is because the div-ε-free modes are eliminated by the operation due to (23) λε⊥∇ · εAε⊥ = 0, and the curl-free modes are recovered due to (24) − 1 ε2 0µ0 ε∇∇ · εA∥ = λ∥εA∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The dimension of the two spaces from the numerical exam- ple described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(b) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The number of nonzero singular values indicate the dimension of the space spanned by a set of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be seen that the sum of the dimension of the space of (71) and (72) equals the dimension of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Thus it proves that the div-ε-free and curl-free modes as defined in the main text form a complete basis of the discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Otherwise, if there exists an eigenmode that is nonzero after both (71) and (72) operations, it must contribute one “extra dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Degenerate modes with the same eigenvalue can also be separated based on this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Given n degenerate modes A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' , An with the same eigenvalue, we first learn the number of independent div-ε-free and curl-free modes by evaluating the dimension of the subspace expanded by (71) and (72), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The div-ε-free modes are isolated by (71) because of (22), and the curl-free modes are isolated by (72) because of (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Gram-Schmidt orthogonalization is then performed within each subspace to extract the independent div-ε-free and curl-free modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The eigenvalues and the resulting solution space of the geometry described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 3(b) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (a) Eigenvalues are shown in blue curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Orange upward spikes denote a cluster of degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (b) The singular values of the matrix constructed by lining up ε−1 ·∇× 1 µ∇×Ai are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be seen that there are 4049 nonzero singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' (c) The singular values of the matrix constructed by lining up −∇∇·ε·Ai are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' It can be seen that there are 1773 nonzero singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The number of nonzero singular values indicate the dimension of the space spanned by a set of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The total number of nonzero singular values from (b) and (c) is 5822, equal to the dimension of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' A closer look at the first cluster of degenerate modes (index 1428 to 1451) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Similar results are observed for the other clusters of degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Singular values of (a) and (b) are obtained in the same way as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' 8(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' The total number of nonzero singular values from (a) and (b) is 24, equal to the dimension of the degenerate subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Na, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} +page_content=' Zhu, W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfzgXI/content/2301.03446v1.pdf'} diff --git a/mtAyT4oBgHgl3EQflPh9/content/tmp_files/2301.00449v1.pdf.txt b/mtAyT4oBgHgl3EQflPh9/content/tmp_files/2301.00449v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..855e883fe920aaa4693fed3bad4f4b7443f38b4c --- /dev/null +++ b/mtAyT4oBgHgl3EQflPh9/content/tmp_files/2301.00449v1.pdf.txt @@ -0,0 +1,2241 @@ +Magnetized Thick Disks around Boson Stars +Kristian Gjorgjieski,∗ Jutta Kunz,† and Matheus C. Teodoro‡ +Department of Physics, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany +Lucas G. Collodel§ +Theoretical Astrophysics, University of T¨ubingen, 72076 T¨ubingen, Germany, +Petya Nedkova¶ +Department of Theoretical Physics, Sofia University, Sofia 1164, Bulgaria +(Dated: January 3, 2023) +The effects of magnetic fields on accretion disks around compact objects are of high importance +in the study of their general properties and dynamics. Here we analyze the influence of magnetic +fields on thick accretion disks around rotating boson stars. We assume a uniform constant specific +angular momentum distribution and a polytropic equation of state. The purely hydrodynamical +thick disk solutions are extended to magnetized solutions by adding a toroidal magnetic field and +then analyzed in terms of a magnetization parameter βmc. We consider one-centered solutions as +well as two-centered solutions and focus on retrograde tori, since they are more distinctive due to +their unique properties. Our computed solutions indicate that strong magnetic fields influence the +characteristics of thick disks around rotating boson stars and possibly effect their unique features. +1. +INTRODUCTION +Accretion disks are formed around various astrophysical objects ranging from supermassive active galactic nuclei to +coalescent stellar mass black holes and neutron stars binaries. Converting gravitational energy into radiation they +harbor many high-energy astrophysical phenomena and thus can serve as a natural arena for probing the gravitational +physics around compact objects. With the development of the imaging techniques we are now able to observe directly +the accretion disks in the nearby galactic targets M87 and Sgr A*, which provides a unique opportunity for gaining +knowledge about the nature of the compact objects in their centers and the physical processes in the accreting plasma +[1–3]. +The interpretation of the astrophysical observations relies considerably on the accretion disk modeling. Currently gen- +eral relativistic magneto-hydrodynamic (GRMHD) simulations are developed which are able to integrate numerically +the coupled Einstein-Euler equations thus providing a precise description of the self-gravitating accreting system. +However, these simulations produce a complicated picture depending on many parameters which can be hard to +disentangle and interpret. In this respect simplified semi-analytical models can be extremely useful since they may +capture the main effects from the non-linear treatment with less computational cost and offer more physical intuition +and predictability. +Some of the basic constructions which describe geometrically thick accretion disks are the equilibrium tori. They +represent the simplest solutions to the relativistic Euler equations assuming a non-selfgravitating perfect fluid and +neglecting the electromagnetic, viscosity and radiation terms. +Thus, they provide the equilibrium states of the +relativistic matter orbiting on non-geodesic trajectories around the compact object, which result from the balance of +the pressure gradients and the gravitational and centrifugal forces. +The theory of the equilibrium tori has a long history dating back to the classical works [4–6] where the equilibrium +tori for the Kerr black hole were obtained. Assuming an isentropic or barotropic equation of state for the perfect fluid +it was demonstrated that in this case the constant angular velocity and the constant angular momentum surfaces +coincide. This statement, known as the von Zeipel theorem, represents an integrability condition for the relativistic +Euler equation and therefore it allows the construction of analytical solutions. +Various equilibrium tori can be constructed possessing a different profile of the specific angular momentum [7, 8]. +Solutions with a constant angular momentum play a fundamental role since they already capture the main qualitative +∗ kristian.gjorgjieski@uol.de +† jutta.kunz@uni-oldenburg.de +‡ matheus.do.carmo.teodoro@uni-oldenburg.de +§ lucas.gardai-collodel@uni-tuebingen.de +¶ pnedkova@phys.uni-sofia.bg +arXiv:2301.00449v1 [gr-qc] 1 Jan 2023 + +2 +features of the more general configurations and represent the marginally stable case. The physical conditions can be +further extended by adding a toroidal magnetic field which allows for the construction of analytical solutions under +the same integrability conditions [9] (see also [10] and [11] for non-constant angular momentum configurations). This +generalization is particularly important for the astrophysical applications since magnetic fields play a dynamical role +in astrophysical scenarios and the magnetized equilibrium configurations provide suitable initial conditions for the +GRMHD simulations of the accretion process or jet formation (see e. g. [12, 13]). +Although the accretion disk theory was developed considering mainly the Kerr black hole, fundamental physics +suggests that more diverse compact objects may exist in nature. This motivates the construction of accretion disk +models in more exotic spacetimes with the view of confronting the simulations with observational data and searching +for astrophysical signatures of new fundamental objects [14–17]. In this line of research thick accretion disks around +black holes in the modified theories of gravity were studied [18–21], equilibrium configurations in de Sitter background +[22, 23] or interacting with an external matter distribution [24, 25], as well as thick disks in naked singularity spacetimes +[26, 27]. +In our work we concentrate on boson stars. Boson stars represent compact scalar field condensates which form as +equilibrium states resulting from the gravitational collapse of a self-gravitating massive scalar field [28–30]. They +produce a strong gravitational field and negligible electromagnetic emission, in this way they are mimicking the +properties of black holes. Various boson star configurations have been constructed [31–40], and their properties were +investigated such as stability [41–43], geodesic motion [44–49] and tidal effects in the motion of gas clouds around +them [50, 51]. Since they are horizonless compact objects, which are not characterized by a solid boundary but rather +by a decaying profile of the scalar field density towards infinity, the particles and light propagation in their vicinity +is specific leading to phenomenological signatures. For example, boson stars may possess only bound circular orbits +which extend into the inner regions with high density of the scalar field distribution [47]. In addition, photon rings +may be absent which prevents the formation of a shadow in the strict sense. Yet, if we assume the presence of an +accretion disk, a central dark region will be observed similar to black holes which corresponds to the lensed image of +an accretion disk’s inner edge [14–17]. +The perfect fluid equilibrium tori around boson stars also possess distinctive morphology compared to the Kerr black +hole [47, 52]. The equilibrium tori in the Kerr spacetime are typically characterized by a cusp located at their inner +edge with the exception of a small range of configurations limited to specific angular momenta of the disk [5, 53]. +In contrast, the equilibrium configurations for boson stars constructed in the literature do not form an inner cusp. +This distinction has implications on the possible scenarios for dynamical evolution of the accretion disk such as the +formation of run-away instabilities [8, 53, 54]. Boson stars can further support topologically non-trivial configurations +like two-centered tori which can be disjoint or connected by a cusp. These topologies are absent for the Kerr black +hole, although they occur for other exotic compact objects like naked singularities [22, 23]. Another interesting feature +is the formation of static surfaces for counter-rotating equilibrium tori around boson stars [39, 52]. These surfaces are +defined as the cross-sections where the fluid is at rest with respect to a zero angular momentum observer (ZAMO) +at infinity. Thus, they serve as a boundary separating regions where the fluid moves in prograde and retrograde +direction, respectively. +The aim of our work is to study the equilibrium tori in boson star spacetimes in the presence of a toroidal magnetic field. +We construct constant angular momentum configurations and analyse how their properties are modified with respect +to the purely hydrodynamical case. In particular we describe the qualitative effects which are induced by the magnetic +field and may have observational implications. These include modifications in the location of the predominant disk +density and its compactness, as well as variation of the characteristic geometry of the isodensity surfaces. Another +interesting phenomenon is that the magnetic field can trigger topological transitions between different thick disk +configurations. Thus, after a certain magnitude of the magnetic field, two-centered disks may lose their outer center +and become one-centered. +The paper is organized as follows. In the next section we describe the boson star solutions which we consider and +some of their relevant properties. In section 3 we briefly review the equilibrium tori configurations, which are possible +in boson star spacetimes in the purely hydrodynamical case. In section 4 we describe the construction procedure +of magnetized tori for a perfect fluid with polytropic equation of state. In section 5 we present our results and the +analysis of the properties of the magnetized thick disk, which are illustrated on a range of representative examples. +Section 6 contains our conclusions. + +3 +2. +BOSON STARS +We consider a complex scalar field without self-interaction. The boson stars (BSs) are obtained from the Einstein- +Klein-Gordon equations, derived from the action S +S = +� √−g +� +R +16πG − Lm +� +d4x, +(1) +where g is the metric determinant, R is the Ricci scalar, G is Newton’s constant and Lm is the Lagrangian of the +complex scalar field φ with mass m +Lm = |∂µφ|2 + m2|φ|2. +(2) +The action (1) is invariant under transformations of the global U(1)-symmetry group of the complex scalar field, +φ → φeiτ, with constant τ. According to Noether’s theorem this invariance implies the existence of a conserved +current, ∇µjµ = 0, with +jµ = i (φ∂µφ∗ − φ∗∂µφ) , +(3) +and a conserved Noether charge Q, which is the bosonic particle number given by Q = +� √−gjtd3x. +The variation of the action (1) leads to the coupled set of Einstein-Klein-Gordon equations, +Rµν − 1 +2Rgµν = 8πGTµν, +(4) +� +□ − m2� +φ = 0, +(5) +where Tµν ≡ (∂µφ∂νφ∗ + ∂νφ∂µφ∗) − Lmgµν is the stress-energy tensor and □ denotes the covariant d’Alembert +operator. In order to obtain rotating BSs, the scalar field should depend on all four coordinates as follows [36], +φ ≡ φ0(r, θ)ei(ωt−kϕ). +(6) +A harmonic time-dependence is already needed for the classical non-rotating BSs in order to obtain stable localized +solutions. The presence of rotation implies an additional harmonic dependence of the scalar field on the azimuthal +angle ϕ. In the Ansatz (6) ω is the angular frequency of the scalar field and k is the azimuthal winding number. +Due to the condition φ(ϕ = 0) = φ(ϕ = 2π) the winding number k must be an integer (k signals the strength of +the angular excitations since it counts the nodes, 2k, of the real and imaginary parts of the scalar field along the +azimuthal direction). +The angular momentum J of the BSs is given by a quantisation law [36], +J = kQ, +(7) +thus it is an integer multiple of the charge Q. This law follows directly from the relation T t +ϕ = njt with ∂ϕφ = inφ. +The harmonic Ansatz for the scalar field yields a stress-energy tensor that does not depend on the coordinates t and +ϕ. Thus solutions with a stationary and axially symmetric metric result, implying the presence of two Killing vectors +of the metric associated with these two coordinates. The line element of the BS spacetime metric can then be written +as, +ds2 = − α2dt2 + A2 � +dr2 + r2dθ2� +(8) ++ B2r2 sin2 θ (dϕ + βϕdt)2 , +(9) +with α the lapse function, βϕ the shift function, and the functions A, B, α, β which depend only on r and θ. +Substituting the harmonic Ansatz (6) into the field equations leads to a coupled set of partial differential equations for +the functions, which were solved numerically. The BSs constructed are asymptotically flat. Thus the metric approaches +asymptotically Minkowski spacetime and the scalar field vanishes exponentially proportional to e− +√ +m2−ω2r/r. The +solutions were computed with the FIDISOL/CADSOL package, which is a PDE solver which employs a finite differ- +ence method of discretization together with a Newton-Raphson scheme to linearize the resulting system of algebraic +equations [55]. Solutions exist only for a set of angular frequencies ω. + +4 +(a) +(b) +Fig. 1. (a) Mass M of BSs versus boson field frequency ω. The dashed circles mark the solutions used for the computation of +the magnetized disks. The dashed red line marks the threshold value of ω below which the solutions contain ergoregions. (b) +The amplitude of the scalar field φ versus the normalized radial coordinate r/M for the BS solutions. The highlighted curves +show the solutions marked in (a), namely ω = {0.671, 0.798, 0.960}. +Here we consider only rotating BSs with winding number k = 1. Furthermore, we focus only on solutions without +ergoregions. Therefore the minimal angular frequency taken into account is ω = 0.655. Solutions with smaller angular +frequencies would possess ergoregions [46]. It should be noted, that the equations feature a scaling symmetry, which +we exploit to go to dimensionless quantities by scaling with the boson mass m, i.e., ˜r = rm, ˜ω = ω +m and ˜ +M = Mm. +In the following we omit the tilde again for brevity. +The solutions with lower angular frequencies ω are more relativistic solutions, whereas the angular frequencies near 1 +are close to vacuum solutions. Since the amplitude of the scalar field φ becomes higher for lower angular frequencies +and for small values of the radial coordinate r, the corresponding accretion disk solutions will be more compact and +located closer to the center compared to those of BSs with higher angular frequencies. +3. +EFFECTIVE POTENTIAL AND KEPLERIAN ANGULAR MOMENTUM +Unmagnetized thick disks can be computed from an effective potential W, which acts as a combination of the gravi- +tational and centrifugal potential of a fluid particle rotating around a central gravitating object. W can be derived by +integrating the relativistic Euler equations and assuming the von Zeipel theorem as a necessary integrability condition, +W − Win := ln |ut| − ln |(ut)in| − +� ℓ +ℓin +Ω +1 − Ωℓ′ dℓ′ +(10) += − +� p +0 +1 +ρhdp′, +(11) +with ut as the covariant four-velocity (and −ut the mass-normalized energy), ℓ the specific angular momentum, ρ the +rest-mass density, h the specific enthalpy and p the thermodynamic pressure. The effective potential at the inner edge +of the disk, Win, is taken as a free parameter. In general the inner edge of a thick disk is located at the marginally +bound orbit rmb [9, 56, 57]. Therefore we set Win = W(rmb) in all further calculations. Assuming a constant specific +angular momentum distribution the integral term containing ℓ vanishes, and by assuming a polytropic equation of +state, p = KρΓ, the rest-mass density can be rewritten to read +ρ = +�(eWin−W − 1)(Γ − 1) +KΓ +� +1 +1−Γ +, +(12) +where ρ depends only on W. + +m +1.00 +1.2 +0.95 +0.90 +1.0 +0.85 +0.8 +M +0.80 +0.6 +0.75 +i=0.665 +0.4 +0.70 +0.2 +0.65 +0.7 +0.8 +0.9 +1.0 +mw +1.00 +0.4 +0.95 +0.3 +0.90 +0.85 +00.2 +0.80 +0.75 +0.1 +0.70 +0.0 +0.65 +5 +10 +15 +0 +r/M5 +The isodensity surfaces coincide with the equipotential surfaces and therefore the disk geometry can be studied by +analyzing the effective potential. The location of the accretion disk center rc is given by the maximum of the rest-mass +density and therefore the minimum of W. A local maximum of W corresponds to a self-intersection of an equipotential +surface and is called a cusp. Since ∂W +∂α |(r=rc,θ= π +2 ) = 0 the motion at the accretion disk center and at the accretion disk +cusp follows a geodesic on the equatorial plane. Due to the axisymmetry of the BS spacetime, the geodesic corresponds +to a Keplerian circular orbit. The specific angular momentum ℓ at the center and cusp is therefore identical to the +Keplerian specific angular momentum ℓK. Considering the von Zeipel theorem, the specific angular momentum ℓ can +be expressed as, +ℓ± +K(r) = −gtϕ + gϕϕΩ± +K +gtt + gtϕΩ± +K +, +(13) +with Ω± +K being the Keplerian angular velocity. The positive and negative sign refer to prograde and retrograde motion, +respectively. In Fig. 2 examples of Keplerian specific angular momenta are shown for a set of Kerr black holes (a) +and BSs (b). +(a) +(b) +Fig. 2. (a) Distributions of ℓ± +K for a set of Kerr black holes with different spin parameter a. The special case a = 0 is representing +a Schwarzschild black hole. (b) Normalized ℓ± +K distributions for the analyzed set of BS solutions. The black curve sections +display the specific angular momentum range for which no bound orbits are possible. +The specific angular momentum is taken as a free parameter in W, the chosen value ℓ0 determines the position of +the minima and maxima of W with rc = {r : ℓ0 = ℓ± +K(r), ∂2W +∂r2 > 0} and rcusp = {r : ℓ0 = ℓ± +K(r), ∂2W +∂r2 < 0}. Tab. I +displays all possible thick disk morphologies for the various BS solutions in dependence of ℓ0. +Centers Cusp +ℓ0 condition +BS models +|ℓ0| /∈ (ℓmin +K +, ℓin +mb) 0.665 ≤ ω ≤ 0.806 +Type 1 +1 +0 +|ℓ0| /∈ (ℓmin +K +, ℓmax +K +) 0.806 < ω ≤ 0.853 +no condition +0.853 < ω < 1.000 +Type 2 +2 +1 +|ℓ0| ∈ (ℓmin +K +, ℓmb +out) 0.665 ≤ ω ≤ 0.806 +|ℓ0| ∈ (ℓmin +K +, ℓmax +K +) 0.806 < ω ≤ 0.853 +Type 3 +2 +0 +|ℓ0| ∈ (ℓout +in , ℓin +mb) +0.665 < ω < 0.806 +Tab. I. Conditions for the different types of non-magnetized thick disks around BSs [52]. +Some of the retrograde accretion disk solutions, possess locations where the fluid stays at rest for a ZAMO at infinity. +Those locations are called static surfaces and are defined by the three dimensional generalization of the so called +static rings, which represent orbits remaining at rest [49]. They are realised in the shape of toroidal surfaces located +inside the accretion disk. Outside of the surfaces the fluid flows in a retrograde motion, while inside these surfaces +the fluid flows in a prograde motion. Since the fluid stays at rest at these surfaces, the angular velocity Ω is zero and + +a +1.0 +4 +0.8 +2 +0.6 +0 ++K +P +-2 +0.4 +-4 +0.2 +-6 +0.0 +2 +8 +4 +6 +10 +r/mw +5.0 +1.00 +0.95 +2.5 +0.90 +0.0 +0.85 +M +-2.5 +0.80 +-5.0 +0.75 +-7.5 +0.70 +-10.0 +0.65 +5 +10 +15 +20 +0 +r/M6 +therefore the specific angular momentum at the surface is given by the rest specific angular momentum ℓr := − gtφ +gtt . +Disk solutions containing static surfaces occur when ∃ r : ℓ0 = ℓr(r). Fig. 3 shows the equatorial rest specific angular +momentum distribution for various BS solutions. +Fig. 3. log10 of the absolute value of the rest specific angular momentum distribution ℓr in the equatorial plane for the set of +BS solutions. +The rest specific angular momentum is significantly smaller for the less relativistic BS solutions, therefore static +surfaces are more likely to appear for the more relativistic BSs. +4. +MAGNETIZED THICK DISKS +In order to construct magnetized torus solutions we follow the procedure employed in [9, 18]. We extend the hydro- +dynamical thick disk model by adding a toroidal magnetic field. The fundamental conservation laws of relativistic +magnetohydrodynamics need to be solved under the assumption, +∇µ(ρuµ) = 0, +(14) +∇µT µν = 0, +(15) +∇µ(∗F µν) = 0, +(16) +where ρ is the rest-mass density, uµ is the four-velocity of a co-moving observer, T µν is the stress-energy tensor and +∗F µν the dual Faraday tensor, +T µν ≡ (ρh + b2)uµuν + +� +p + b2 +2 +� +gµν − bµbν, +(17) +∗F µν ≡ uµbν − uνbµ. +(18) +Here bµ is the magnetic field with b2 ≡ bµbµ and bµ = (0, B), where B is the three-dimensional magnetic field measured +by a co-moving observer. Due to the axisymmetry, stationarity and azimuthal magnetic field distribution eqs. (14) +and (16) are always satisfied. Contracting eq. (15) with the orthogonal projection tensor hν +α one gets, +(ρh + b2)uν∂αuν + ∂α +� +p + b2 +2 +� +− bν∂αbν = 0, +(19) +where α = r, θ is non-trivial. Expressing the equation in terms of the angular velocity and specific angular momentum +leads to [9] +∂α(ln |ut|) − Ω∂αℓ +1 − ℓΩ + ∂αp +ρh + ∂α(Lpm) +Lρh += 0, +(20) + +w +1.00 +2 +0.95 +0.90 +0 +0.85 +0.80 +-2 +0.75 +0.70 +0.65 +5 +0 +10 +15 +20 +r/M7 +with L ≡ g2 +tϕ − gttgϕϕ and the magnetic pressure pm ≡ b2 +2 . +By assuming a polytropic equation of state (Ω = Ω(ℓ)) and introducing the definitions �pm = Lpm and �ω = Lω with +ω ≡ ρh integration of eq. (20) yields +ln |ut| − +� ℓ +0 +Ω +1 − ℓ′Ωdℓ′ + +� p +0 +1 +ρhdp′ + +� �pm +0 +1 +�ω d˜p′ +m = C. +(21) +The integration constant is defined by the boundary conditions at the edge of the disk and therefore given by +C = ln |(ut)in| = Win. +Since we suppose a constant specific angular momentum distribution the integral term +regarding ℓ vanishes and integration leads to +W − Win + ln(h) + +q +q − 1Km(Lρh)q−1 = 0, +(22) +with Γ and K the polytropic exponent and constant, q and Km the polytropic magnetic exponent and constant, +respectively. +Since eq. (22) is a transcendental equation for the rest-mass density ρ, it needs to be solved numerically at every point +of the numerical grid. In order to fix the gauge, the rest-mass density will be normalized at the densest center of the +disk to ρc = 1. Rewriting eq. (22) with respect to the torus center and expressing the specific enthalpy h in terms of +the rest-mass density, we obtain an expression where K is the only unknown parameter, +Wc − Win + ln +� +1 + ΓK +Γ − 1ρΓ−1 +c +� ++ +q +q − 1 +KρΓ +c +βmc +� +ρc + KΓρΓ +c +Γ−1 +� = 0. +(23) +Here we have defined the magnetization parameter βmc = +pc +pmc as the ratio between the thermodynamic and magnetic +pressure at the densest center of the disk. Therefore, a high magnetization parameter βmc corresponds to an essentially +non-magnetized disk, βmc ∼ 1 describes a mildly magnetized disk, and a low magnetization parameter βmc implies a +strongly magnetized disk. The polytropic exponents are chosen as usual for accretion disks, Γ = q = 4 +3. +With βmc as the only variable, eq. (23) can be solved for K for different degrees of magnetization represented by +βmc. After computation of K the general eq. (22) for the rest-mass density can be solved at every point of the +numerical grid. As a numerical solving algorithm the bisection method was chosen with an absolute convergence error +of ϵ = 10−15 between the last and second to last iteration step. + +8 +5. +RESULTS +We now present our results for magnetized thick tori, discussing first the one-centered disks and then the two-centered +ones. In pursuance of a more precise study of the one-centered solutions we define an effective equatorial columnar +radius ˜R and the corresponding mean equatorial columnar density ¯ρ, +˜R = +� +R : +� R +0 ρ(r, θ = π +2 )dr +� ∞ +0 +ρ(r, θ = π +2 )dr = 0.99 +� +; +¯ρ = 1 +˜R +� +˜ +R +0 +ρ(r, θ = π +2 )dr. +(24) +5.1. +One-centered Disks +For the one-centered solutions, we have selected 3 exemplary solutions representing less and moderately relativistic +BS solutions: a. ω = 0.960, ℓ0 = −4.5M, b. ω = 0.960, ℓ0 = −0.1M, and c. ω = 0.798, ℓ0 = −0.4666M, which are +discussed in the following. +a. +ω = 0.960, ℓ0 = −4.5M : +This solution is representative for the less relativistic BS solutions, the accretion disk +consists of a one-centered far-reaching torus, since the slope of the density curve flattens with increasing radial value +(Fig. 4). Since the highest density value is realized at the center of the torus, the maximum in Fig. 4 represents the +location of the disk center. With a higher magnetization the maximum density increases and the location of the disk +center shifts closer to the center of the BS. The equatorial density of the magnetized disks is higher for all radial +values smaller than the initial center location compared to the non-magnetized disk. In contrast to this, the density +in the magnetized solutions becomes smaller for larger values of r compared to the non-magnetized case, scaling up +to several orders of magnitude difference for large values of r. The effective equatorial columnar radius ˜R decreases +with a higher magnetisation, implying a more compact mass distribution. Whereas the mean equatorial columnar +density ¯ρ remains similar, which is a consequence of the denser center and steeper slope of the density curve. As seen +in Fig. 5, the disk gets compressed for a strong magnetization, leading to a more elliptic shape of the equidensity +surfaces. Most of the accretion disk mass is located in a smaller volume around the disk center, since the mass +distribution decreases even more rapidly with increasing magnetization. We conclude that strong magnetic fields +lead to a compactification of the torus, while the general shape and geometry of the torus is preserved for the less +relativistic BS solutions. +b. +ω = 0.960, ℓ0 = −0.1M : +This example represents a special set of solutions, which occur for small values of the +specific angular momentum and are also known as fat tori. Since the specific angular momentum is sufficiently small, +they are capable of possessing static surfaces. As seen in Fig. 6 the location of the center is for strong magnetized +disks located close to the center of the BS and is approximately two orders of magnitude higher compared to the +non-magnetized disk. The mean columnar density ¯ρ and effective columnar radius ˜R are significantly higher/lower +for the strong magnetized case, indicating a very compressed disk and a more longitudinal mass distribution. Fur- +thermore the density at the static surface decreases, resulting in less matter located inside and at the static surface. +In the magnetized solution the static surface lies completely outside the volume in which approximately 50% of +the mass is contained (Fig. 7). As seen in Fig. 7 strong magnetic fields compress the equidensity surfaces parallel +to the equatorial plane in the direction of the BS center, resulting in extremal tori with a cylindrical shape cen- +tered around the rotational axis. +The geometry mimics a sharp ellipsoid centered around the BS center, where +most of the mass is located close to the BS center and alongside the rotational axis. We conclude that strong mag- +netic fields have a significant effect on fat tori, highly compactifying and elongating them alongside the rotational axis. +c. +ω = 0.798, ℓ0 = −0.4666M : +For the mildly relativistic BS solutions the non-magnetized disk is already relatively +compact compared to the previous solutions and located close to the center of the BS. The static surface is located +close to the center, with the inner intersection with the equatorial plane being located at the center. As in the other +solutions, magnetization compresses the disk leading to a higher density around the torus center and a shift towards the +BS center. This shift is relatively small, since the non-magnetized center location is already close to the BS center. The +decrease of ˜R and increase of ¯ρ indicate again a further compactification and elongated density distribution (Fig. 8). +The density at the outer intersection of the static surface with the equatorial plane is approximately one order of +magnitude lower compared to the non-magnetized solution (Fig. 8), in the high magnetized case it lies only partially in +the volume which contains half the accretion disk mass, as shown in Fig. 9. In general the disk gets characteristically +more compressed, with sharp edged contour lines forming equidensity surfaces with a smaller diameter, resulting in +a torus geometry similar to a sharp ellipsoid which is centered around the BS center with most of the mass located +close to the center (Fig. 9). + +9 +log10 βmc ρmax +rc +∆rc +˜R +¯ρ +log10 (pmax) log10 (pmmax) hmax +5 +1.000 19.190 +- +220.867 0.102 +-2.098 +-7.098 +1.032 +0 +1.041 17.759 1.431 84.341 0.184 +-2.378 +-2.378 +1.016 +-5 +1.165 16.678 2.511 61.774 0.224 +-7.017 +-2.017 +1.000 +Fig. 4. a. ω = 0.960, ℓ0 = −4.5M: log10 of the density in the equatorial plane for the different magnetization parameters +βmc. Vertical dashed lines represent the position of the torus center. Vertical dotted lines represent the effective columnar +radius ˜R. Horizontal dotted lines correspond to the density value ρ( ˜R, θ +2). Dashed dotted horizontal blue lines represent the +mean equatorial columnar density ¯ρ. The table presents properties of the accretion disks, where ρc describes the density at the +center, rc the center location, ∆r the distance between the center locations of magnetized and non-magnetized tori, pmax and +pmmax the maximum of the thermodynamic and magnetic pressure and hmax the maximum of the specific enthalpy. +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 172.05 +-2.30 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 459.35 +-3.52 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 483.68 +-3.62 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +53.13 +-1.41 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 230.47 +-4.12 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 348.11 +-4.94 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +36.49 +-0.16 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 151.24 +-3.92 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 268.08 +-5.12 +Fig. 5. a. ω = 0.960, ℓ0 = −4.5M: Density distribution visualized for the different magnetization parameters. The black solid +contour lines represent the equidensity surfaces. The minimum of the density is set to log10 ρ = −6.37 for all figures and the +maximum is set to the density value at the center of the highly magnetized solution. The light blue, violet and red contour +lines represent each the equidensity surface within which 50%, 95% and 99% of the total mass lies. The tables below each +figure contain the corresponding radial value in the equatorial plane of these equidensity surfaces and their density. It should +be noted, that owing to the numerical nature of the solutions (and all further solutions), the computed values of these surfaces +are only representing approximations. + +0 +H +0 +0 +..... +βmc= 1 +βmc = 10-5 +: +. +- +-1 +- +-1 +- +: +: +- +: +2 +2 +-2 +log10( +1 +-3 +-3 +- +irc +irc +.4 +...... +- +100 +0 +50 +100 +150 +200 +250 +0 +50 +150 +200 +250 +0 +50 +100 +150 +200 +250 +r/M +r/M +r/Mlog10p +0.002 +200 +-0.645 +150 +-1.291 +100 +-1.938 +50 +-2.584 +Z/M +0 +-3.231 +-50 +-3.878 +-4.524 +-100 +-5.171 +-150 +-5.817 +-200 +0 +100 +200 +300 +400 +X/Mlog10p +0.002 +200 +-0.645 +150 +-1.291 +100 +-1.938 +50 +-2.584 +W/z +0 +-3.231 +-50 +-3.878 +-4.524 +-100 +-5.171 +-150 +-5.817 +-200 +0 +100 +200 +300 +400 +X/Mlog10p +0.002 +200 +-0.645 +150 +-1.291 +100 +-1.938 +50 +-2.584 +Z/M +0 +0 +-3.231 +-50 +-3.878 +-4.524 +-100 +-5.171 +-150 +-5.817 +-200 +0 +100 +200 +300 +400 +X/M10 +log10 βmc +ρmax +rc +∆rc +˜R +¯ρ +log10 (pmax) log10 (pmmax) hmax +5 +1.000 7.474 0.000 82.741 0.240 +-1.751 +-6.751 +1.071 +0 +3.135 0.960 6.513 30.083 0.651 +-1.394 +-1.394 +1.051 +-5 +94.798 0.490 6.983 14.307 6.705 +-4.130 +0.870 +1.000 +Fig. 6. b. ω = 0.960, ℓ0 = −0.1M: log10 of the density in the equatorial plane for the different magnetization parameters. +Vertical dotted indigo lines represent the radial value corresponding to the intersection of the static surface with the equatorial +plane. Vertical dashed lines represent the position of the torus center. Vertical dotted lines represent the effective columnar +radius ˜R. Horizontal dotted lines correspond to the density value ρ( ˜R, θ +2). The dashed dotted horizontal blue lines represent +the mean equatorial density ˜ρ. +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +76.82 +-2.20 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 421.09 +-4.42 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 478.55 +-4.61 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +18.34 +-0.74 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 68.82 +-3.30 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 112.83 +-4.31 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +6.23 +0.16 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 34.09 +-2.39 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 66.42 +-3.87 +Fig. 7. b. ω = 0.960, ℓ0 = −0.1M: Density distribution visualized for different magnetization parameters. Black solid contour +lines represent the equidensity surfaces, the white circle represents the static surface. The minimum of the density is set to +log10 ρ = −5.79 in all figures and the maximum is set to the density at the center of the highly magnetized solution. The light +blue, violet and red lines represent each the equidensity surface within which 50%, 95% and 99% of the total mass lies. The +tables below each figure contain the corresponding radial value in the equatorial plane of these equidensity surfaces and their +density value. + +2 +2 +2 +1: +.......... +..... +βmc = 1 +βmc = 10-5 +-- +I : +! : +..: +: +0 +0 +0 +1 : +....... +(d)0T601 +1: +i : +I : +-2 +2 +-2 +1: +:: +.............. +I: +R +rci: +rc +rc +1 : +4 +βmc= 105 +4 +4 +! : +I: +0 +20 +60 +80 +40 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +r/M +r/M +r/Mlog10p +1.548 +60 +0.767 +40 +-0.013 +-0.794 +20 +-1.574 +W/z +0 +-2.355 +-20 +-3.135 +-3.916 +-40 +-4.696 +-60 +-5.477 +0 +25 +50 +75 +100 +125 +150 +X/Mlog1op +1.548 +60 +0.767 +40 +-0.013 +-0.794 +20 +-1.574 +W/z +0 +-2.355 +-20 +-3.135 +-3.916 +-40 +-4.696 +-60 +-5.477 +0 +25 +50 +75 +100 +125 +150 +X/Mlog10p +1.548 +60 +0.767 +40 +-0.013 +-0.794 +20 +-1.574 +W/z +0 +-2.355 +-20 +-3.135 +-3.916 +-40 +-4.696 +-60- +-5.477 +0 +25 +50 +75 +100 +125 +150 +X/M11 +log10 βmc ρmax +rc +∆rc +˜R +¯ρ +log10 (pmax) log10 (pmmax) hmax +5 +1.000 1.321 0.00 10.555 0.226 +-0.833 +-5.833 +1.588 +0 +1.403 0.670 0.65 5.273 0.413 +-0.962 +-0.962 +1.311 +-5 +4.101 0.450 0.87 3.492 0.891 +-5.120 +-0.120 +1.000 +Fig. 8. c. ω = 0.798, ℓ0 = −0.4666M: log10 of the density in the equatorial plane for the different magnetization parameters. +Vertical dotted indigo lines represent the radial value corresponding to the intersection of the static surface with the equatorial +plane. Vertical dashed lines represent the position of the torus center. Vertical dotted lines represent the effective columnar +radius ˜R. Horizontal dotted lines correspond to the density value ρ( ˜R, θ +2). The dashed dotted horizontal blue lines represent +the mean equatorial density ˜ρ. +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +21.60 +-3.29 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 376.12 +-7.00 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 413.57 +-7.34 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +3.12 +-0.84 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 15.65 +-3.99 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 44.01 +-6.10 +R +log10 ρ +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.5 +1.88 +-0.36 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.95 7.35 +-3.11 +� R +0 +ρdV +� ∞ +0 +ρdV ≈ 0.99 27.20 +-5.99 +Fig. 9. c. ω = 0.798, ℓ0 = −0.4666M: Torus solutions visualized for different magnetization parameters. Black solid contour +lines represent the equidensity surfaces and the white circle represents the static surface. The minimum of the density is set to +log10 ρ = −6.37 in all figures and the maximum is set to the density at the center of the highly magnetized solution. The light +blue, violet and red lines represent each the equidensity surface within which 50%, 95% and 99% of the total mass lies. The +tables below each figure contain the corresponding radial value in the equatorial plane of these equidensity surfaces and their +density value. + +1 +7 +1 +.... +βmc = 1 +βmc = 10-5 +0 +0 +0 +........... +(d)0T60l +-1 +-1 +-2 +-2 +Irei +irc +3· +-3 +βmc = 105 +- +4 +4 +2.5 +7.5 +0.0 +2.5 +5.0 +7.5 +10.0 +2.5 +5.0 +7.5 +10.0 +0.0 +5.0 +10.0 +0.0 +r/M +r/M +r/Mlog1op +15 +0.016 +10 +-0.686 +-1.388 +5 +-2.090 +W/z +-2.792 +0 +-3.494 +-5 +-4.196 +-4.897 +-10 +-5.599 +-6.301 +-15 +0 +5 +10 +15 +20 +25 +30 +X/Mlog1op +15 +0.016 +10 +-0.686 +-1.388 +5 +-2.090 +W/z +-2.792 +0 +-3.494 +-5 +-4.196 +-4.897 +-10 +-5.599 +-6.301 +-15 +0 +5 +10 +15 +20 +25 +30 +x/Mlog1op +15 +0.016 +10 +-0.686 +-1.388 +5 +-2.090 +W/z +-2.792 +0 +-3.494 +-5 +-4.196 +-4.897 +-10 +-5.599 +-6.301 +-15 +5 +0 +10 +15 +20 +25 +30 +x/M12 +5.2. +Two-centered Disks +In contrast to black holes, it is possible for mildly and highly relativistic BSs to shelter accretion disks solutions with +more than one center. These two-centered solutions differ quite strongly from those analyzed so far, they either possess +a cusp, which connects the two centers of the disk or they have no cusp. In the latter case the accretion disk consists +of two separated tori, one inner torus and one outer torus. For the two-centered disks we have selected the following +4 representative examples: a. ω = 0.798, ℓ0 = −4.5M, and b. ω = 0.671, ℓ0 = −4.5M, c. ω = 0.671, ℓ0 = −5M, +d. ω = 0.798, ℓ0 = −4.75M. +a. +ω = 0.798, ℓ0 = −4.5M : +This solution is in the non-magnetized case composed of a two-centered disk connected +by a cusp. As shown in Fig. 10, the equatorial density of the mildly and strong magnetized solution only possesses +one maximum and no minimum. Since maxima of the density curve mark disk centers and minima mark cusps, +the two-centered disk in the non-magnetized solution becomes a one-centered disk without a cusp for a mildly and +strong magnetization. For all radial values greater than the non-magnetized inner center the density is monotonically +decreasing in correspondence to the magnetization parameter. Since the local extreme points vanish for stronger +magnetized disks, there must be a threshold value β0, below which only one centered solutions exist, the disk topol- +ogy is therefore dependent on the magnetization. +Fig. 11 shows an analysis of density curves for magnetization +parameters close to this threshold value. The cusp and outer center converge to one location for the threshold value +β0 ≈ 1.757, which marks a saddle point of the density curve. Since the pressure gradients vanish, the motion at this +location is geodesic and of unstable nature (Fig. 11 (b)). As seen in Fig. 12 the disk gets compactified towards the +inner center and the contour lines of the equidensity surfaces close to the center are of circular shape and smaller in +diameter compared to the contour lines of lower density, which have a greater extent and possess a teardrop-like shape. +b. +ω = 0.671, ℓ0 = −4.5M: +The solution presented in Fig. 13 is highly relativistic and composed of a two-centered +disk connected by a cusp with a static surface. A high magnetization does not effect the torus geometry around +the inner center considerably. Similar to the other solutions it gets denser and more compressed, thus it is located +closer to the BS center as compared to the non-magnetized case. In contrast, the geometry around the cusp and +outer center is more affected by strong magnetic fields. +The cusp is located slightly further away from the BS +center, whereas the outer center moves closer to it, therefore the distance between cusp and outer center decreases +significantly, as shown in Fig. 14. +The distance between outer center and cusp converges for low magnetization +parameters to ∆r ≡ rco − rcusp = 1.09. Furthermore the densities at the cusp and outer center are similar for a +high magnetization. The difference between them converges to ∆ρ ≡ ρc0 − ρcusp = 6.93 · 10−10. Since the difference +of the densities is significantly small, the physical properties of the disk between cusp and outer center would be +similar and it would be hard to distinguish between them. It should be noted that the density in general is very +small around the outer center, having a magnitude around 10−7 in the strong magnetized case. Equidensity surfaces +around the outer center become smaller in diameter for a low magnetization parameter, as seen in Fig. 15. +In +general we conclude that strong magnetic fields are suppressing the outer center. Since the static surface is located +close to the inner center, the properties in and at the static surface are only slightly influenced by strong magnetization. +c. +ω = 0.671, ℓ0 = −5M: +Considering the same BS solution as in b. and setting the specific angular momentum +to ℓ0 = −5M leads to a two-centered solution without a cusp, as presented in Fig. 16. The outer center moves closer +to the BS center for a low magnetization parameter and the density values of the outer torus are for the most part +more than two orders of magnitude lower compared to the non-magnetized case, meaning there is almost no matter +in the outer torus (in comparison to the inner torus), with density levels around ∼ 10−9 (Fig. 15). The inner center +with the static surface behaves similarly to the solution with the cusp. +d. +ω = 0.798, ℓ0 = −4.75M: +Looking at the ω = 0.798 BS solution, there exist also two-centered solutions with a +cusp, which are composed of a denser outer center, as seen in Fig. 18. For the high magnetized disks the inner center +as well as the outer center become denser compared to the non-magnetized solution, with the inner center being denser +than the outer one. In general the equatorial density of the magnetized disks is for all radial values smaller than +the non-magnetized outer center location higher compared to the non-magnetized disk. As a consequence there are +higher density values at and around the cusp. Fig. 20 shows the 2-dimensional density distribution. The locations of +the inner center and the cusp are not significantly influenced by strong magnetization. The outer center moves closer +to the BS center and the cusp. Considering that the inner center becomes denser for a strong magnetization, there +must be a threshold value β0, below which the outer center has lower density. Fig. 19 presents solutions close to this +threshold value. Since β0 marks the intersection between the density of the inner and outer center, a two-centered +magnetized disk solution with the same density at both centers is possible for this magnetization parameter. + +13 +log10 βmc +ρci +rci +ρco +rco +ρcusp rcusp +ρci +ρco +log10 (pmax) log10 (pmmax) hmax +5 +1.000 2.921 0.102 11.372 0.068 6.061 9.813 +-1.733 +-6.733 +1.074 +0 +1.011 2.861 +- +- +- +- +- +-2.031 +-2.031 +1.037 +-5 +1.044 2.811 +- +- +- +- +- +-6.724 +-1.724 +1.000 +Fig. 10. a. ω = 0.798, ℓ0 = −4.5M: log10 of the density in the equatorial plane for different magnetization parameters. Dashed +vertical lines represent center positions, with rci referring to the inner center and rco to the outer center. The dotted vertical +line represents the cusp. The outer center (and therefore also the cusp) vanishes in the magnetized solutions. +(a) 1 ≤ βmc ≤ 100 +(b) 1.72 ≤ βmc ≤ 1.78 +Fig. 11. a. ω = 0.798, ℓ0 = −4.5M: (a) Solutions for βmc in the range of 1 to 100, representing mildly magnetized disks. (b) +Closeup of solutions for βmc around the threshold value β0. Dashed lines represent the location of the outer center rc0, dotted +lines represent the location of the cusp. The dashed-dotted black line in (b) shows the location of the saddle point, which exists +for the density curve corresponding to the threshold value of β0 ≈ 1.757. For the lowest two curves in (a) and for the lowest +five curves in (b) βmc is below β0, therefore there exists no cusp and no outer center. + +0.0 +βmc = 105 +-0.5 +βmc = 1 +βmc = 10-5 +-1.0 +(d)0T601 +I +-1.5 +I +I +2.0 +-2.5 +Irco +:rcusp +-3.0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +r/Mβmc +100 +0.0 +80 +-0.5 +(d)0160l +60 +-1.0 +40 +20 +-1.5 +2 +4 +6 +8 +10 +12 +r/Mβmc +-1.430 +1.78 +-1.435 +-1.440 +(d)0160l +1.76 +-1.445 +-1.450 +1.74 +-1.455 +1.72 +-1.460 +6.5 +7.0 +7.5 +8.0 +8.5 +r/M14 +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +(d) βmc = 10−5 +Fig. 12. a. ω = 0.798, ℓ0 = −4.5M: Density distribution visualized for different magnetization parameters. The minimum of the +density is set to log10 ρ = −3.8 for (a) - (c) and the maximum to the density value at the inner center of the strong magnetized +solution. Black solid lines represent the isodensity surfaces, dotted red lines represent the isodensity surface corresponding to +the cusp. The dashed-dotted blue line in (d) represents the location of the saddle point for the solution with the threshold +value β0 = 1.757, the minimum of the density is set to log10 ρ = −2 in (d). + +l0g10p +15 +-0.001 +-0.384 +10 +-0.768 +-1.152 +5 . +-1.536 +W/z +0 +-1.919 +-2.303 +-5 +-2.687 +-3.071 +-10 +-3.455 +-15 +0 +5 +10 +15 +20 +25 +30 +x/Ml0g10g +15 +-0.001 +-0.384 +10 +-0.768 +-1.152 +-1.536 +W/z +0 +-1.919 +-2.303 +-5 +-2.687 +-3.071 +-10 +-3.455 +-15 +0 +5 +10 +15 +20 +25 +30 +X/Ml0g10p +15 +-0.001 +-0.384 +10 +-0.768 +-1.152 +-1.536 +z/M +0 +-1.919 +-2.303 +-5 +-2.687 +-3.071 +-10 +-3.455 +-15 +0 +5 +10 +15 +20 +25 +30 +X/Mlog10P +3 +-0.001 +-0.384 +-0.768 +-1.152 +1 +-1.536 +z/M +-1.919 +-2.303 +-1- +-2.687 +-3.071 +-21 +-3.455 +-3 +2 +3 +4 +5 +6 +7 +8 +X/M15 +log10 βmc log10 ρci +rci +log10 ρco +rco +log10 ρcusp rcusp log10 +� ρci +ρco +� +log10 (pmax) log10 (pmmax) hmax +5 +0.000 +0.66 +-5.903 +11.802 +-6.356 +5.261 +5.903 +-0.101 +-5.101 +4.172 +0 +0.031 +0.57 +-6.359 +8.512 +-6.461 +5.631 +6.389 +-0.431 +-0.431 +2.382 +-5 +0.144 +0.50 +-7.377 +7.241 +-7.384 +6.151 +7.522 +-5.255 +-0.255 +1.000 +Fig. 13. b. ω = 0.671, ℓ0 = −4.5M: The left panel shows the log10 of the density in the equatorial plane around the inner +center of the disk, with the dashed lines marking the position of the inner center and the indigo dotted line the position of +the inner intersection of the static surface with the equatorial plane. The right panel shows the density in the equatorial plane +around the outer center, with the dotted lines representing the location of the cusp and the dashed lines the location of the +outer center. +(a) +(b) +Fig. 14. b. ω = 0.671, ℓ0 = −4.5M: (a) Locations of the outer center and cusp versus βmc. (b) Density at the outer center and +cusp versus βmc. + +0.2 +4 +βmc = 105 +βmc = 105 +βmc = 1 +βmc = 1 +0.0 +-5 +βmc = 10-5 +βmc = 10-5 +(d)0T60l +: +-0.2 +-6 +一 +: : +-0.4 +-7 +:: +Irci +irc. +Ici! +Ircoirc. +.... +-0.6 +-8 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +r/M +r/M12 +10 +Center +W/ +Cusp +8 +6 +-5.0 +-2.5 +0.0 +2.5 +5.0 +log10βmc-6.0 +(d)0160l +-6.5 +-7.0 +Center +Cusp +7.5 +-2.5 +0.0 +2.5 +5.0 +5.0 +log10βmc16 +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +Fig. 15. b. ω = 0.671, ℓ0 = −4.5M: Density distribution visualized for different magnetization parameters, the minimum of +the density is set to log10 ρ = −8.9 and the maximum to the density at the inner center of the high magnetized solution. The +black solid lines represent the isodensity surfaces, the red dotted lines represent the cusp. +log10 βmc log10 ρci +rci +log10 ρco +rco +log10 ρcusp rcusp log10 +� ρci +ρco +� +log10 (pmax) log10 (pmmax) hmax +5 +0.000 +0.67 +-6.337 +17.744 +- +- +6.337 +-0.092 +-5.092 +4.237 +0 +0.028 +0.59 +-7.046 +14.053 +- +- +7.074 +-0.427 +-0.427 +2.404 +-5 +0.131 +0.52 +-8.207 +13.293 +– +- +8.338 +-5.268 +-0.268 +1.000 +Fig. 16. c. ω = 0.671, ℓ0 = −5M: The left panel shows the equatorial density around the inner center of the disk, the right +panel shows the equatorial density around the outer center of the disk. Dotted vertical lines are marking the positions of the +inner and outer center. The indigo dotted line represents the location of the inner intersection of the static surfaces. + +log10p +0.008 +10 +-0.901 +-1.810 +5 +-2.719 +-3.628 +z/M +0 +-4.537 +-5.446 +-5 +-6.355 +-7.264 +-10 +-8.173 +0 +5 +10 +15 +20 +25 +X/Mlog10p +0.008 +10 +-0.901 +-1.810 +5 +-2.719 +-3.628 +z/M +0 +-4.537 +-5.446 +-5 +-6.355 +-7.264 +-10 +-8.173 +0 +5 +10 +15 +20 +25 +X/Mlog10p +0.008 +10 +-0.901 +-1.810 +5 +-2.719 +-3.628 +z/M +0 +-4.537 +-5.446 +-5 +-6.355 +-7.264 +-10 +-8.173 +0 +5 +10 +15 +20 +25 +W/X0.2 +βmc = 105 +二二二 +βmc = 105 +- +βmc = 1 +-7 +βmc = 1 +0.0 +βmc = 10-5 +βmc = 10-5 +(d)0T60) +- +-0.2 +-8 +- +-0.4 +(ci +Irci +Irco +rco +-- +-0.6 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +10 +15 +20 +25 +30 +r/M +r/M17 +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +Fig. 17. c. ω = 0.671, ℓ0 = −5m: Density distribution visualized for different magnetization parameters, the minimum of the +density is set to log10 ρ = −10 and the maximum to the density at the inner center of the high magnetized solution. The black +solid lines represent the isodensity surfaces. +log10 βmc +ρci +rci +ρco +rco +ρcusp rcusp +ρci +ρco +log10 (pmax) log10 (pmmax) hmax +5 +0.297 3.151 1.000 14.653 0.005 4.851 0.297 +-2.843 +-7.843 +1.019 +0 +0.969 3.141 1.104 12.142 0.014 4.871 0.878 +-2.460 +-2.460 +1.014 +-5 +6.998 3.101 1.461 10.442 0.051 4.931 4.790 +-6.019 +-1.019 +1.000 +Fig. 18. d. ω = 0.798, ℓ0 = −4.75M: log10 of the density in the equatorial plane. Dashed lines show the locations of the inner +and outer center, the dotted lines show the position of the cusp. + +log10p +15 +-0.022 +-1.040 +10 +-2.058 +-3.076 +5 +-4.094 +W/z +0 +-5.113 +-6.131 +-5 +-7.149 +-8.167 +-10 +-9.185 +-15 +0 +5 +10 +15 +20 +25 +30 +X/Mlog10P +15 +-0.022 +-1.040 +10 +-2.058 +-3.076 +5 +-4.094 +W/z +0 +-5.113 +-6.131 +-5 +-7.149 +-8.167 +-10 +-9.185 +-15 +0 +5 +10 +15 +20 +25 +30 +X/Mlog10P +15 +F-0.022 +-1.040 +10 +-2.058 +-3.076 +5 +-4.094 +W/z +0 +-5.113 +-6.131 +-5 +-7.149 +-8.167 +-10 +-9.185 +-15 +0 +5 +10 +15 +20 +25 +30 +X/M1.0 +88888888888838888888888 +βmc = 105 +0.5 +1 +βmc= 1 +0.0 +(d)0T60) +-0.5 +-1.0 +-1.5 +: +Irco +Irco +-2.0 +-2.5 +10 +12 +4 +6 +8 +14 +16 +18 +20 +r/M18 +(a) +(b) +Fig. 19. d. ω = 0.798, ℓ0 = −4.75M: (a) Equatorial density curves for βmc in the range of 0.1 ≤ βmc ≤ 10. Horizontal dotted +lines mark the density value at the inner center. (b) Density of the inner and outer center versus βmc. The dashed vertical line +represents the threshold value β0 = 0.814. +(a) βmc = 105 +(b) βmc = 105 +(c) βmc = 10−5 +Fig. 20. d. ω = 0.798, ℓ0 = −4.75M: Density distribution visualized for different magnetization parameters. The minimum +density is set to log10 ρ = −2 and the maximum to the density value at the outer center of the high magnetized solution. The +black solid lines represent the isodensity surfaces, the dotted red line represents the isodensity surface corresponding to the +cusp. +6. +CONCLUSION +In this work we investigate the properties of non-selfgravitating magnetized thick disks in boson star spacetimes. +We construct the equilibrium configurations as solutions to the relativistic Euler equations assuming the presence +of a toroidal magnetic field and constant angular momentum of the disk. We further describe the accreting plasma +by a perfect fluid with polytropic equation of state. The influence of the magnetic field is evaluated by considering +models with different degree of magnetization quantified by the ratio of the thermal and magnetic pressure, which +serves as a magnetization parameter. +In order to compare to the purely hydrodynamical case we choose several +representative boson star solutions ranging from mildly to highly relativistic, which are characterized by qualitatively +different morphology of the equilibrium tori. Then we analyse the impact of the magnetic field by exploring certain +distinctive features such as modifications in the disk compactness, location of the cusps and centers, distribution of +the predominant fluid density, transitions in the geometry or topology of the equilibrium configurations. +We observe the following systematic behavior. Mildly relativistic boson stars, which in the purely hydrodynamical +case support equilibrium tori with a single center and no cusp preserve the disk morphology, when magnetic field is + +log10βmc +1.0 +0.5 +0.0 +0.5 +(d)0T60l +-0.5 +0.0 +-1.0 +-1.5 +-0.5 +-2.0 +1.0 +5 +10 +15 +r/MInner center +0.75 +Outer center +0.50 +(d)0T601 +BO +0.25 +0.00 +-0.25 +-0.50 +-2 +-1 +0 +1 +2 +log10βmclog10p +15 +0.001 +10 +-0.216 +-0.434 +5 - +-0.651 +W/z +-0.869 +0 +-1.086 +-5 +-1.304 +-1.521 +-10 +-1.739 +-1.956 +-15 +5 +10 +15 +20 +25 +0 +30 +X/Mlog10p +15 +0.001 +10 +-0.216 +-0.434 +5 - +-0.651 +W/z +-0.869 +0 +-1.086 +-5 +-1.304 +-1.521 +-10 +-1.739 +-1.956 +-15 +0 +5 +10 +15 +20 +25 +30 +X/Mlog10p +15 +0.001 +10 +-0.216 +-0.434 +5 +-0.651 +Z/M +-0.869 +0 +-1.086 +-5 +-1.304 +-1.521 +-10 +-1.739 +-1.956 +-15 +0 +5 +10 +15 +20 +25 +30 +X/M19 +present. However, when the magnetization increases the disks become more compact and the predominant matter +distribution shifts inwards towards the boson star center. This is manifested quantitatively by a range of features. +In particular, the location of the torus center moves closer to the boson star, the maximum density at the torus +center increases, and the predominant fluid density concentrates in smaller regions of the disk. In addition, for strong +magnetic fields the disk can become highly compressed towards the rotation axis and the geometry of the equidensity +surfaces can change from oblate to prolate. These effects are consistent with the studies of magnetized equilibrium +tori around black holes where similar qualitative behavior is observed [18, 19]. +In the case of highly relativistic boson stars we observe some qualitatively new features induced by the magnetic +field. In the purely hydrodynamical case these solutions are characterized by two-centered equilibrium tori which +can be either disjoint or connected by a cusp. We demonstrate that a sufficiently strong magnetic field can cause +a transition in the disk topology. After a certain critical value of the magnetization parameter, some configurations +lose their outer center and become one-centered. +For other configurations the outer center does not disintegrate +in the strict sense but it moves inwards and the fluid density decreases so much in this region that it becomes +insignificant. In these cases the ratio of the maximum fluid density at the inner and outer center reaches eight orders +of magnitude. Another phenomenon which we observed is a shift of the predominant fluid density between the torus +centers. 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Abramowicz, Acta Astron. 21, 81 (1971). + diff --git a/mtAyT4oBgHgl3EQflPh9/content/tmp_files/load_file.txt b/mtAyT4oBgHgl3EQflPh9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85df8d0082e1d756ba7cc482b0d4c0d102bb15d8 --- /dev/null +++ b/mtAyT4oBgHgl3EQflPh9/content/tmp_files/load_file.txt @@ -0,0 +1,1596 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf,len=1595 +page_content='Magnetized Thick Disks around Boson Stars Kristian Gjorgjieski,∗ Jutta Kunz,† and Matheus C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Teodoro‡ Department of Physics, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany Lucas G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Collodel§ Theoretical Astrophysics, University of T¨ubingen, 72076 T¨ubingen, Germany, Petya Nedkova¶ Department of Theoretical Physics, Sofia University, Sofia 1164, Bulgaria (Dated: January 3, 2023) The effects of magnetic fields on accretion disks around compact objects are of high importance in the study of their general properties and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Here we analyze the influence of magnetic fields on thick accretion disks around rotating boson stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We assume a uniform constant specific angular momentum distribution and a polytropic equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The purely hydrodynamical thick disk solutions are extended to magnetized solutions by adding a toroidal magnetic field and then analyzed in terms of a magnetization parameter βmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We consider one-centered solutions as well as two-centered solutions and focus on retrograde tori, since they are more distinctive due to their unique properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Our computed solutions indicate that strong magnetic fields influence the characteristics of thick disks around rotating boson stars and possibly effect their unique features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' INTRODUCTION Accretion disks are formed around various astrophysical objects ranging from supermassive active galactic nuclei to coalescent stellar mass black holes and neutron stars binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Converting gravitational energy into radiation they harbor many high-energy astrophysical phenomena and thus can serve as a natural arena for probing the gravitational physics around compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' With the development of the imaging techniques we are now able to observe directly the accretion disks in the nearby galactic targets M87 and Sgr A*, which provides a unique opportunity for gaining knowledge about the nature of the compact objects in their centers and the physical processes in the accreting plasma [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The interpretation of the astrophysical observations relies considerably on the accretion disk modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Currently gen- eral relativistic magneto-hydrodynamic (GRMHD) simulations are developed which are able to integrate numerically the coupled Einstein-Euler equations thus providing a precise description of the self-gravitating accreting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' However, these simulations produce a complicated picture depending on many parameters which can be hard to disentangle and interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In this respect simplified semi-analytical models can be extremely useful since they may capture the main effects from the non-linear treatment with less computational cost and offer more physical intuition and predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Some of the basic constructions which describe geometrically thick accretion disks are the equilibrium tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' They represent the simplest solutions to the relativistic Euler equations assuming a non-selfgravitating perfect fluid and neglecting the electromagnetic, viscosity and radiation terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Thus, they provide the equilibrium states of the relativistic matter orbiting on non-geodesic trajectories around the compact object, which result from the balance of the pressure gradients and the gravitational and centrifugal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The theory of the equilibrium tori has a long history dating back to the classical works [4–6] where the equilibrium tori for the Kerr black hole were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Assuming an isentropic or barotropic equation of state for the perfect fluid it was demonstrated that in this case the constant angular velocity and the constant angular momentum surfaces coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This statement, known as the von Zeipel theorem, represents an integrability condition for the relativistic Euler equation and therefore it allows the construction of analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Various equilibrium tori can be constructed possessing a different profile of the specific angular momentum [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Solutions with a constant angular momentum play a fundamental role since they already capture the main qualitative ∗ kristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='gjorgjieski@uol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='de † jutta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='kunz@uni-oldenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='de ‡ matheus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='carmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='teodoro@uni-oldenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='de § lucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='gardai-collodel@uni-tuebingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='de ¶ pnedkova@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='uni-sofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='bg arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00449v1 [gr-qc] 1 Jan 2023 2 features of the more general configurations and represent the marginally stable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The physical conditions can be further extended by adding a toroidal magnetic field which allows for the construction of analytical solutions under the same integrability conditions [9] (see also [10] and [11] for non-constant angular momentum configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This generalization is particularly important for the astrophysical applications since magnetic fields play a dynamical role in astrophysical scenarios and the magnetized equilibrium configurations provide suitable initial conditions for the GRMHD simulations of the accretion process or jet formation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [12, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Although the accretion disk theory was developed considering mainly the Kerr black hole, fundamental physics suggests that more diverse compact objects may exist in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This motivates the construction of accretion disk models in more exotic spacetimes with the view of confronting the simulations with observational data and searching for astrophysical signatures of new fundamental objects [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In this line of research thick accretion disks around black holes in the modified theories of gravity were studied [18–21], equilibrium configurations in de Sitter background [22, 23] or interacting with an external matter distribution [24, 25], as well as thick disks in naked singularity spacetimes [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In our work we concentrate on boson stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Boson stars represent compact scalar field condensates which form as equilibrium states resulting from the gravitational collapse of a self-gravitating massive scalar field [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' They produce a strong gravitational field and negligible electromagnetic emission, in this way they are mimicking the properties of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Various boson star configurations have been constructed [31–40], and their properties were investigated such as stability [41–43], geodesic motion [44–49] and tidal effects in the motion of gas clouds around them [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since they are horizonless compact objects, which are not characterized by a solid boundary but rather by a decaying profile of the scalar field density towards infinity, the particles and light propagation in their vicinity is specific leading to phenomenological signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' For example, boson stars may possess only bound circular orbits which extend into the inner regions with high density of the scalar field distribution [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In addition, photon rings may be absent which prevents the formation of a shadow in the strict sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Yet, if we assume the presence of an accretion disk, a central dark region will be observed similar to black holes which corresponds to the lensed image of an accretion disk’s inner edge [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The perfect fluid equilibrium tori around boson stars also possess distinctive morphology compared to the Kerr black hole [47, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The equilibrium tori in the Kerr spacetime are typically characterized by a cusp located at their inner edge with the exception of a small range of configurations limited to specific angular momenta of the disk [5, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In contrast, the equilibrium configurations for boson stars constructed in the literature do not form an inner cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This distinction has implications on the possible scenarios for dynamical evolution of the accretion disk such as the formation of run-away instabilities [8, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Boson stars can further support topologically non-trivial configurations like two-centered tori which can be disjoint or connected by a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' These topologies are absent for the Kerr black hole, although they occur for other exotic compact objects like naked singularities [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Another interesting feature is the formation of static surfaces for counter-rotating equilibrium tori around boson stars [39, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' These surfaces are defined as the cross-sections where the fluid is at rest with respect to a zero angular momentum observer (ZAMO) at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Thus, they serve as a boundary separating regions where the fluid moves in prograde and retrograde direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The aim of our work is to study the equilibrium tori in boson star spacetimes in the presence of a toroidal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We construct constant angular momentum configurations and analyse how their properties are modified with respect to the purely hydrodynamical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In particular we describe the qualitative effects which are induced by the magnetic field and may have observational implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' These include modifications in the location of the predominant disk density and its compactness, as well as variation of the characteristic geometry of the isodensity surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Another interesting phenomenon is that the magnetic field can trigger topological transitions between different thick disk configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Thus, after a certain magnitude of the magnetic field, two-centered disks may lose their outer center and become one-centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the next section we describe the boson star solutions which we consider and some of their relevant properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In section 3 we briefly review the equilibrium tori configurations, which are possible in boson star spacetimes in the purely hydrodynamical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In section 4 we describe the construction procedure of magnetized tori for a perfect fluid with polytropic equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In section 5 we present our results and the analysis of the properties of the magnetized thick disk, which are illustrated on a range of representative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Section 6 contains our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' BOSON STARS We consider a complex scalar field without self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The boson stars (BSs) are obtained from the Einstein- Klein-Gordon equations, derived from the action S S = � √−g � R 16πG − Lm � d4x, (1) where g is the metric determinant, R is the Ricci scalar, G is Newton’s constant and Lm is the Lagrangian of the complex scalar field φ with mass m Lm = |∂µφ|2 + m2|φ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (2) The action (1) is invariant under transformations of the global U(1)-symmetry group of the complex scalar field, φ → φeiτ, with constant τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' According to Noether’s theorem this invariance implies the existence of a conserved current, ∇µjµ = 0, with jµ = i (φ∂µφ∗ − φ∗∂µφ) , (3) and a conserved Noether charge Q, which is the bosonic particle number given by Q = � √−gjtd3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The variation of the action (1) leads to the coupled set of Einstein-Klein-Gordon equations, Rµν − 1 2Rgµν = 8πGTµν, (4) � □ − m2� φ = 0, (5) where Tµν ≡ (∂µφ∂νφ∗ + ∂νφ∂µφ∗) − Lmgµν is the stress-energy tensor and □ denotes the covariant d’Alembert operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In order to obtain rotating BSs, the scalar field should depend on all four coordinates as follows [36], φ ≡ φ0(r, θ)ei(ωt−kϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (6) A harmonic time-dependence is already needed for the classical non-rotating BSs in order to obtain stable localized solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The presence of rotation implies an additional harmonic dependence of the scalar field on the azimuthal angle ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the Ansatz (6) ω is the angular frequency of the scalar field and k is the azimuthal winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Due to the condition φ(ϕ = 0) = φ(ϕ = 2π) the winding number k must be an integer (k signals the strength of the angular excitations since it counts the nodes, 2k, of the real and imaginary parts of the scalar field along the azimuthal direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The angular momentum J of the BSs is given by a quantisation law [36], J = kQ, (7) thus it is an integer multiple of the charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This law follows directly from the relation T t ϕ = njt with ∂ϕφ = inφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The harmonic Ansatz for the scalar field yields a stress-energy tensor that does not depend on the coordinates t and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Thus solutions with a stationary and axially symmetric metric result, implying the presence of two Killing vectors of the metric associated with these two coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The line element of the BS spacetime metric can then be written as, ds2 = − α2dt2 + A2 � dr2 + r2dθ2� (8) + B2r2 sin2 θ (dϕ + βϕdt)2 , (9) with α the lapse function, βϕ the shift function, and the functions A, B, α, β which depend only on r and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Substituting the harmonic Ansatz (6) into the field equations leads to a coupled set of partial differential equations for the functions, which were solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The BSs constructed are asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Thus the metric approaches asymptotically Minkowski spacetime and the scalar field vanishes exponentially proportional to e− √ m2−ω2r/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The solutions were computed with the FIDISOL/CADSOL package, which is a PDE solver which employs a finite differ- ence method of discretization together with a Newton-Raphson scheme to linearize the resulting system of algebraic equations [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Solutions exist only for a set of angular frequencies ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 4 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) Mass M of BSs versus boson field frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed circles mark the solutions used for the computation of the magnetized disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed red line marks the threshold value of ω below which the solutions contain ergoregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (b) The amplitude of the scalar field φ versus the normalized radial coordinate r/M for the BS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The highlighted curves show the solutions marked in (a), namely ω = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Here we consider only rotating BSs with winding number k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Furthermore, we focus only on solutions without ergoregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Therefore the minimal angular frequency taken into account is ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Solutions with smaller angular frequencies would possess ergoregions [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' It should be noted, that the equations feature a scaling symmetry, which we exploit to go to dimensionless quantities by scaling with the boson mass m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=', ˜r = rm, ˜ω = ω m and ˜ M = Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the following we omit the tilde again for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The solutions with lower angular frequencies ω are more relativistic solutions, whereas the angular frequencies near 1 are close to vacuum solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the amplitude of the scalar field φ becomes higher for lower angular frequencies and for small values of the radial coordinate r, the corresponding accretion disk solutions will be more compact and located closer to the center compared to those of BSs with higher angular frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' EFFECTIVE POTENTIAL AND KEPLERIAN ANGULAR MOMENTUM Unmagnetized thick disks can be computed from an effective potential W, which acts as a combination of the gravi- tational and centrifugal potential of a fluid particle rotating around a central gravitating object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' W can be derived by integrating the relativistic Euler equations and assuming the von Zeipel theorem as a necessary integrability condition, W − Win := ln |ut| − ln |(ut)in| − � ℓ ℓin Ω 1 − Ωℓ′ dℓ′ (10) = − � p 0 1 ρhdp′, (11) with ut as the covariant four-velocity (and −ut the mass-normalized energy), ℓ the specific angular momentum, ρ the rest-mass density, h the specific enthalpy and p the thermodynamic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The effective potential at the inner edge of the disk, Win, is taken as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In general the inner edge of a thick disk is located at the marginally bound orbit rmb [9, 56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Therefore we set Win = W(rmb) in all further calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Assuming a constant specific angular momentum distribution the integral term containing ℓ vanishes, and by assuming a polytropic equation of state, p = KρΓ, the rest-mass density can be rewritten to read ρ = �(eWin−W − 1)(Γ − 1) KΓ � 1 1−Γ , (12) where ρ depends only on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='8 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75 i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 mw 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='85 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='65 5 10 15 0 r/M5 The isodensity surfaces coincide with the equipotential surfaces and therefore the disk geometry can be studied by analyzing the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The location of the accretion disk center rc is given by the maximum of the rest-mass density and therefore the minimum of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' A local maximum of W corresponds to a self-intersection of an equipotential surface and is called a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since ∂W ∂α |(r=rc,θ= π 2 ) = 0 the motion at the accretion disk center and at the accretion disk cusp follows a geodesic on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Due to the axisymmetry of the BS spacetime, the geodesic corresponds to a Keplerian circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The specific angular momentum ℓ at the center and cusp is therefore identical to the Keplerian specific angular momentum ℓK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Considering the von Zeipel theorem, the specific angular momentum ℓ can be expressed as, ℓ± K(r) = −gtϕ + gϕϕΩ± K gtt + gtϕΩ± K , (13) with Ω± K being the Keplerian angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The positive and negative sign refer to prograde and retrograde motion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 2 examples of Keplerian specific angular momenta are shown for a set of Kerr black holes (a) and BSs (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) Distributions of ℓ± K for a set of Kerr black holes with different spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The special case a = 0 is representing a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (b) Normalized ℓ± K distributions for the analyzed set of BS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The black curve sections display the specific angular momentum range for which no bound orbits are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The specific angular momentum is taken as a free parameter in W, the chosen value ℓ0 determines the position of the minima and maxima of W with rc = {r : ℓ0 = ℓ± K(r), ∂2W ∂r2 > 0} and rcusp = {r : ℓ0 = ℓ± K(r), ∂2W ∂r2 < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' I displays all possible thick disk morphologies for the various BS solutions in dependence of ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Centers Cusp ℓ0 condition BS models |ℓ0| /∈ (ℓmin K , ℓin mb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='665 ≤ ω ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='806 Type 1 1 0 |ℓ0| /∈ (ℓmin K , ℓmax K ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='806 < ω ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='853 no condition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='853 < ω < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Type 2 2 1 |ℓ0| ∈ (ℓmin K , ℓmb out) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='665 ≤ ω ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='806 |ℓ0| ∈ (ℓmin K , ℓmax K ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='806 < ω ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='853 Type 3 2 0 |ℓ0| ∈ (ℓout in , ℓin mb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='665 < ω < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='806 Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Conditions for the different types of non-magnetized thick disks around BSs [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Some of the retrograde accretion disk solutions, possess locations where the fluid stays at rest for a ZAMO at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Those locations are called static surfaces and are defined by the three dimensional generalization of the so called static rings, which represent orbits remaining at rest [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' They are realised in the shape of toroidal surfaces located inside the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Outside of the surfaces the fluid flows in a retrograde motion, while inside these surfaces the fluid flows in a prograde motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the fluid stays at rest at these surfaces, the angular velocity Ω is zero and a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='6 0 +K P 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2 8 4 6 10 r/mw 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='85 M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='65 5 10 15 20 0 r/M6 therefore the specific angular momentum at the surface is given by the rest specific angular momentum ℓr := − gtφ gtt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Disk solutions containing static surfaces occur when ∃ r : ℓ0 = ℓr(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 3 shows the equatorial rest specific angular momentum distribution for various BS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' log10 of the absolute value of the rest specific angular momentum distribution ℓr in the equatorial plane for the set of BS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The rest specific angular momentum is significantly smaller for the less relativistic BS solutions, therefore static surfaces are more likely to appear for the more relativistic BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' MAGNETIZED THICK DISKS In order to construct magnetized torus solutions we follow the procedure employed in [9, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We extend the hydro- dynamical thick disk model by adding a toroidal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The fundamental conservation laws of relativistic magnetohydrodynamics need to be solved under the assumption, ∇µ(ρuµ) = 0, (14) ∇µT µν = 0, (15) ∇µ(∗F µν) = 0, (16) where ρ is the rest-mass density, uµ is the four-velocity of a co-moving observer, T µν is the stress-energy tensor and ∗F µν the dual Faraday tensor, T µν ≡ (ρh + b2)uµuν + � p + b2 2 � gµν − bµbν, (17) ∗F µν ≡ uµbν − uνbµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (18) Here bµ is the magnetic field with b2 ≡ bµbµ and bµ = (0, B), where B is the three-dimensional magnetic field measured by a co-moving observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Due to the axisymmetry, stationarity and azimuthal magnetic field distribution eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (14) and (16) are always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Contracting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (15) with the orthogonal projection tensor hν α one gets, (ρh + b2)uν∂αuν + ∂α � p + b2 2 � − bν∂αbν = 0, (19) where α = r, θ is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Expressing the equation in terms of the angular velocity and specific angular momentum leads to [9] ∂α(ln |ut|) − Ω∂αℓ 1 − ℓΩ + ∂αp ρh + ∂α(Lpm) Lρh = 0, (20) w 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='90 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='80 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='65 5 0 10 15 20 r/M7 with L ≡ g2 tϕ − gttgϕϕ and the magnetic pressure pm ≡ b2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' By assuming a polytropic equation of state (Ω = Ω(ℓ)) and introducing the definitions �pm = Lpm and �ω = Lω with ω ≡ ρh integration of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (20) yields ln |ut| − � ℓ 0 Ω 1 − ℓ′Ωdℓ′ + � p 0 1 ρhdp′ + � �pm 0 1 �ω d˜p′ m = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (21) The integration constant is defined by the boundary conditions at the edge of the disk and therefore given by C = ln |(ut)in| = Win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since we suppose a constant specific angular momentum distribution the integral term regarding ℓ vanishes and integration leads to W − Win + ln(h) + q q − 1Km(Lρh)q−1 = 0, (22) with Γ and K the polytropic exponent and constant, q and Km the polytropic magnetic exponent and constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (22) is a transcendental equation for the rest-mass density ρ, it needs to be solved numerically at every point of the numerical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In order to fix the gauge, the rest-mass density will be normalized at the densest center of the disk to ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (22) with respect to the torus center and expressing the specific enthalpy h in terms of the rest-mass density, we obtain an expression where K is the only unknown parameter, Wc − Win + ln � 1 + ΓK Γ − 1ρΓ−1 c � + q q − 1 KρΓ c βmc � ρc + KΓρΓ c Γ−1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (23) Here we have defined the magnetization parameter βmc = pc pmc as the ratio between the thermodynamic and magnetic pressure at the densest center of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Therefore, a high magnetization parameter βmc corresponds to an essentially non-magnetized disk, βmc ∼ 1 describes a mildly magnetized disk, and a low magnetization parameter βmc implies a strongly magnetized disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The polytropic exponents are chosen as usual for accretion disks, Γ = q = 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' With βmc as the only variable, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (23) can be solved for K for different degrees of magnetization represented by βmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' After computation of K the general eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (22) for the rest-mass density can be solved at every point of the numerical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As a numerical solving algorithm the bisection method was chosen with an absolute convergence error of ϵ = 10−15 between the last and second to last iteration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' RESULTS We now present our results for magnetized thick tori, discussing first the one-centered disks and then the two-centered ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In pursuance of a more precise study of the one-centered solutions we define an effective equatorial columnar radius ˜R and the corresponding mean equatorial columnar density ¯ρ, ˜R = � R : � R 0 ρ(r, θ = π 2 )dr � ∞ 0 ρ(r, θ = π 2 )dr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ¯ρ = 1 ˜R � ˜ R 0 ρ(r, θ = π 2 )dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (24) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' One-centered Disks For the one-centered solutions, we have selected 3 exemplary solutions representing less and moderately relativistic BS solutions: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1M, and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4666M, which are discussed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M : This solution is representative for the less relativistic BS solutions, the accretion disk consists of a one-centered far-reaching torus, since the slope of the density curve flattens with increasing radial value (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the highest density value is realized at the center of the torus, the maximum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 4 represents the location of the disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' With a higher magnetization the maximum density increases and the location of the disk center shifts closer to the center of the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The equatorial density of the magnetized disks is higher for all radial values smaller than the initial center location compared to the non-magnetized disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In contrast to this, the density in the magnetized solutions becomes smaller for larger values of r compared to the non-magnetized case, scaling up to several orders of magnitude difference for large values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The effective equatorial columnar radius ˜R decreases with a higher magnetisation, implying a more compact mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Whereas the mean equatorial columnar density ¯ρ remains similar, which is a consequence of the denser center and steeper slope of the density curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 5, the disk gets compressed for a strong magnetization, leading to a more elliptic shape of the equidensity surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Most of the accretion disk mass is located in a smaller volume around the disk center, since the mass distribution decreases even more rapidly with increasing magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We conclude that strong magnetic fields lead to a compactification of the torus, while the general shape and geometry of the torus is preserved for the less relativistic BS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1M : This example represents a special set of solutions, which occur for small values of the specific angular momentum and are also known as fat tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the specific angular momentum is sufficiently small, they are capable of possessing static surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 6 the location of the center is for strong magnetized disks located close to the center of the BS and is approximately two orders of magnitude higher compared to the non-magnetized disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The mean columnar density ¯ρ and effective columnar radius ˜R are significantly higher/lower for the strong magnetized case, indicating a very compressed disk and a more longitudinal mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Fur- thermore the density at the static surface decreases, resulting in less matter located inside and at the static surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the magnetized solution the static surface lies completely outside the volume in which approximately 50% of the mass is contained (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 7 strong magnetic fields compress the equidensity surfaces parallel to the equatorial plane in the direction of the BS center, resulting in extremal tori with a cylindrical shape cen- tered around the rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The geometry mimics a sharp ellipsoid centered around the BS center, where most of the mass is located close to the BS center and alongside the rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We conclude that strong mag- netic fields have a significant effect on fat tori, highly compactifying and elongating them alongside the rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4666M : For the mildly relativistic BS solutions the non-magnetized disk is already relatively compact compared to the previous solutions and located close to the center of the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The static surface is located close to the center, with the inner intersection with the equatorial plane being located at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As in the other solutions, magnetization compresses the disk leading to a higher density around the torus center and a shift towards the BS center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This shift is relatively small, since the non-magnetized center location is already close to the BS center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The decrease of ˜R and increase of ¯ρ indicate again a further compactification and elongated density distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The density at the outer intersection of the static surface with the equatorial plane is approximately one order of magnitude lower compared to the non-magnetized solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 8), in the high magnetized case it lies only partially in the volume which contains half the accretion disk mass, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In general the disk gets characteristically more compressed, with sharp edged contour lines forming equidensity surfaces with a smaller diameter, resulting in a torus geometry similar to a sharp ellipsoid which is centered around the BS center with most of the mass located close to the center (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 9 log10 βmc ρmax rc ∆rc ˜R ¯ρ log10 (pmax) log10 (pmmax) hmax 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='190 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='102 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='098 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='098 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='032 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='041 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='759 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='431 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='184 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='378 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='378 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='016 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='165 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='678 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='511 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='224 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='017 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: log10 of the density in the equatorial plane for the different magnetization parameters βmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dashed lines represent the position of the torus center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dotted lines represent the effective columnar radius ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Horizontal dotted lines correspond to the density value ρ( ˜R, θ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Dashed dotted horizontal blue lines represent the mean equatorial columnar density ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The table presents properties of the accretion disks, where ρc describes the density at the center, rc the center location, ∆r the distance between the center locations of magnetized and non-magnetized tori, pmax and pmmax the maximum of the thermodynamic and magnetic pressure and hmax the maximum of the specific enthalpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) βmc = 105 (b) βmc = 105 (c) βmc = 10−5 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='30 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='52 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='62 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='41 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='12 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='94 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='16 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='92 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: Density distribution visualized for the different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The black solid contour lines represent the equidensity surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The minimum of the density is set to log10 ρ = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='37 for all figures and the maximum is set to the density value at the center of the highly magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The light blue, violet and red contour lines represent each the equidensity surface within which 50%, 95% and 99% of the total mass lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The tables below each figure contain the corresponding radial value in the equatorial plane of these equidensity surfaces and their density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' It should be noted, that owing to the numerical nature of the solutions (and all further solutions), the computed values of these surfaces are only representing approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 0 H 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' βmc= 1 βmc = 10-5 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 1 1 : : : 2 2 2 log10( 1 3 3 irc irc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='. 100 0 50 100 150 200 250 0 50 150 200 250 0 50 100 150 200 250 r/M r/M r/Mlog10p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='002 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='645 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='291 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='938 50 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='071 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='513 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='651 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='394 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='394 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='051 5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='490 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='983 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='307 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='705 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='870 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1M: log10 of the density in the equatorial plane for the different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dotted indigo lines represent the radial value corresponding to the intersection of the static surface with the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dashed lines represent the position of the torus center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dotted lines represent the effective columnar radius ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Horizontal dotted lines correspond to the density value ρ( ˜R, θ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed dotted horizontal blue lines represent the mean equatorial density ˜ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) βmc = 105 (b) βmc = 105 (c) βmc = 10−5 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='20 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='42 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='61 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='74 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='30 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='83 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='31 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='16 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='39 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='87 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='960, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1M: Density distribution visualized for different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Black solid contour lines represent the equidensity surfaces, the white circle represents the static surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The minimum of the density is set to log10 ρ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='79 in all figures and the maximum is set to the density at the center of the highly magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The light blue, violet and red lines represent each the equidensity surface within which 50%, 95% and 99% of the total mass lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The tables below each figure contain the corresponding radial value in the equatorial plane of these equidensity surfaces and their density value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 2 2 2 1: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' βmc = 1 βmc = 10-5 -- I : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='.: : 0 0 0 1 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (d)0T601 1: i : I : 2 2 2 1: :: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='. I: R rci: rc rc 1 : 4 βmc= 105 4 4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' : I: 0 20 60 80 40 0 20 40 60 80 0 20 40 60 80 r/M r/M r/Mlog10p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='548 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='767 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='794 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='574 W/z 0 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='355 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='135 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='916 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='696 60- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='477 0 25 50 75 100 125 150 X/M11 log10 βmc ρmax rc ∆rc ˜R ¯ρ log10 (pmax) log10 (pmmax) hmax 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='65 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='962 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='311 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='891 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4666M: log10 of the density in the equatorial plane for the different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dotted indigo lines represent the radial value corresponding to the intersection of the static surface with the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dashed lines represent the position of the torus center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vertical dotted lines represent the effective columnar radius ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Horizontal dotted lines correspond to the density value ρ( ˜R, θ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed dotted horizontal blue lines represent the mean equatorial density ˜ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) βmc = 105 (b) βmc = 105 (c) βmc = 10−5 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='29 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='34 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='84 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='10 R log10 ρ � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='36 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='11 � R 0 ρdV � ∞ 0 ρdV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='99 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4666M: Torus solutions visualized for different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Black solid contour lines represent the equidensity surfaces and the white circle represents the static surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The minimum of the density is set to log10 ρ = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='37 in all figures and the maximum is set to the density at the center of the highly magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The light blue, violet and red lines represent each the equidensity surface within which 50%, 95% and 99% of the total mass lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The tables below each figure contain the corresponding radial value in the equatorial plane of these equidensity surfaces and their density value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 1 7 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='. βmc = 1 βmc = 10-5 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (d)0T60l 1 1 2 2 Irei irc 3· 3 βmc = 105 4 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 r/M r/M r/Mlog1op 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='016 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='388 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='090 W/z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='792 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='494 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='196 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='897 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='599 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='301 15 0 5 10 15 20 25 30 X/Mlog1op 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='016 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='388 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='090 W/z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='792 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='494 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='196 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='897 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='599 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='301 15 0 5 10 15 20 25 30 x/Mlog1op 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='016 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='388 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='090 W/z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='792 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='494 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='196 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='897 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='599 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='301 15 5 0 10 15 20 25 30 x/M12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Two-centered Disks In contrast to black holes, it is possible for mildly and highly relativistic BSs to shelter accretion disks solutions with more than one center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' These two-centered solutions differ quite strongly from those analyzed so far, they either possess a cusp, which connects the two centers of the disk or they have no cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the latter case the accretion disk consists of two separated tori, one inner torus and one outer torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' For the two-centered disks we have selected the following 4 representative examples: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M, and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −5M, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M : This solution is in the non-magnetized case composed of a two-centered disk connected by a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 10, the equatorial density of the mildly and strong magnetized solution only possesses one maximum and no minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since maxima of the density curve mark disk centers and minima mark cusps, the two-centered disk in the non-magnetized solution becomes a one-centered disk without a cusp for a mildly and strong magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' For all radial values greater than the non-magnetized inner center the density is monotonically decreasing in correspondence to the magnetization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the local extreme points vanish for stronger magnetized disks, there must be a threshold value β0, below which only one centered solutions exist, the disk topol- ogy is therefore dependent on the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 11 shows an analysis of density curves for magnetization parameters close to this threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The cusp and outer center converge to one location for the threshold value β0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='757, which marks a saddle point of the density curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the pressure gradients vanish, the motion at this location is geodesic and of unstable nature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 11 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 12 the disk gets compactified towards the inner center and the contour lines of the equidensity surfaces close to the center are of circular shape and smaller in diameter compared to the contour lines of lower density, which have a greater extent and possess a teardrop-like shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: The solution presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 13 is highly relativistic and composed of a two-centered disk connected by a cusp with a static surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' A high magnetization does not effect the torus geometry around the inner center considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Similar to the other solutions it gets denser and more compressed, thus it is located closer to the BS center as compared to the non-magnetized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In contrast, the geometry around the cusp and outer center is more affected by strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The cusp is located slightly further away from the BS center, whereas the outer center moves closer to it, therefore the distance between cusp and outer center decreases significantly, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The distance between outer center and cusp converges for low magnetization parameters to ∆r ≡ rco − rcusp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Furthermore the densities at the cusp and outer center are similar for a high magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The difference between them converges to ∆ρ ≡ ρc0 − ρcusp = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='93 · 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the difference of the densities is significantly small, the physical properties of the disk between cusp and outer center would be similar and it would be hard to distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' It should be noted that the density in general is very small around the outer center, having a magnitude around 10−7 in the strong magnetized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Equidensity surfaces around the outer center become smaller in diameter for a low magnetization parameter, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In general we conclude that strong magnetic fields are suppressing the outer center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since the static surface is located close to the inner center, the properties in and at the static surface are only slightly influenced by strong magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −5M: Considering the same BS solution as in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' and setting the specific angular momentum to ℓ0 = −5M leads to a two-centered solution without a cusp, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The outer center moves closer to the BS center for a low magnetization parameter and the density values of the outer torus are for the most part more than two orders of magnitude lower compared to the non-magnetized case, meaning there is almost no matter in the outer torus (in comparison to the inner torus), with density levels around ∼ 10−9 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The inner center with the static surface behaves similarly to the solution with the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75M: Looking at the ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798 BS solution, there exist also two-centered solutions with a cusp, which are composed of a denser outer center, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' For the high magnetized disks the inner center as well as the outer center become denser compared to the non-magnetized solution, with the inner center being denser than the outer one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In general the equatorial density of the magnetized disks is for all radial values smaller than the non-magnetized outer center location higher compared to the non-magnetized disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' As a consequence there are higher density values at and around the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 20 shows the 2-dimensional density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The locations of the inner center and the cusp are not significantly influenced by strong magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The outer center moves closer to the BS center and the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Considering that the inner center becomes denser for a strong magnetization, there must be a threshold value β0, below which the outer center has lower density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 19 presents solutions close to this threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Since β0 marks the intersection between the density of the inner and outer center, a two-centered magnetized disk solution with the same density at both centers is possible for this magnetization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 13 log10 βmc ρci rci ρco rco ρcusp rcusp ρci ρco log10 (pmax) log10 (pmmax) hmax 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='102 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='068 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='061 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='813 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='733 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='733 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='074 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='011 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='861 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='031 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='031 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='037 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='044 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='811 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='724 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='724 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: log10 of the density in the equatorial plane for different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Dashed vertical lines represent center positions, with rci referring to the inner center and rco to the outer center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dotted vertical line represents the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The outer center (and therefore also the cusp) vanishes in the magnetized solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) 1 ≤ βmc ≤ 100 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='72 ≤ βmc ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='78 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: (a) Solutions for βmc in the range of 1 to 100, representing mildly magnetized disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (b) Closeup of solutions for βmc around the threshold value β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Dashed lines represent the location of the outer center rc0, dotted lines represent the location of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed-dotted black line in (b) shows the location of the saddle point, which exists for the density curve corresponding to the threshold value of β0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' For the lowest two curves in (a) and for the lowest five curves in (b) βmc is below β0, therefore there exists no cusp and no outer center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 βmc = 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 βmc = 1 βmc = 10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 (d)0T601 I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 I I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 Irco :rcusp 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2 4 6 8 10 12 14 16 18 20 r/Mβmc 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 (d)0160l 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 40 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 2 4 6 8 10 12 r/Mβmc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='430 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='435 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='440 (d)0160l 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='445 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='450 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='455 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='460 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 r/M14 (a) βmc = 105 (b) βmc = 105 (c) βmc = 10−5 (d) βmc = 10−5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: Density distribution visualized for different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The minimum of the density is set to log10 ρ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='8 for (a) - (c) and the maximum to the density value at the inner center of the strong magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Black solid lines represent the isodensity surfaces, dotted red lines represent the isodensity surface corresponding to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed-dotted blue line in (d) represents the location of the saddle point for the solution with the threshold value β0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='757, the minimum of the density is set to log10 ρ = −2 in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' l0g10p 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='384 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='768 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='303 1- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='687 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='071 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='455 3 2 3 4 5 6 7 8 X/M15 log10 βmc log10 ρci rci log10 ρco rco log10 ρcusp rcusp log10 � ρci ρco � log10 (pmax) log10 (pmmax) hmax 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='66 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='241 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='384 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='151 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='522 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='255 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: The left panel shows the log10 of the density in the equatorial plane around the inner center of the disk, with the dashed lines marking the position of the inner center and the indigo dotted line the position of the inner intersection of the static surface with the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The right panel shows the density in the equatorial plane around the outer center, with the dotted lines representing the location of the cusp and the dashed lines the location of the outer center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: (a) Locations of the outer center and cusp versus βmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (b) Density at the outer center and cusp versus βmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 4 βmc = 105 βmc = 105 βmc = 1 βmc = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 5 βmc = 10-5 βmc = 10-5 (d)0T60l : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 6 一 : : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 7 :: Irci irc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Ici!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Ircoirc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='. 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 r/M r/M12 10 Center W/ Cusp 8 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 log10βmc-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 (d)0160l 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 Center Cusp 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 log10βmc16 (a) βmc = 105 (b) βmc = 105 (c) βmc = 10−5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5M: Density distribution visualized for different magnetization parameters, the minimum of the density is set to log10 ρ = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='9 and the maximum to the density at the inner center of the high magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The black solid lines represent the isodensity surfaces, the red dotted lines represent the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' log10 βmc log10 ρci rci log10 ρco rco log10 ρcusp rcusp log10 � ρci ρco � log10 (pmax) log10 (pmmax) hmax 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='67 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −5M: The left panel shows the equatorial density around the inner center of the disk, the right panel shows the equatorial density around the outer center of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Dotted vertical lines are marking the positions of the inner and outer center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The indigo dotted line represents the location of the inner intersection of the static surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' log10p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='008 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='901 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='810 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='719 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='628 z/M 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='537 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='446 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='355 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='264 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='173 0 5 10 15 20 25 W/X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 βmc = 105 二二二 βmc = 105 βmc = 1 7 βmc = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 βmc = 10-5 βmc = 10-5 (d)0T60) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 (ci Irci Irco rco -- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': 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+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='671, ℓ0 = −5m: Density distribution visualized for different magnetization parameters, the minimum of the density is set to log10 ρ = −10 and the maximum to the density at the inner center of the high magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The black solid lines represent the isodensity surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' log10 βmc ρci rci ρco rco ρcusp rcusp ρci ρco log10 (pmax) log10 (pmmax) hmax 5 0.' metadata={'source': 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plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Dashed lines show the locations of the inner and outer center, the dotted lines show the position of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' log10p 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='022 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='040 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='058 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='076 5 4.' metadata={'source': 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+page_content='149 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='167 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='185 15 0 5 10 15 20 25 30 X/Mlog10P 15 F-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='022 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='040 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='058 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='076 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='094 W/z 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='113 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='131 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='149 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='167 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='185 15 0 5 10 15 20 25 30 X/M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 88888888888838888888888 βmc = 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 1 βmc= 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 (d)0T60) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 : Irco Irco 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 10 12 4 6 8 14 16 18 20 r/M18 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75M: (a) Equatorial density curves for βmc in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='1 ≤ βmc ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Horizontal dotted lines mark the density value at the inner center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (b) Density of the inner and outer center versus βmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The dashed vertical line represents the threshold value β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (a) βmc = 105 (b) βmc = 105 (c) βmc = 10−5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='798, ℓ0 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75M: Density distribution visualized for different magnetization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The minimum density is set to log10 ρ = −2 and the maximum to the density value at the outer center of the high magnetized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The black solid lines represent the isodensity surfaces, the dotted red line represents the isodensity surface corresponding to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' CONCLUSION In this work we investigate the properties of non-selfgravitating magnetized thick disks in boson star spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We construct the equilibrium configurations as solutions to the relativistic Euler equations assuming the presence of a toroidal magnetic field and constant angular momentum of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We further describe the accreting plasma by a perfect fluid with polytropic equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The influence of the magnetic field is evaluated by considering models with different degree of magnetization quantified by the ratio of the thermal and magnetic pressure, which serves as a magnetization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In order to compare to the purely hydrodynamical case we choose several representative boson star solutions ranging from mildly to highly relativistic, which are characterized by qualitatively different morphology of the equilibrium tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Then we analyse the impact of the magnetic field by exploring certain distinctive features such as modifications in the disk compactness, location of the cusps and centers, distribution of the predominant fluid density, transitions in the geometry or topology of the equilibrium configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We observe the following systematic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Mildly relativistic boson stars, which in the purely hydrodynamical case support equilibrium tori with a single center and no cusp preserve the disk morphology, when magnetic field is log10βmc 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='0 5 10 15 r/MInner center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='75 Outer center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='50 (d)0T601 BO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='50 2 1 0 1 2 log10βmclog10p 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='001 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='434 5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} 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Z/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='869 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='086 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='304 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='521 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='739 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='956 15 0 5 10 15 20 25 30 X/M19 present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' However, when the magnetization increases the disks become more compact and the predominant matter distribution shifts inwards towards the boson star center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' This is manifested quantitatively by a range of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In particular, the location of the torus center moves closer to the boson star, the maximum density at the torus center increases, and the predominant fluid density concentrates in smaller regions of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In addition, for strong magnetic fields the disk can become highly compressed towards the rotation axis and the geometry of the equidensity surfaces can change from oblate to prolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' These effects are consistent with the studies of magnetized equilibrium tori around black holes where similar qualitative behavior is observed [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the case of highly relativistic boson stars we observe some qualitatively new features induced by the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In the purely hydrodynamical case these solutions are characterized by two-centered equilibrium tori which can be either disjoint or connected by a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' We demonstrate that a sufficiently strong magnetic field can cause a transition in the disk topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' After a certain critical value of the magnetization parameter, some configurations lose their outer center and become one-centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' For other configurations the outer center does not disintegrate in the strict sense but it moves inwards and the fluid density decreases so much in this region that it becomes insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' In these cases the ratio of the maximum fluid density at the inner and outer center reaches eight orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Another phenomenon which we observed is a shift of the predominant fluid density between the torus centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Configurations in which the fluid density is concentrated at the outer center in the purely hydrodynamical case can turn into configurations with dominant matter distribution at the inner center under a sufficiently strong magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' The described effects can be interpreted as a manifestation of the same physical behavior as for the mildly relativistic boson stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' They illustrate further the trend that the presence of magnetic field leads to denser and more strongly compactified equilibrium configurations which are concentrated closer to the gravitational field center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to gratefully acknowledge support by the DFG Research Training Group 1620 Models of Gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' gratefully 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+page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' (Event Horizon Telescope), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 930, L17 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Gimeno-Soler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Font, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Herdeiro, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Radu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' D 99, 043002 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Cruz-Osorio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Gimeno-Soler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Font, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' De Laurentis, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Mendoza, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' D 103, 124009 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Teodoro, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Collodel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Doneva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Kunz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Nedkova, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Yazadjiev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' D 104, 103008 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [22] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Stuchl´ık, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Slan`y, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Hled´ık, A&A 363, 425 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Palenzuela, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 20, 5 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Kaup, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Schunck and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Mielke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' A 249, 389 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [37] B.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Collodel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Kleihaus, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Kunz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' D 99, 104076 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [43] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Sanchis-Gual, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Di Giovanni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Zilh˜ao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Herdeiro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Cerd´a-Dur´an, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Eilers, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Hartmann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Schaffer, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Toma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' D 88, 044025 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 32, 235022 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Grould, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Meliani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Vincent, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Grandcl´ement, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Gourgoulhon, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' 34, 215007 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQflPh9/content/2301.00449v1.pdf'} +page_content=' [49] L.' metadata={'source': 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in Small Datasets +Sheng Kuanga, Henry C. Woodruffa,b, Renee Granzierc, Thiemo J.A. van Nijnattenb,d, Marc B.I. Lobbesb,d,e, Marjolein L. Smidtc,d, +Philippe Lambina,b, Siamak Mehrkanoonf,∗ +aThe D-Lab, Department of Precision Medicine, GROW – School or Oncology and Reproduction, Maastricht University, Maastricht, The Netherlands +bDepartment of Radiology and Nuclear Medicine, Maastricht University Medical Centre+, Maastricht, The Netherlands +cDepartment of Surgery, Maastricht University Medical Centre+, Maastricht, The Netherlands +dGROW – School for Oncology and Reproduction, Maastricht University, Maastricht, The Netherlands +eDepartment of Medical Imaging, Zuyderland Medical Center, Sittard-Geleen, The Netherlands +fDepartment of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands +Abstract +Deep learning (DL) applied to breast tissue segmentation in magnetic resonance imaging (MRI) has received increased attention +in the last decade, however, the domain shift which arises from different vendors, acquisition protocols, and biological heterogene- +ity, remains an important but challenging obstacle on the path towards clinical implementation. Recently, unsupervised domain +adaptation (UDA) methods have attempted to mitigate this problem by incorporating self-training with contrastive learning. To +better exploit the underlying semantic information of the image at different levels, we propose a Multi-level Semantic-guided Con- +trastive Domain Adaptation (MSCDA) framework to align the feature representation between domains. In particular, we extend the +contrastive loss by incorporating pixel-to-pixel, pixel-to-centroid, and centroid-to-centroid contrasts to integrate semantic informa- +tion of images. We utilize a category-wise cross-domain sampling strategy to sample anchors from target images and build a hybrid +memory bank to store samples from source images. Two breast MRI datasets were retrospectively collected: The source dataset +contains non-contrast MRI examinations from 11 healthy volunteers and the target dataset contains contrast-enhanced MRI exami- +nations of 134 invasive breast cancer patients. We set up experiments from source T2W image to target dynamic contrast-enhanced +(DCE)-T1W image (T2W-to-T1W) and from source T1W image to target T2W image (T1W-to-T2W). The proposed method +achieved Dice similarity coefficient (DSC) of 89.2% and 84.0% in T2W-to-T1W and T1W-to-T2W, respectively, outperforming +state-of-the-art methods. Notably, good performance is still achieved with a smaller source dataset, proving that our framework is +label-efficient. +Keywords: Breast segmentation, Unsupervised domain adaptation, Contrastive learning +1. Introduction +Breast cancer is the most commonly diagnosed cancer in +women and contributes to 15% of mortality worldwide, rank- +ing as a leading cause of death in many countries (Francies et al. +(2020); Sung et al. (2021)). The significantly increasing mor- +tality rates of breast cancer, especially in developing countries +and low-income regions, lead to increased burdens for patients, +their families, and society, highlighting the need for early de- +tection and intervention (Azamjah et al. (2019)). In the past +decades, breast magnetic resonance imaging (MRI) has been +recommended to supplement conventional mammography and +ultrasound techniques to screen women at a high risk of breast +cancer and determine the extent of breast cancer after diagnosis +(Saslow et al. (2007); Gupta and Billadello (2017)). +∗Corresponding author +Email address: s.mehrkanoon@uu.nl (Siamak Mehrkanoon ) +A further step towards advanced MRI-based diagnosis is ac- +curate breast segmentation. In clinical routine, whole-breast +segmentation and analysis are conducted manually relying on +the expertise of clinicians, which is a challenging and time- +consuming process. With the advent of computer vision, atlas- +and statistical-based methods with high accuracy compared to +manual segmentation have been proposed (Wu et al. (2013)). +Recently, numerous deep learning (DL) approaches have been +developed to further improve the performance of breast seg- +mentation. These tools extract salient features directly from the +images and automatically segment the breast boundary (Dalmıs¸ +et al. (2017); Hu et al. (2018); Ivanovska et al. (2019)), breast +fibroglandular tissue (FGT) (Dalmıs¸ et al. (2017); Ivanovska +et al. (2019)) and breast lesions (Zhang et al. (2018b); Gallego- +Ortiz and Martel (2017); Negi et al. (2020)), which are less +prone to errors and have achieved encouraging Dice similarity +coefficients (DSCs) in various datasets. +Preprint submitted to arXiv +January 9, 2023 +arXiv:2301.02554v1 [q-bio.QM] 4 Jan 2023 + +Despite the high popularity of DL approaches, there are +some barriers on the path to clinical implementation. +One +main concern is performance degradation due to large inhomo- +geneities present in MRI datasets, leading to differing imaging +feature distributions between training (source domain) and test- +ing (target domain) datasets, also known as the domain shift +problem. For instance, most available imaging datasets from +different medical centers are acquired with different acquisi- +tion parameters, protocols and hardware vendors (Perone et al. +(2019)). Moreover, patient heterogeneity, stemming from dif- +ferent breast sizes, menstrual cycle effects and stage of disease +progression, leads to inconsistencies within the images, which +may result in unstable performance of DL models (Granzier +et al. (2022)). Although this problem could be addressed by +acquiring large and varied datasets of accurately annotated tar- +get images for training, this exercise would be labor-consuming +and expensive, and is further hindered by legal and ethical con- +siderations regarding the sharing of patient data. Thus, recent +published studies (Hoffman et al. (2018); Hoyer et al. (2022); +Hoffman et al. (2016)) focus on developing unsupervised do- +main adaptation (UDA) methods to mitigate the distribution +discrepancy without target labels. +Recent advances in UDA methods have enabled its applica- +tion to medical images by transferring the knowledge from the +source domain to the target domain (Perone et al. (2019); Sha- +nis et al. (2019); Liu et al. (2022); Chaitanya et al. (2020)). For +instance, adversarial learning (e.g. Cycle-GAN) adopts the dis- +criminator to align the distribution in the latent feature space +(Dou et al. (2018); Hoffman et al. (2018)) or label space (Tsai +et al. (2018)). +In practice, these methods suffer from train- +ing instability and model collapse (Zhu et al. (2017)). Self- +training (Tarvainen and Valpola (2017); Perone et al. (2019)) +is another promising method which combines entropy mini- +mizing (Vu et al. (2019)), pseudo-label denoising (Zhang et al. +(2021)), adversarial learning (Shanis et al. (2019)), generative +methods (Zhao et al. (2021b)) or contrastive learning (Zhang +et al. (2022); Liu et al. (2022); Chaitanya et al. (2020)). More +recently, incorporating self-training with complementary con- +trastive learning shows remarkable performance improvement +by utilizing the ground truth and pseudo labels as supervised +semantic signals to guide the training. +Contrastive learning in this case explicitly computes the +inter-category similarity between pixel representation pairs (Liu +et al. (2022)) (refers to pixel-to-pixel (P2P) contrast) aiming +to learn an invariant representation in feature space. However, +it still suffers from the following two major concerns that are +not taken into account: (i) P2P contrast skips the structure con- +text of adjacent pixels, so it does not extensively exploit the +semantic information present in the MRI scan. To alleviate this +problem, we propose a method to integrate different levels of +semantic information into a contrastive loss function. More +specifically, the mean value of the pixel representations of a +specific category, i.e., the centroid, should be similar to the pix- +els contained in the region. Likewise, centroids, regardless of +whether they are from the same domain, should also be close +to centroids of the same category and far away from centroids +of other categories. We denote these two relations as pixel-to- +centroid (P2C) and centroid-to-centroid (C2C) respectively. (ii) +A common practice to perform inter-category contrast is to gen- +erate positive and negative pairs by sampling partial pixel rep- +resentations in a mini-batch (Chaitanya et al. (2020)). However, +the imbalanced proportion between background and regions of +interest (ROIs) in the breast MRIs poses a challenge to obtain +adequate pairs during training. To address this problem, we +build a hybrid memory bank and optimize the sampling strat- +egy to ensure enough cross-domain positive and negative pairs +especially for the highly imbalanced mini-batches. Addition- +ally, we also explore the impact of anchors and samples from +different domains on model performance. +In summary, we extend the contrastive UDA framework for +breast segmentation to further mitigate the domain shift prob- +lem. To the best of our knowledge, this is the first attempt to +apply contrastive UDA in breast MRI. We briefly provide the +novel contributions of our work as follows: +1. To solve the domain shift problem in breast MRI, we +develop a novel Multi-level Semantic-guided Contrastive +Domain Adaptation (MSCDA) framework for cross- +domain breast tissue segmentation. +2. To exploit the semantic information present in source la- +bels, we propose a method that combines pixel-to-pixel, +pixel-to-centroid and centroid-to-centroid contrasts into +the loss function. +3. To resolve the data imbalance problem, we develop a hy- +brid memory bank that saves both category-wise pixel and +centroid samples. We further investigate a category-wise +cross-domain sampling strategy to form adequate con- +trastive pairs. +4. To validate the performance of the UDA framework, we +replicate our experiment under multiple source datasets +of different sizes. The results show robust performance +and label-efficient learning ability. We further show that +our framework achieves comparable performance to su- +pervised learning. +2. Related Works +2.1. Semantic Segmentation +Semantic segmentation is an essential and hot topic in com- +puter vision, achieving automatic categorization of each pixel +(or voxel) into one or more categories. In recent years, con- +volutional neural networks (CNNs) have shown significant re- +sults in multiple fields. +Fully convolutional network (FCN) +(Long et al. (2015)), as one of the most remarkable early-stage +segmentation architectures, demonstrated the pixel-level repre- +sentation learning ability of CNNs. However, CNNs are still +far from maturity in terms of accuracy and efficiency. There- +fore, many mechanisms have been proposed to improve seg- +mentation performance. For instance, U-Net Ronneberger et al. +(2015) introduced skip connections in an encoder-decoder de- +sign to solve the vanishing gradient problem; DeepLab v3+ +(Chen et al. (2018)) proposed Atrous Spatial Pyramid Pool- +ing (ASPP) to capture more context information in multi-scale +2 + +receptive fields. Meanwhile, inspired by the effectiveness of +residual blocks, ResNet (He et al. (2016)) was adopted as the +backbone in many encoder-decoder segmentation frameworks +(Chen et al. (2018); Wu et al. (2019); Zhang et al. (2018a); He +et al. (2019)) to provide deep feature representations. +DL techniques have also been widely adopted in the field +of medical imaging that pave the way towards more precise +and automated clinical diagnosis and prognosis. +Common +DL methods focus on the segmentation of contours (Zhang +et al. (2020); Dalmıs¸ et al. (2017); Zhang et al. (2019); Pi- +antadosi et al. (2018)) and lesions (Dalmıs¸ et al. (2017); Zhang +et al. (2018c); El Adoui et al. (2019)) in breast magnetic res- +onance imaging (MRI). Despite the promising performance, +these methods require large datasets with expert annotations, +expensive and hard to realize the current data-sharing paradigm. +2.2. Contrastive Learning +Contrastive learning (CL) was introduced as a self- +supervised learning framework, allowing the model to learn +representations without labels (Oord et al. (2018); He et al. +(2020); Chen et al. (2020b,a); Grill et al. (2020)). An essen- +tial step of early CL methods is to build a pretext task, such +as instance discrimination (Wu et al. (2018); He et al. (2020); +Chen et al. (2020a)), to discriminate a positive pair (two aug- +mented views of an identical image) from negative pairs (aug- +mented view of other images). Based on this pioneering ap- +proach, many subsequent advanced mechanisms have been pro- +posed to improve the representation learning ability. For exam- +ple, Moco v1 (He et al. (2020)) and v2 (Chen et al. (2020b)) +combined a momentum encoder with a first-in-first-out queue +to maintain more negative samples. This results in an improved +classification performance e.g., ImageNet (Deng et al. (2009)) +and enables training the network on normal graphics process- +ing units (GPUs). Afterwards, the projection head (Chen et al. +(2020a)) and the prediction head (Grill et al. (2020)) were in- +troduced respectively to improve the classification accuracy on +downstream tasks. +For semantic segmentation tasks, recent CL works lever- +age the pixel-level labels as supervised signals (Zhao et al. +(2021a); Zhong et al. (2021); Hu et al. (2021); Wang et al. +(2021); Chaitanya et al. (2020)). +The underlying idea is to +group the pixel representations from the same category and to +separate pixel representations from different categories. Zhao +et al. (2021a) introduced a label-efficient two-stage method that +pre-trained the network by using P2P contrastive loss and then +fine-tuned the network using cross-entropy (CE) loss (Bishop +and Nasrabadi (2006)). PC2Seg (Zhong et al. (2021)) improved +this method in a one-stage semi-supervised learning (SSL) ap- +proach by jointly updating the network weights with pixel con- +trastive loss and consistency loss. ContrastiveSeg (Wang et al. +(2021)) combined pixel-to-region contrastive loss to explicitly +leverage the context relation across images. Similar to Zhong +et al. (2021); Wang et al. (2021), the authors in Chaitanya +et al. (2020) validated the effectiveness of different sampling +strategies on contrastive learning for multiple medical MRI seg- +mentation tasks (Bernard et al. (2018); Antonelli et al. (2022); +Zhuang and Shen (2016)). +2.3. Unsupervised Domain Adaptation +Unsupervised Domain Adaptation (UDA) is used to general- +ize learned knowledge from a labeled source domain to an unla- +beled target domain. The key challenge of UDA is domain shift, +i.e., the inconsistent data distribution across domains, which +usually causes performance degradation of models. Early ma- +chine learning methods utilized different feature transforma- +tions or regularizations to overcome this problem (Kouw and +Loog (2018); Mehrkanoon and Suykens (2017); Mehrkanoon +(2019)). +A number of existing DL methods solve the domain shift +problem using adversarial learning or self-training-based ap- +proaches. Adversarial learning utilizes generative adversarial +networks (GANs) (Goodfellow et al. (2014)) to align the distri- +bution of the feature space (Tzeng et al. (2017); Hoffman et al. +(2016); Chen et al. (2019b); Dou et al. (2018)) or label space +(Vu et al. (2019); Dou et al. (2018); Tsai et al. (2018)). In par- +ticular, CycleGAN (Zhu et al. (2017); Kim et al. (2017); Yi et al. +(2017)) has been extensively explored and adopted in medical +image UDA (Jiang et al. (2018); Zhang et al. (2018d); Chen +et al. (2019a); Guan and Liu (2021)) because of its ability to +translate the ‘style’ of the source domain to the target domain +in an unpaired way. +Self-training, frequently used in SSL, uses the predictions of +the target domain as pseudo-labels and retrains the model iter- +atively. A typical self-training network (Tarvainen and Valpola +(2017)) generates pseudo-labels from a momentum teacher net- +work and distills knowledge to the student network by using +the consistency loss. The authors in Perone et al. (2019); Per- +one and Cohen-Adad (2018) improved the self-training method +by aligning the geometrical transformation between the student +and teacher networks. DART (Shanis et al. (2019)) and MT- +UDA (Zhao et al. (2021b)) combined self-training with adver- +sarial learning in different ways, both receiving promising re- +sults. For imbalanced datasets, different denoising methods and +sampling strategies have been proposed to improve the qual- +ity of pseudo-labels (Zhang et al. (2021); Hoyer et al. (2022); +Xie et al. (2022)). Similar to Chaitanya et al. (2020), recent +self-training approaches (Xie et al. (2022); Zhang et al. (2022)) +incorporated CL, i.e. unsupervised contrastive domain adap- +tation, to align cross-domain features by sampling or merging +contrastive feature embeddings across categories. In this study, +we integrate three kinds of contrastive losses and a category- +wise cross-domain sampling strategy to accomplish the UDA +segmentation task for breast MRI. +3. Method +3.1. Problem Definition +Source domain data and target domain data are two sets of +data used in the domain adaptation problem. The source do- +main data Xs = {xs}M +i=1 have pixel-level labels whereas the target +domain image data Xt = {xt}N +i=1 are unlabeled. We aim at devel- +oping a method that can learn from the labeled source domain +and be applied to the target domain. In particular, the learned +3 + +Table 1: Important notations in our proposed method. +Notations +Description +xs, xt, ys, ˆys +Source image, target image, source image +ground truth and corresponding one-hot repre- +sentation respectively; +ps, pt +Student network probability map of the source +and target images respectively; +p′ +s, p′ +t +Teacher network probability map of the source +and target images respectively; +zt +Student network feature embedding of the tar- +get image; +z′ +s +Teacher network feature embedding of the +source image; +ˆyt, ˆy′ +t +One-hot pseudo-label of pt and p′ +t respectively +(y=argmax(p)); +vk +s, vk +t +Pixel feature embedding of category k of the +source and target images respectively; +ck +s, ck +t +Centroid feature embedding of category k of the +source and target images respectively; +Qpixel, Qcentroid +Pixel queue and centroid queue in the memory +bank. +network is used to classify each pixel of the target domain im- +age into K categories. A direct approach is to train the net- +work in a supervised manner on the source domain and apply +it directly to the target domain. However, the performance of +the network often drops because of the aforementioned domain +gap between source and target domains. To address this con- +cern, we propose a new domain adaptation approach, named +MSCDA, based on the combination of self-training and con- +trastive learning. +3.2. Overall Framework +The proposed domain adaptation framework is depicted in +Fig. 1. It consists of a student network and a momentum teacher +network. The student network consists of four main compo- +nents, a feature encoder fe, a feature decoder fd, a projection +head fproj, and an additional prediction head fpred. These com- +ponents are correspondingly mapped in the teacher network +with the only exception of the last component (i.e., the pre- +diction head). The three components in the teacher network are +called f ′ +e, f ′ +d and f ′ +pred. The important notations are listed in +Table 1. +In the student network, the feature encoder fe maps the in- +put MRI image x ∈ RH×W×1 into a high dimension feature map +h ∈ RH′×W′×C. Next, h is transferred into a segmentation prob- +ability map p ∈ RH×W×K and a low dimension feature embed- +ding z ∈ RH′×W′×D through two forward passes, hereafter re- +ferred to as segmentation and contrast paths, respectively. In +the first forward pass (segmentation path), the decoder fd gen- +erates the segmentation probability map p of the input h. In the +second forward pass (contrast path), the projection head fpro j +and prediction head fpred jointly reduce the feature map into a +low-dimension projected feature embedding z = fpred( fpro j(h)). +Similar steps are conducted in the teacher network, yielding the +momentum probability map p′ and feature embedding z′. Fi- +nally, the probability map p and p′ are used for self-training +while the projected feature embeddings z and z′ are used for +semantic-guided contrastive learning to diminish the discrep- +ancy between the two domains. The overall loss function is +given by: +L = Lseg + λ1Lcon + λ2Lctr, +(1) +where Lseg is the supervised segmentation loss, Lcon is the +consistency loss, Lctr is the contrastive loss, and λ1 and λ2 +are the regularization parameters of the corresponding losses. +The summation of segmentation and consistency loss is hence- +forth referred to as the self-training loss. We elaborate the self- +training loss in Section 3.3 and our proposed contrastive loss in +Section 3.4. +3.3. Self-training +Following the self-training paradigm (Perone et al. (2019)), +two optimization goals were established. The first goal is to per- +form supervised learning on the student network from source +image labels. The second goal is that the student network learns +the pseudo labels generated by the teacher network to distill +knowledge from target images. Only the weights in the seg- +mentation path of both networks are updated in this phase. +3.3.1. Supervised Learning +In supervised learning, we employ a hybrid segmentation +loss (Isensee et al. (2018)) that combines Dice loss (Sudre et al. +(2017)) and CE loss, and is formulated as: +Lseg = 1 +2LDice(ps, ˆys) + 1 +2Lce(ps, ˆys), +(2) +where ˆys is the one-hot ground truth and ps is the probability +map of the source domain image in the student network. +3.3.2. Distilling Knowledge from Pseudo Labels +The pseudo label of the target image is generated by the seg- +mentation path in the momentum teacher network iteratively: +ˆy′ +t = argmax(p′ +t), +(3) +where p′ +t is the probability map of the target domain image in +the teacher network. In order to distill knowledge from the +pseudo label, an extra consistency loss is added between the +two networks. In other words, the target image segmentation pt +generated by the student network is guided by the pseudo label +ˆy′ +t. The consistency loss is formulated as: +Lcon = +1 +H × W × K +H×W +� +i=1 +K−1 +� +k=0 +���p(i,k) +t +− ˆy′(i,k) +t +��� +2 , +(4) +where i is the pixel index of the image and k is the category. +Here, we update the weights of the student network by means +of back propagation. However, in the teacher network, a stop- +gradient operation is applied, and the network weights are up- +dated by exponential moving average (EMA): +Θ′ ← αΘ′ + (1 − α)Θ, +(5) +4 + +Student Network +Teacher Network +stop-grad +stop-grad +projection head +feature encoder +feature decoder +prediction head +feature embedding +projected embedding +one-hot prediction +memory bank +loss function +Figure 1: The proposed framework of the unsupervised domain adaptation method, named MSCDA. A source image xs and a target image xt are the inputs to the +student network and the momentum teacher networks, respectively. Each network consists of a segmentation path and a contrast path. The student network is trained +by a supervised segmentation loss, an inter-network consistency loss and a multi-level contrastive loss. The teacher network updates the weights by exponential +moving average (EMA). The training procedure is layer out in Sections 3.3 and 3.4. +where Θ and Θ′ are the weights of the student network and +teacher network respectively, and α ∈ (0, 1) is the momentum +coefficient. +Combining data augmentation with self-training has been +shown to improve the domain adaptation performance (Tar- +vainen and Valpola (2017); Chen et al. (2020a)). The student +network receives strongly-augmented images, and the teacher +network receives weekly-augmented images during the training +process. Random resized cropping is used as the weak augmen- +tation method, and random brightness, contrast and Gaussian +blur are used as strong augmentation methods. The strongly- +augmented path learns a robust feature representation from the +weakly-augmented path that has less disruption. +3.4. Semantic-guided Contrastive Loss +In order to improve the performance of our UDA framework +even further, we incorporate a multi-level semantic-guided con- +trast to the self-training framework. The idea is to leverage the +ground truth of the source domain as supervised signals to en- +force the encoder to learn a well-aligned feature representation +that mitigates the domain discrepancy. A common way is to cat- +egorize the feature embedding and conduct contrastive learning +using the pixels or centroids between domains. In our approach, +we develop the contrastive loss at P2P, P2C and C2C levels to +directly utilize multi-level semantic information to guide the +feature alignment. The data flow of our proposed contrastive +loss is depicted in Fig. 2. +3.4.1. Preliminaries +In unsupervised contrastive segmentation approaches, the +contrast is performed using a randomly selected sample (called +the anchor) v, a positive sample v+ and n negative samples +V− = {v− +1, v− +2, ..., v− +n}. The aim is to learn a feature representation +that yields high similarity in positive pairs (v, v+) and low simi- +larity in negative pairs (v, v−). Following He et al. (2020); Chen +et al. (2020b); Zhong et al. (2021), we utilize the InfoNCE as +our loss function, which is given as follows: +Lctr = − log +exp(v · v+/τ) +exp(v · v+/τ) + �n +i=1 exp(v · v− +i /τ), +(6) +where n is the number of negative samples per anchor, ‘·’ is +the dot product between two samples, and τ is a temperature +hyperparameter that controls the gradient penalty of hard nega- +tive samples, which is empirically set to 0.07 (He et al. (2020)). +Here, samples are selected from D-dimensional feature embed- +ding followed by l2-normalization. +3.4.2. Feature Categorization +Feature categorization is a necessary step required for super- +vised contrastive learning in the feature space. To utilize the +semantic information effectively, we categorize the feature em- +bedding from both domains. For the source image, the feature +embedding in the teacher network and its ground truth are re- +quired. Given the l2-normalized target network feature embed- +ding of a source image z′ +s ∈ RH′×W′×D and the one-hot ground +truth ˆys ∈ RH×W×K, we first down-sample the one-hot ground +truth into ¯ys ∈ RH′×W′×K to fit the embedding size, then assign +the category label index k ∈ {0, K − 1} of ¯ys to each pixel of z′ +s +(Fig. 2(a)). Similarly, the target image embedding zt can also +be categorized using the pseudo label ˆyt. Based on the catego- +rized feature embedding, we further compute the category-wise +mean value of pixels of the feature embedding as the centroid +C={ck}K−1 +k=0 , which is given as follows: +ck = +1 +���Yk��� +H′×W′ +� +i=1 +1 +� +¯y(i,k) = k +� +· zi, +(7) +where 1 [·] is an indicator function that returns 1 when the con- +dition holds and 0 otherwise, zi is the ith pixel of the feature +embedding and ¯y(i,k) is the down-sampled label which belongs +to the ith pixel and category k, Yk is the set of labels of category +k. +5 + +student network +teacher network +sampling +sampling +pixels +centroids +pixels +centroids +pixel queue +centroid queue +mean +mean +anchors +(c) memory bank +(b) category-wise anchor +sampling & computing +(a) feed-forward & +feature categorization +... +... +... +... +... +... +source image +target image +... +... +(d) cross-domain contrasts +Category0-Source +Category0-Target +Category1-Target +Category1-Source +anchors +pos. +neg. +pixel-to-pixel +neg. +pos. +pixel-to-centroid +pos. +pos. +neg. +neg. +centroid-to-centroid +pos. +pos. +neg. +neg. +anchors +anchors +Figure 2: The data flow of our proposed multi-level contrastive loss. (a) +feed-forward and feature categorization. (b) category-wise anchor sampling +and computing. (c) memory bank. (d) proposed multi-level cross-domain con- +trasts. +3.4.3. Memory Bank & Sampling Strategy +Although the number of negative samples is critical for learn- +ing the feature representation (He et al. (2020)), insufficient +negative pairs of each batch in the breast MRI segmentation +task may occur because of the highly imbalanced ration be- +tween foreground and background pixels. To overcome this +problem, we utilize two category-wise queues as a memory +bank to maintain the pixel and centroid samples from the source +images in the teacher network. However, keeping all pixels in +the queue is not feasible because of GPU memory limitations. +Thus, we uniformly sample b pixels from each category in the +feature embedding to the pixel queue (Fig. 2(b,c)). This under- +sampling strategy enables the queue to maintain enough bal- +anced, but not redundant, pixel samples. The pixel queue Qpixel +and the centroid queue Qcentroid can be represented as: +Qpixel = {Qk +pixel}K−1 +k=0 , +Qk +pixel = {vk +(s,i)} +Bp +i=1, +(8) +Qcentroid = {Qk +centroid}K−1 +k=0 , +Qk +centroid = {ck +(s,i)}Bc +i=1, +(9) +where Qk +pixel is the pixel queue of category k, vk +(s,i) is the ith +source pixel sample of category k, Qk +centroid is the centroid queue +of category k, ck +(s,i) is the ith source centroid sample of category +k, and Bp and Bc are the size of the queue respectively. +3.4.4. Pixel-to-pixel Contrast +We perform the pixel-to-pixel (P2P) contrastive loss to align +the cross-domain feature representation of the same category. +To resolve this problem, we first sample m anchors from each +category of the target feature embedding zt in the student net- +work, denoted as set Vk +t . Then, for each anchor vk +t ∈ Vk +t with +category label k, we sample a source pixel of the same category +from the pixel queue Qpixel to form a positive pair (vk +t , vk+ +s ), and +sample n source pixels of category q ∈ K\{k} to form n negative +pairs (vk +t , vq− +s ). Based on these positive and negative pairs, the +InfoNCE loss of a single target anchor is computed by using +Eq.(6). Overall, the P2P loss is defined as: +LP2P +ctr = +1 +�K−1 +k=0 +���Vk +t +��� +K−1 +� +k=0 +� +vk +t ∈Vk +t +Lctr(vk +t , vk+ +s , Vq− +s ), +(10) +where |·| is the number of elements in a set, and Vq− +s +is the set of +negative source pixels. Note that the number of pixels labeled +as foreground categories might be less than m (or even 0) if the +model predicts a few (or no) breast tissue labels in a mini-batch. +Nevertheless, benefiting from the category-wise memory bank, +the contrast loss can still be computed even if all pixels in a +mini-batch belong to the same category. +3.4.5. Pixel-to-centroid Contrast +Due to the under-sampling strategy in selecting anchors and +updating the memory bank, the network may suffer from inade- +quate semantic knowledge and thereby be difficult to converge. +This issue is further addressed by incorporating P2C and C2C +contrasts to P2P contrast. +For P2C contrast, we force the pixel representation to learn +a more general representation with the guidance of the centroid +(Wang et al. (2021); Xie et al. (2022)). Specifically, a pixel and +a centroid from the same category are considered as a positive +pair (vk, ck+), while a pixel and a centroid from different cate- +gories are considered as a negative pair (vk, cq−). We reuse the +anchors in Section 3.4.4 and sample all positive and negative +centroids from the centroid queue Qcentroid. Similar to P2P loss, +the P2C loss is defined as: +LP2C +ctr += +1 +�K−1 +k=0 +���Vk +t +��� +K−1 +� +k=0 +� +vk +t ∈Vt +Lctr(vk +t , ck+ +s ,Cq− +s ), +(11) +where Cq− +s +is the set of negative source centroids. +3.4.6. Centroid-to-centroid Contrast +For C2C contrast, the ideal situation is that the centroids from +the same category are located near to one another, whereas cen- +troids from other categories are located far apart. Unlike P2C +contrast, the total number of centroids p (BK ≤ p ≤ 2BK) +is much smaller than the pixel number in a mini-batch. Be- +sides, calculating centroids is computationally efficient. There- +fore, the centroids of the whole mini-batch can be fully involved +as anchors in C2C contrast. Similar to P2P and P2C contrast, +the positive pairs (ck, ck+) and negative pairs (ck, cq−) are de- +fined according to whether centroids are from the same cate- +gory. Thus, the C2C loss is defined as: +LC2C +ctr += +1 +�K−1 +k=0 +���Ck +t +��� +K−1 +� +k=0 +� +ck +t ∈Ct +Lctr(ck +t , ck+ +s ,Cq− +s ), +(12) +where Ct is the set of target centroid anchors. +6 + +Table 2: Dataset description and acquisition parameters of dataset 1 and 2. +Subject +Number +Type +Scanner +Sequence +Acquisition Parameters +TR (ms) +TE (ms) +PS (mm) +ST (mm) +Dataset 1 +11 +Healthy +volunteers +Philips 1.5T +(Ingenia) +T1W +5.3 +3 +0.36×0.36 +2 +T2W +2000 +223 +0.79×0.79 +2 +Dataset 2 +134 +Patients with +invasive breast +cancer +Philips 1.5T +(Ingenia/Intera) +DCE-T1W +6.5-7.6 +2.9-3.5 +0.85×0.85- +0.97×0.97 +1 +T2W +2000 +170-259 +0.65×0.65- +0.97×0.97 +1 +Abbreviations: TR=Repetition time; TE=Echo time; PS=Pixel spacing; ST=Slice thickness; T1W=T1-weighted; +T2W=T2-weighted; DCE=Dynamic contrast-enhanced; T=Tesla. +Finally, we combine three above-mentioned contrasts (Fig. +2(d)) as our proposed multi-level semantic-guided contrastive +loss: +Lctr = LP2P +ctr + LP2C +ctr + LC2C +ctr . +(13) +The overall training process of our proposed MSCDA is pre- +sented in Algorithm 1. +4. Experiments +4.1. Datasets +Dataset 1. Dataset 1 consists of test-retest breast T1-weighted +(T1W) and T2-weighted (T2W) MRI images and correspond- +ing right-breast masks of eleven healthy female volunteers, +which is described in Granzier et al. (2022). The images of +each subject were collected in two separate sessions (interval<7 +days), during which three 3D scans were collected. Subjects +were asked to lay in the prone position and remain still in the +MRI scanner while both modalities are sequentially acquired. +All images were acquired with an identical 1.5T MRI scan- +ner (Philips Ingenia, Philips Healthcare, Best, the Netherlands) +using a fixed clinical breast protocol without contrast. +The +detailed acquisition parameters are listed in Table 2. In pre- +processing, we first resize all MRI slices and corresponding +masks to 256×256 pixels using cubic interpolation and nearest- +neighbor interpolation respectively, and then normalize images +with z-score transformation. In total, dataset 1 contains 14520 +(11 subjects × 2 sessions × 3 scans × 220 slices) T1W slices +and 11220 (11 subjects × 2 sessions × 3 scans × 170 slices) +T2W slices. +Dataset 2. Dataset 2 consists of the images from 134 subjects +with histologically confirmed invasive breast cancer imaged be- +tween 2011 and 2017 in Maastricht University Medical Cen- +ter+ and collected retrospectively (Granzier et al. (2020, 2021)). +The images contain dynamic contrast-enhanced breast T1W +and T2W MRI and corresponding right-breast masks. Simi- +lar to Dataset 1, each subject underwent the examinations with +1.5T MRI scanners (Philips Intera and Philips Ingenia (idem)) +in a prone position. In particular, T1W images were acquired +before and after the intravenous injection of gadolinium-based +contrast Gadobutrol (Gadovist, Bayer Healthcare, Berlin, Ger- +many (EU)) with a volume of 15 cc and a flow rate of 2 ml/s. +The acquisition parameters are also listed in Table 2. We con- +duct the same image pre-processing as in Dataset 1. In total, +Dataset 2 contains 21793 T2W and 28540 T1W slices and they +are split into three folds with 45, 45 and 44 subjects for the +cross-validation depicted in Section 4.2. +4.2. Experiment Setup +As shown in Table 2, the subject population, machine vendor +and acquisition parameters between the two datasets are het- +erogeneous, indicating the common domain shift problem in +clinical practice. We set up the experiment on both Dataset 1 +and 2 to transfer the knowledge of breast segmentation from +healthy women to patients. In particular, the experiment con- +sists of two scenarios: (1) Utilizing the T2W images of Dataset +1 as the source domain and the T1W images of Dataset 2 as the +target domain; (2) utilizing the T1W images of Dataset 1 as the +source domain and the T2W images of Dataset 2 as the target +domain. +In each scenario, we establish three tasks with a different +number of subjects in the source domain to validate the label- +efficient learning ability of our framework. The three tasks con- +tain four, eight and eleven (i.e., the whole dataset) randomly se- +lected subjects respectively, and are denoted as S4, S8 and S11. +To further verify the robustness of UDA performance, we split +the target domain into three folds to perform a three-fold cross- +validation. In each run of the cross-validation, two folds are +used as the target domain for training and the remaining fold +for testing. +4.3. Model Evaluation +The DSC is used as the main evaluation metric. Addition- +ally, we use the Jaccard Similarity Coefficient (JSC) as well as +precision (PRC) and sensitivity (SEN) as auxiliary evaluation +metrics. These metrics are formulated as follows: +DSC = +2 × TP +2 × TP + FP + FN × 100%, +(14) +7 + +Algorithm 1: MSCDA for Breast MRI +Input: Source domain image xs and label ys; Target +domain image xt; +1 Initialize the weights of the student network Θe, Θd with +pre-trained weights, Θproj and Θpred via He et al. +(2015). Initialize the teacher network by copying +weights from the student network and applying +stop-gradient; Initialize the memory bank Qpixel and +Qcentroid ; +2 for epoch = 1, Emax do +3 +foreach mini-batch do +4 +Apply weak and strong data augmentation; +5 +Forward propagate weak-augmented batch in the +student network to get ps, pt and zt; +6 +Forward propagate strong-augmented batch in +the teacher network to get p′ +t and z′ +s; +7 +Compute loss Lseg using ps and ys via Eq.(2); +8 +Compute loss Lcon using pt and p′ +t via Eq.(4); +9 +Categorize the feature embedding z′ +s and zt; +10 +foreach category do +11 +Sample pixel anchors and compute centroid +anchors from zt; +12 +Sample corresponding positive and negative +pairs from Qpixel and Qcentroid; +13 +Update Qpixel and Qcentroid using z′ +s; +14 +end +15 +Compute loss LP2P +ctr , LP2C +ctr and LC2C +ctr +via +Eq.(10)-(12) respectively; +16 +Update the student network via Eq.(4); +17 +Update the teacher network by Eq.(5); +18 +end +19 end +Output: Weights of the student network Θe and Θd. +JSC = +TP +TP + FP + FN × 100%, +(15) +PRC = +TP +TP + FP × 100%, +(16) +SEN = +TP +TP + FN × 100%, +(17) +where TP, FP and FN are the number of true positive, false +positive and false negative pixels of the prediction respectively. +Note that we show the mean value of each metric of the three- +fold cross-validation. +4.4. Implementation Details +4.4.1. Architecture +Encoder & decoder. We conduct our experiment by adopting +DeepLab-v3+ (Chen et al. (2018)) with ResNet-50 (He et al. +(2016)) as backbone. +Benefiting from the encoder-decoder +architecture, the encoder and decoder of DeepLab-v3+ are +adopted in our framework. +Specifically, the hidden dimen- +sion of ResNet-50 is set to (16, 32, 64, 128), yielding a 512- +dimension feature map. +Projection/Prediction Head. The projection head fpro j is a +shallow network that contains two 1 × 1 convolutional layers +with BatchNorm and ReLU. It projects the 512-d feature map +into a 128-dimension l2-normalized feature embedding. The +prediction head fpred shares the same architecture setting with +fpro j with the exception that the fpred does not change the di- +mension of the features. +Memory bank. The size of the pixel queue and the centroid +queue are respectively set to 32768 and 4096. In each mini- +batch, we randomly sample eight pixels per category of each +feature embedding to the queue and discard the oldest samples. +4.4.2. Training Settings +To accelerate the training procedure, we pre-train the +DeepLab-v3+ on the source domain and then use the weights to +initialize the encoder fe and decoder fd of our UDA framework. +Additionally, the projection and prediction heads are initialized +by He et al. (2015). By default, the number of pixel anchors for +P2P loss is set to 32 and the number of negative pairs of (P2P, +P2C, C2C) loss is set to (32768, 4096, 4096). The regulariza- +tion parameters λ1 and λ2 are set to 1. The Adam (Kingma +and Ba (2014)) optimizer is used for training the framework for +Emax=100 epochs with a fixed learning rate of 0.01, batch size +24. Note that only fe and fd participate in inference, while fpro j, +fpred, f ′ +e, f ′ +d, f ′ +pro j and Qp/c are discarded after training. All +networks are implemented based on Python 3.8.8 and Pytorch +1.7.1 and are trained on an NVIDIA GeForce GTX 2080Ti +GPU. +5. Results +5.1. Quantitative Comparison with Other Start-of-art Ap- +proaches +The performance of our proposed MSCDA is depicted in Ta- +ble 3. We compared our proposed method with two state-of- +art UDA approaches: CyCADA (Hoffman et al. (2018)) using +adversarial learning methods and SEDA (Perone et al. (2019)) +using self-training methods which are frequently used for med- +ical images. Additionally, the two selected methods were both +trained with two different domain labels, i.e. source domain +labels (denoted as “Src-Only”) and target domain labels (de- +noted as “Supervised”). In summary, we compare MSCDA to +four methods and each has two different types of backbones (U- +Net (Ronneberger et al. (2015)) or DeepLab v3+ (Chen et al. +(2018))), yielding eight combinations. Note that plain U-Net is +not applicable for our method because the very small (e.g., 8 × +8) resolution in latent space leads to the inaccurate classification +of embeddings. +MSCDA outperforms the other examined methods under the +same task. More specifically, the DSC reaches over 83% in task +S4 in both T2W-to-T1W and T1W-to-T2W scenarios (T2W-to- +T1W: 87.2%, T1W-to-T2W: 83.4%), while the DSC of other +8 + +Table 3: The evaluation results of the proposed UDA framework compared with source-only, supervised training and two other UDA methods. The best performance +in each metric is in bold. +Method +Backbone +Task +Scenario 1: T2W to T1W +Scenario 2: T1W to T2W +DSC (%) +JSC (%) +PRC (%) +SEN (%) +DSC (%) +JSC (%) +PRC (%) +SEN (%) +Src-Only +U-Net +S11 +65.8 +56.7 +70.7 +79.8 +74.3 +63.7 +88.9 +69.3 +S8 +64.0 +53.9 +82.7 +67.8 +74.2 +65.1 +87.1 +71.9 +S4 +21.0 +16.0 +96.5 +17.4 +60.7 +45.2 +96.3 +46.0 +DeepLab v3+ +S11 +71.9 +58.4 +83.1 +69.2 +70.0 +58.0 +90.5 +63.7 +S8 +69.1 +56.1 +90.9 +61.8 +74.3 +65.4 +88.5 +73.4 +S4 +54.9 +41.3 +94.1 +44.3 +70.3 +57.2 +95.7 +60.0 +Supervised +U-Net +- +94.8 +91.7 +94.4 +94.8 +95.7 +92.7 +96.9 +95.5 +DeepLab v3+ +- +95.8 +92.8 +98.0 +94.7 +96.0 +93.0 +96.2 +96.5 +CyCADA +U-Net +S11 +78.7 +68.8 +86.9 +79.4 +79.7 +69.3 +90.5 +75.6 +S8 +77.0 +66.4 +83.5 +79.8 +78.5 +66.8 +91.5 +72.1 +S4 +54.5 +42.6 +94.7 +45.1 +63.0 +49.7 +97.8 +50.7 +DeepLab v3+ +S11 +80.0 +68.0 +78.6 +86.0 +73.8 +61.3 +85.8 +70.6 +S8 +77.2 +64.5 +81.1 +79.2 +70.3 +59.0 +92.1 +64.5 +S4 +64.0 +50.4 +92.4 +54.1 +67.6 +53.8 +92.8 +57.4 +SEDA +U-Net +S11 +79.0 +67.1 +81.4 +82.6 +81.2 +70.4 +96.2 +72.7 +S8 +79.4 +70.0 +83.3 +84.3 +80.2 +70.3 +92.0 +75.0 +S4 +69.0 +56.1 +93.4 +60.0 +73.5 +60.8 +98.9 +61.3 +DeepLab v3+ +S11 +81.7 +70.6 +88.6 +79.2 +82.5 +73.7 +94.0 +78.5 +S8 +80.3 +68.4 +88.0 +77.9 +82.4 +71.4 +83.9 +83.2 +S4 +71.4 +57.9 +95.4 +60.1 +75.5 +62.5 +98.5 +63.5 +MSCDA +DeepLab v3+ +S11 +88.6 +79.9 +86.5 +92.3 +83.1 +71.8 +88.7 +79.5 +S8 +89.2 +81.0 +89.3 +89.9 +84.0 +73.2 +91.7 +78.8 +S4 +87.2 +78.0 +92.4 +83.6 +83.4 +72.5 +98.0 +73.8 +methods are below 76% (e.g., CyCADA, T2W-to-T1W: 64.0%, +T1W-to-T2W: 67.6%; SEDA, T2W-to-T1W: 71.4%, T1W-to- +T2W: 75.5%). This result is supported by other evaluation met- +rics, such as JSC and SEN. As it can be seen in the bottom of +Table 3, in both scenarios, MSCDA achieved better results in all +evaluated metrics except in PRC although it reaches over 92%. +For the other two tasks (S8 and S11), the proposed method in +general outperforms other approaches. The box plot (see Fig. +3) also indicates that MSCDA method not only performs better +but also has a smaller interquartile range than Src-Only and the +other two methods. The segmentation results of MSCDA and +other methods are shown in Fig. 4. +From Table 3, one can observe that when comparing the +performance between different tasks (i.e., S11, S8 and S4), +MSCDA shows high label-efficient learning ability. +More +precisely, the DSC of our methods in T2W-to-T1W scenar- +ios only drops 2.0% from 89.2% to 87.2% while CyCACA +and SEDA drop 16.0% and 10.3% respectively; The DSC +of our method in T1W-to-T2W scenario remains relatively +stable across three tasks with the difference of 0.9% across +tasks. +Compared to our model, the performance of other +methods drops significantly as the number of the source sub- +jects decreases. Therefore, the obtained results show that our +method is less sensitive to the size of source domain com- +pared to other UDA methods. Notably, the performance of our +method is very close to that of supervised learning (MSCDA: +DSC=89.2%, JSC=81.0%, PRC=89.3% SEN=89.9%; Super- +vised: DSC=95.8%, JSC=92.8%, PRC=98.0%, SEN=94.7%) +when training with the eight source subjects (task S8) in T2W- +to-T1W scenario, demonstrating the potential of contrastive +representation learning and self-training framework. +Figure 3: +The box plot comparison of the DSC between our proposed +MSCDA and other methods. All methods are equipped with DeepLab v3+ as +the backbone. The plots show the distribution of model performance at a sub- +ject level. The DSC of each subject is the mean value of all slices containing +foreground pixels. +9 + +MSCDA +MSCDA +Figure 4: Segmentation results compared with previous methods on scenario +1/2 task S4. All methods are equipped with DeepLab v3+ as the backbone. In +each scenario, the subplots from left to right indicate the original image, ground +truth, Src-Only, SEDA, CyCADA, our proposed MSCDA and supervised train- +ing respectively. +5.2. Ablation Study of Different Loss Function & Augmentation +In order to investigate the contribution of augmentation and +different loss function, we conduct an ablation experiment by +removing/adding each component separately. We test the net- +work on scenario 1 task S4 fold 1 with combinations of self- +training, data augmentation, P2P, P2C and C2C contrast. All +the networks are trained under the same experimental settings +as Section 4.4. As illustrated in Table 4, adding data augmenta- +tion (see case 2) to self-training can increase the DSC by 21.3% +compared to plain self-training (see case 1). Combining case +2 with P2P (see case 3) or P2C (see case 4) contrast increase +the DSC to 80.2% and 76.0% respectively. However, when +adding C2C contrast into case 2 (see case 5), the network per- +formance deteriorates to a DSC of 67.3%, indicating centroid- +level contrastive learning does not benefit feature embeddings +in our breast segmentation task. Nonetheless, this shortcom- +ing is canceled out by adding P2P or P2C contrast, as shown +in case 6 and 8. When integrating all contrasts together (see +case 9), the DSC reaches highest score of 82.2%, an increment +of 31.9% compared to the simple case 1. In summary, these +results suggest that our proposed contrastive loss can improve +the self-training framework to achieve promising segmentation +performance. +5.3. Ablation Study of Contrast Between Domains +As mentioned in Section 3.4, we compute three types of con- +trasts between the student and teacher networks. In particular, +only the target feature embeddings in the student network are +sampled as anchors, while only the source feature embeddings +in the teacher network are sampled to update the memory bank. +To further elaborate our selection, we conduct an additional, +complementary ablation study by selecting different domains +Table 4: Ablation study of each proposed component on scenario 1 task S4 fold +1. A check mark indicates that a specific component is applied. The DSC is +utilized to evaluate the performance and the extra points gained compared to +the baseline (case 1) are listed. +case +Self- +training +Aug. +P2P +P2C +C2C +DSC +(%) +gain +(%) +1 +✓ +50.3 +2 +✓ +✓ +71.6 ++21.3 +3 +✓ +✓ +✓ +80.2 ++29.9 +4 +✓ +✓ +✓ +76.0 ++25.7 +5 +✓ +✓ +✓ +67.3 ++17.0 +6 +✓ +✓ +✓ +✓ +79.1 ++28.8 +7 +✓ +✓ +✓ +✓ +81.6 ++31.3 +8 +✓ +✓ +✓ +✓ +81.5 ++31.2 +9 +✓ +✓ +✓ +✓ +✓ +82.2 ++31.9 +Table 5: Ablation study of contrast between domains on scenario 1 task S4 fold +1. A check mark indicates that a specific component is applied. The DSC is +utilized to evaluate the performance and the extra points gained compared to +the lowest value (case 1) are listed. The best performed combination is in bold. +case +Student Network +(anchor) +Teacher Network +(queue) +DSC +(%) +gain +(%) +Source +Target +Source +Target +1 +✓ +✓ +✓ +✓ +77.0 ++0.5 +2 +✓ +✓ +✓ +76.5 +3 +✓ +✓ +✓ +77.5 ++1.0 +4 +✓ +✓ +✓ +78.1 ++1.6 +5 +✓ +✓ +✓ +82.0 ++5.5 +6 +✓ +✓ +78.7 ++2.2 +7 +✓ +✓ +82.2 ++5.7 +for computing contrast. Note that all other experimental set- +tings remained unchanged. +As shown in Table 5, we observe that the best candidate (see +case 7, DSC=82.2%) is the combination of the target samples +in the student network and the source sample in the teacher net- +work. More specifically, we adopt source samples from the +teacher network to create the memory bank and to guide the +target samples from the student network. As expected, when +adding target samples to the memory bank (see case 5), the per- +formance shows a minor decrease of 0.2%, indicating that the +pseudo label brings uncertainty to the model. It is worth notic- +ing that we observe 5.7% of degradation when adopting addi- +tional source samples as anchor (see case 2). It might be due to +the overfitting of the model on the source domain. +5.4. Visualization of Feature Maps +To visualize the effect of our proposed method on domain +shift, we plot the learned features from the source and target +testing images with t-SNE (Van der Maaten and Hinton (2008)). +The learned features are obtained by using DeepLab v3+ (Chen +et al. (2018)) as the backbone. At the pixel level (Fig. 5), +when no domain adaptation method is applied, the breast pixels +of Src-Only highly overlap with non-breast pixels (Fig. 5(a)), +making them indistinguishable. +Compared to Src-Only, the +10 + +(a) Src-Only +(b) w/o contrast (self-training) +(c) w/ P2P contrast +(d) w/ P2P+P2C+C2C contrasts (MSCDA) +Figure 5: t-SNE visualization of the pixel representations on scenario 1 task +S4. Each colored point indicates a categorized pixel representation in the high +dimension feature map. Note that we only partially visualize the testing images +of the target domain due to the large dataset size. All methods are equipped +with DeepLab v3+ as the backbone. +self-training (Fig. 5(b)) makes it possible to align part of the +breast pixels between domains but fails to separate them from +non-breast pixels. Incorporating P2P contrast (Fig. 5(c)) highly +aligns the breast pixels; however, a number of breast pixels are +contaminated by non-breast pixels which may increase the er- +ror. In contrast to the above-mentioned methods, our method +nicely aligns the breast pixels and separates them from non- +breast pixels. +The visualization of the centroid level (Fig. 6) further illus- +trates the effect of our method on the feature space. Compared +to the pixel level, the uneven distribution caused by the imbal- +anced dataset is alleviated at the centroid level, making the visu- +alization clearer. We can observe that the learned centroids of +different categories in all methods are linearly separable. Be- +fore self-training, the centroids of the same category are com- +pletely separable by domain, as can be observed in Fig. 6(a). +When self-training is applied (Fig. 6(b)), the non-breast cen- +troids are clustered together while the breast centroids are still +not aligned. The P2P contrast (Fig. 6(c)) improves the centroid +alignment between domains but is still not fully overlapped. In +our method (Fig. 6(d)), the centroids of the same category share +a well-aligned tight representation space. In summary, the t- +SNE visualization demonstrates the effect of domain shift in +the feature space, an effect that can be mitigated by applying +our method. +(a) Src-Only +(b) w/o contrast (self-training) +(c) w/ P2P contrast +(d) w/ P2P+P2C+C2C contrasts (MSCDA) +Figure 6: t-SNE visualization of the centroid representations on scenario 1 +task S4. Each colored point indicates a categorized centroid representation in +the high dimension feature map. All testing images of the target domain are +included in the visualization. All methods are equipped with DeepLab v3+ as +the backbone. +6. Conclusion +In this paper, a novel multi-level semantic-guided con- +trastive UDA framework for breast MRI segmentation, named +MSCDA, is introduced. +We found that by combining self- +training with multi-level contrastive loss, the semantic infor- +mation can be further exploited to improve segmentation per- +formance on the unlabeled target domain. +Furthermore, we +built a hybrid memory bank for sample storage and proposed +a category-wise cross-domain sampling strategy to balance the +contrastive pairs. The proposed model shows a robust and clini- +cally relevant performance in a cross-sequence label-sparse sce- +nario of breast MRI segmentation. The code of our MSCDA +model is available at https://github.com/ShengKuangCN/ +MSCDA. +Acknowledgements +The authors disclosed receipt of the following financial sup- +port for the research, authorship, and/or publication of this ar- +ticle: Authors acknowledge financial support from ERC ad- +vanced grant (ERC-ADG-2015 n° 694812 - Hypoximmuno), +ERC-2020-PoC: 957565-AUTO.DISTINCT. Authors also ac- +knowledge financial support from the European Union’s Hori- +zon 2020 research and innovation programme under grant +agreement: +ImmunoSABR n° 733008, CHAIMELEON n° +952172, EuCanImage n° 952103. This work was also partially +supported by the Dutch Cancer Society (KWF Kankerbestrijd- +ing), project number 14449. +11 + +References +Antonelli, M., Reinke, A., Bakas, S., Farahani, K., Kopp-Schneider, A., Land- +man, B.A., Litjens, G., Menze, B., Ronneberger, O., Summers, R.M., et al., +2022. The medical segmentation decathlon. 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Medical image analysis 31, 77–87. +13 + diff --git a/pNE0T4oBgHgl3EQfqgGa/content/tmp_files/load_file.txt b/pNE0T4oBgHgl3EQfqgGa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae9525c5d4fe6df35ad812cbb7cc361786d9c27e --- /dev/null +++ b/pNE0T4oBgHgl3EQfqgGa/content/tmp_files/load_file.txt @@ -0,0 +1,1715 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf,len=1714 +page_content='MSCDA: Multi-level Semantic-guided Contrast Improves Unsupervised Domain Adaptation for Breast MRI Segmentation in Small Datasets Sheng Kuanga, Henry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Woodruffa,b, Renee Granzierc, Thiemo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' van Nijnattenb,d, Marc B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Lobbesb,d,e, Marjolein L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Smidtc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Philippe Lambina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Siamak Mehrkanoonf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='∗ aThe D-Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Department of Precision Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' GROW – School or Oncology and Reproduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Netherlands bDepartment of Radiology and Nuclear Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht University Medical Centre+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Netherlands cDepartment of Surgery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht University Medical Centre+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Netherlands dGROW – School for Oncology and Reproduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Maastricht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Netherlands eDepartment of Medical Imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zuyderland Medical Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Sittard-Geleen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Netherlands fDepartment of Information and Computing Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Utrecht University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Utrecht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Netherlands Abstract Deep learning (DL) applied to breast tissue segmentation in magnetic resonance imaging (MRI) has received increased attention in the last decade,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' the domain shift which arises from different vendors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' acquisition protocols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' and biological heterogene- ity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' remains an important but challenging obstacle on the path towards clinical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Recently, unsupervised domain adaptation (UDA) methods have attempted to mitigate this problem by incorporating self-training with contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To better exploit the underlying semantic information of the image at different levels, we propose a Multi-level Semantic-guided Con- trastive Domain Adaptation (MSCDA) framework to align the feature representation between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In particular, we extend the contrastive loss by incorporating pixel-to-pixel, pixel-to-centroid, and centroid-to-centroid contrasts to integrate semantic informa- tion of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We utilize a category-wise cross-domain sampling strategy to sample anchors from target images and build a hybrid memory bank to store samples from source images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Two breast MRI datasets were retrospectively collected: The source dataset contains non-contrast MRI examinations from 11 healthy volunteers and the target dataset contains contrast-enhanced MRI exami- nations of 134 invasive breast cancer patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We set up experiments from source T2W image to target dynamic contrast-enhanced (DCE)-T1W image (T2W-to-T1W) and from source T1W image to target T2W image (T1W-to-T2W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The proposed method achieved Dice similarity coefficient (DSC) of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2% and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0% in T2W-to-T1W and T1W-to-T2W, respectively, outperforming state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Notably, good performance is still achieved with a smaller source dataset, proving that our framework is label-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Keywords: Breast segmentation, Unsupervised domain adaptation, Contrastive learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Introduction Breast cancer is the most commonly diagnosed cancer in women and contributes to 15% of mortality worldwide, rank- ing as a leading cause of death in many countries (Francies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The significantly increasing mor- tality rates of breast cancer, especially in developing countries and low-income regions, lead to increased burdens for patients, their families, and society, highlighting the need for early de- tection and intervention (Azamjah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In the past decades, breast magnetic resonance imaging (MRI) has been recommended to supplement conventional mammography and ultrasound techniques to screen women at a high risk of breast cancer and determine the extent of breast cancer after diagnosis (Saslow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Gupta and Billadello (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ∗Corresponding author Email address: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='mehrkanoon@uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='nl (Siamak Mehrkanoon ) A further step towards advanced MRI-based diagnosis is ac- curate breast segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In clinical routine, whole-breast segmentation and analysis are conducted manually relying on the expertise of clinicians, which is a challenging and time- consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' With the advent of computer vision, atlas- and statistical-based methods with high accuracy compared to manual segmentation have been proposed (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Recently, numerous deep learning (DL) approaches have been developed to further improve the performance of breast seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' These tools extract salient features directly from the images and automatically segment the breast boundary (Dalmıs¸ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Ivanovska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)), breast fibroglandular tissue (FGT) (Dalmıs¸ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Ivanovska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)) and breast lesions (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Gallego- Ortiz and Martel (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Negi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)), which are less prone to errors and have achieved encouraging Dice similarity coefficients (DSCs) in various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Preprint submitted to arXiv January 9, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='02554v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='QM] 4 Jan 2023 Despite the high popularity of DL approaches, there are some barriers on the path to clinical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' One main concern is performance degradation due to large inhomo- geneities present in MRI datasets, leading to differing imaging feature distributions between training (source domain) and test- ing (target domain) datasets, also known as the domain shift problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For instance, most available imaging datasets from different medical centers are acquired with different acquisi- tion parameters, protocols and hardware vendors (Perone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Moreover, patient heterogeneity, stemming from dif- ferent breast sizes, menstrual cycle effects and stage of disease progression, leads to inconsistencies within the images, which may result in unstable performance of DL models (Granzier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Although this problem could be addressed by acquiring large and varied datasets of accurately annotated tar- get images for training, this exercise would be labor-consuming and expensive, and is further hindered by legal and ethical con- siderations regarding the sharing of patient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Thus, recent published studies (Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2016)) focus on developing unsupervised do- main adaptation (UDA) methods to mitigate the distribution discrepancy without target labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Recent advances in UDA methods have enabled its applica- tion to medical images by transferring the knowledge from the source domain to the target domain (Perone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Sha- nis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chaitanya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For instance, adversarial learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Cycle-GAN) adopts the dis- criminator to align the distribution in the latent feature space (Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) or label space (Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In practice, these methods suffer from train- ing instability and model collapse (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Self- training (Tarvainen and Valpola (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Perone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)) is another promising method which combines entropy mini- mizing (Vu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)), pseudo-label denoising (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021)), adversarial learning (Shanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)), generative methods (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021b)) or contrastive learning (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chaitanya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' More recently, incorporating self-training with complementary con- trastive learning shows remarkable performance improvement by utilizing the ground truth and pseudo labels as supervised semantic signals to guide the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Contrastive learning in this case explicitly computes the inter-category similarity between pixel representation pairs (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022)) (refers to pixel-to-pixel (P2P) contrast) aiming to learn an invariant representation in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, it still suffers from the following two major concerns that are not taken into account: (i) P2P contrast skips the structure con- text of adjacent pixels, so it does not extensively exploit the semantic information present in the MRI scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To alleviate this problem, we propose a method to integrate different levels of semantic information into a contrastive loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' More specifically, the mean value of the pixel representations of a specific category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', the centroid, should be similar to the pix- els contained in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Likewise, centroids, regardless of whether they are from the same domain, should also be close to centroids of the same category and far away from centroids of other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We denote these two relations as pixel-to- centroid (P2C) and centroid-to-centroid (C2C) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (ii) A common practice to perform inter-category contrast is to gen- erate positive and negative pairs by sampling partial pixel rep- resentations in a mini-batch (Chaitanya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, the imbalanced proportion between background and regions of interest (ROIs) in the breast MRIs poses a challenge to obtain adequate pairs during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To address this problem, we build a hybrid memory bank and optimize the sampling strat- egy to ensure enough cross-domain positive and negative pairs especially for the highly imbalanced mini-batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Addition- ally, we also explore the impact of anchors and samples from different domains on model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In summary, we extend the contrastive UDA framework for breast segmentation to further mitigate the domain shift prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To the best of our knowledge, this is the first attempt to apply contrastive UDA in breast MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We briefly provide the novel contributions of our work as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To solve the domain shift problem in breast MRI, we develop a novel Multi-level Semantic-guided Contrastive Domain Adaptation (MSCDA) framework for cross- domain breast tissue segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To exploit the semantic information present in source la- bels, we propose a method that combines pixel-to-pixel, pixel-to-centroid and centroid-to-centroid contrasts into the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To resolve the data imbalance problem, we develop a hy- brid memory bank that saves both category-wise pixel and centroid samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We further investigate a category-wise cross-domain sampling strategy to form adequate con- trastive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To validate the performance of the UDA framework, we replicate our experiment under multiple source datasets of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The results show robust performance and label-efficient learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We further show that our framework achieves comparable performance to su- pervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Semantic Segmentation Semantic segmentation is an essential and hot topic in com- puter vision, achieving automatic categorization of each pixel (or voxel) into one or more categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In recent years, con- volutional neural networks (CNNs) have shown significant re- sults in multiple fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Fully convolutional network (FCN) (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2015)), as one of the most remarkable early-stage segmentation architectures, demonstrated the pixel-level repre- sentation learning ability of CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, CNNs are still far from maturity in terms of accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' There- fore, many mechanisms have been proposed to improve seg- mentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For instance, U-Net Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2015) introduced skip connections in an encoder-decoder de- sign to solve the vanishing gradient problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' DeepLab v3+ (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) proposed Atrous Spatial Pyramid Pool- ing (ASPP) to capture more context information in multi-scale 2 receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Meanwhile, inspired by the effectiveness of residual blocks, ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2016)) was adopted as the backbone in many encoder-decoder segmentation frameworks (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)) to provide deep feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' DL techniques have also been widely adopted in the field of medical imaging that pave the way towards more precise and automated clinical diagnosis and prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Common DL methods focus on the segmentation of contours (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Dalmıs¸ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Pi- antadosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) and lesions (Dalmıs¸ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' El Adoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)) in breast magnetic res- onance imaging (MRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Despite the promising performance, these methods require large datasets with expert annotations, expensive and hard to realize the current data-sharing paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Contrastive Learning Contrastive learning (CL) was introduced as a self- supervised learning framework, allowing the model to learn representations without labels (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020b,a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' An essen- tial step of early CL methods is to build a pretext task, such as instance discrimination (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020a)), to discriminate a positive pair (two aug- mented views of an identical image) from negative pairs (aug- mented view of other images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Based on this pioneering ap- proach, many subsequent advanced mechanisms have been pro- posed to improve the representation learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For exam- ple, Moco v1 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)) and v2 (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020b)) combined a momentum encoder with a first-in-first-out queue to maintain more negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' This results in an improved classification performance e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2009)) and enables training the network on normal graphics process- ing units (GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Afterwards, the projection head (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020a)) and the prediction head (Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)) were in- troduced respectively to improve the classification accuracy on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For semantic segmentation tasks, recent CL works lever- age the pixel-level labels as supervised signals (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chaitanya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The underlying idea is to group the pixel representations from the same category and to separate pixel representations from different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021a) introduced a label-efficient two-stage method that pre-trained the network by using P2P contrastive loss and then fine-tuned the network using cross-entropy (CE) loss (Bishop and Nasrabadi (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' PC2Seg (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021)) improved this method in a one-stage semi-supervised learning (SSL) ap- proach by jointly updating the network weights with pixel con- trastive loss and consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ContrastiveSeg (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021)) combined pixel-to-region contrastive loss to explicitly leverage the context relation across images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Similar to Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021), the authors in Chaitanya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020) validated the effectiveness of different sampling strategies on contrastive learning for multiple medical MRI seg- mentation tasks (Bernard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Antonelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhuang and Shen (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Unsupervised Domain Adaptation Unsupervised Domain Adaptation (UDA) is used to general- ize learned knowledge from a labeled source domain to an unla- beled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The key challenge of UDA is domain shift, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', the inconsistent data distribution across domains, which usually causes performance degradation of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Early ma- chine learning methods utilized different feature transforma- tions or regularizations to overcome this problem (Kouw and Loog (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Mehrkanoon and Suykens (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Mehrkanoon (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A number of existing DL methods solve the domain shift problem using adversarial learning or self-training-based ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Adversarial learning utilizes generative adversarial networks (GANs) (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2014)) to align the distri- bution of the feature space (Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) or label space (Vu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In par- ticular, CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017)) has been extensively explored and adopted in medical image UDA (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Guan and Liu (2021)) because of its ability to translate the ‘style’ of the source domain to the target domain in an unpaired way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Self-training, frequently used in SSL, uses the predictions of the target domain as pseudo-labels and retrains the model iter- atively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A typical self-training network (Tarvainen and Valpola (2017)) generates pseudo-labels from a momentum teacher net- work and distills knowledge to the student network by using the consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The authors in Perone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Per- one and Cohen-Adad (2018) improved the self-training method by aligning the geometrical transformation between the student and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' DART (Shanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)) and MT- UDA (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021b)) combined self-training with adver- sarial learning in different ways, both receiving promising re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For imbalanced datasets, different denoising methods and sampling strategies have been proposed to improve the qual- ity of pseudo-labels (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Hoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Similar to Chaitanya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020), recent self-training approaches (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022)) incorporated CL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' unsupervised contrastive domain adap- tation, to align cross-domain features by sampling or merging contrastive feature embeddings across categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In this study, we integrate three kinds of contrastive losses and a category- wise cross-domain sampling strategy to accomplish the UDA segmentation task for breast MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Problem Definition Source domain data and target domain data are two sets of data used in the domain adaptation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The source do- main data Xs = {xs}M i=1 have pixel-level labels whereas the target domain image data Xt = {xt}N i=1 are unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We aim at devel- oping a method that can learn from the labeled source domain and be applied to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In particular, the learned 3 Table 1: Important notations in our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Notations Description xs, xt, ys, ˆys Source image, target image, source image ground truth and corresponding one-hot repre- sentation respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ps, pt Student network probability map of the source and target images respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' p′ s, p′ t Teacher network probability map of the source and target images respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' zt Student network feature embedding of the tar- get image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' z′ s Teacher network feature embedding of the source image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ˆyt, ˆy′ t One-hot pseudo-label of pt and p′ t respectively (y=argmax(p));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' vk s, vk t Pixel feature embedding of category k of the source and target images respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ck s, ck t Centroid feature embedding of category k of the source and target images respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Qpixel, Qcentroid Pixel queue and centroid queue in the memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' network is used to classify each pixel of the target domain im- age into K categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A direct approach is to train the net- work in a supervised manner on the source domain and apply it directly to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, the performance of the network often drops because of the aforementioned domain gap between source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To address this con- cern, we propose a new domain adaptation approach, named MSCDA, based on the combination of self-training and con- trastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Overall Framework The proposed domain adaptation framework is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' It consists of a student network and a momentum teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The student network consists of four main compo- nents, a feature encoder fe, a feature decoder fd, a projection head fproj, and an additional prediction head fpred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' These com- ponents are correspondingly mapped in the teacher network with the only exception of the last component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', the pre- diction head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The three components in the teacher network are called f ′ e, f ′ d and f ′ pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The important notations are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In the student network, the feature encoder fe maps the in- put MRI image x ∈ RH×W×1 into a high dimension feature map h ∈ RH′×W′×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Next, h is transferred into a segmentation prob- ability map p ∈ RH×W×K and a low dimension feature embed- ding z ∈ RH′×W′×D through two forward passes, hereafter re- ferred to as segmentation and contrast paths, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In the first forward pass (segmentation path), the decoder fd gen- erates the segmentation probability map p of the input h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In the second forward pass (contrast path), the projection head fpro j and prediction head fpred jointly reduce the feature map into a low-dimension projected feature embedding z = fpred( fpro j(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Similar steps are conducted in the teacher network, yielding the momentum probability map p′ and feature embedding z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Fi- nally, the probability map p and p′ are used for self-training while the projected feature embeddings z and z′ are used for semantic-guided contrastive learning to diminish the discrep- ancy between the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The overall loss function is given by: L = Lseg + λ1Lcon + λ2Lctr, (1) where Lseg is the supervised segmentation loss, Lcon is the consistency loss, Lctr is the contrastive loss, and λ1 and λ2 are the regularization parameters of the corresponding losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The summation of segmentation and consistency loss is hence- forth referred to as the self-training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We elaborate the self- training loss in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 and our proposed contrastive loss in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Self-training Following the self-training paradigm (Perone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)), two optimization goals were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The first goal is to per- form supervised learning on the student network from source image labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The second goal is that the student network learns the pseudo labels generated by the teacher network to distill knowledge from target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Only the weights in the seg- mentation path of both networks are updated in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Supervised Learning In supervised learning, we employ a hybrid segmentation loss (Isensee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) that combines Dice loss (Sudre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2017)) and CE loss, and is formulated as: Lseg = 1 2LDice(ps, ˆys) + 1 2Lce(ps, ˆys), (2) where ˆys is the one-hot ground truth and ps is the probability map of the source domain image in the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Distilling Knowledge from Pseudo Labels The pseudo label of the target image is generated by the seg- mentation path in the momentum teacher network iteratively: ˆy′ t = argmax(p′ t), (3) where p′ t is the probability map of the target domain image in the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In order to distill knowledge from the pseudo label, an extra consistency loss is added between the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In other words, the target image segmentation pt generated by the student network is guided by the pseudo label ˆy′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The consistency loss is formulated as: Lcon = 1 H × W × K H×W � i=1 K−1 � k=0 ���p(i,k) t − ˆy′(i,k) t ��� 2 , (4) where i is the pixel index of the image and k is the category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Here, we update the weights of the student network by means of back propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, in the teacher network, a stop- gradient operation is applied, and the network weights are up- dated by exponential moving average (EMA): Θ′ ← αΘ′ + (1 − α)Θ, (5) 4 Student Network Teacher Network stop-grad stop-grad projection head feature encoder feature decoder prediction head feature embedding projected embedding one-hot prediction memory bank loss function Figure 1: The proposed framework of the unsupervised domain adaptation method, named MSCDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A source image xs and a target image xt are the inputs to the student network and the momentum teacher networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Each network consists of a segmentation path and a contrast path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The student network is trained by a supervised segmentation loss, an inter-network consistency loss and a multi-level contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The teacher network updates the weights by exponential moving average (EMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The training procedure is layer out in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' where Θ and Θ′ are the weights of the student network and teacher network respectively, and α ∈ (0, 1) is the momentum coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Combining data augmentation with self-training has been shown to improve the domain adaptation performance (Tar- vainen and Valpola (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The student network receives strongly-augmented images, and the teacher network receives weekly-augmented images during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Random resized cropping is used as the weak augmen- tation method, and random brightness, contrast and Gaussian blur are used as strong augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The strongly- augmented path learns a robust feature representation from the weakly-augmented path that has less disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Semantic-guided Contrastive Loss In order to improve the performance of our UDA framework even further, we incorporate a multi-level semantic-guided con- trast to the self-training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The idea is to leverage the ground truth of the source domain as supervised signals to en- force the encoder to learn a well-aligned feature representation that mitigates the domain discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A common way is to cat- egorize the feature embedding and conduct contrastive learning using the pixels or centroids between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In our approach, we develop the contrastive loss at P2P, P2C and C2C levels to directly utilize multi-level semantic information to guide the feature alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The data flow of our proposed contrastive loss is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Preliminaries In unsupervised contrastive segmentation approaches, the contrast is performed using a randomly selected sample (called the anchor) v, a positive sample v+ and n negative samples V− = {v− 1, v− 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', v− n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The aim is to learn a feature representation that yields high similarity in positive pairs (v, v+) and low simi- larity in negative pairs (v, v−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Following He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021), we utilize the InfoNCE as our loss function, which is given as follows: Lctr = − log exp(v · v+/τ) exp(v · v+/τ) + �n i=1 exp(v · v− i /τ), (6) where n is the number of negative samples per anchor, ‘·’ is the dot product between two samples, and τ is a temperature hyperparameter that controls the gradient penalty of hard nega- tive samples, which is empirically set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='07 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Here, samples are selected from D-dimensional feature embed- ding followed by l2-normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Feature Categorization Feature categorization is a necessary step required for super- vised contrastive learning in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To utilize the semantic information effectively, we categorize the feature em- bedding from both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For the source image, the feature embedding in the teacher network and its ground truth are re- quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Given the l2-normalized target network feature embed- ding of a source image z′ s ∈ RH′×W′×D and the one-hot ground truth ˆys ∈ RH×W×K, we first down-sample the one-hot ground truth into ¯ys ∈ RH′×W′×K to fit the embedding size, then assign the category label index k ∈ {0, K − 1} of ¯ys to each pixel of z′ s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Similarly, the target image embedding zt can also be categorized using the pseudo label ˆyt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Based on the catego- rized feature embedding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' we further compute the category-wise mean value of pixels of the feature embedding as the centroid C={ck}K−1 k=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' which is given as follows: ck = 1 ���Yk��� H′×W′ � i=1 1 � ¯y(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='k) = k � zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (7) where 1 [·] is an indicator function that returns 1 when the con- dition holds and 0 otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' zi is the ith pixel of the feature embedding and ¯y(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='k) is the down-sampled label which belongs to the ith pixel and category k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Yk is the set of labels of category k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5 student network teacher network sampling sampling pixels centroids pixels centroids pixel queue centroid queue mean mean anchors (c) memory bank (b) category-wise anchor sampling & computing (a) feed-forward & feature categorization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' source image target image .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (d) cross-domain contrasts Category0-Source Category0-Target Category1-Target Category1-Source anchors pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' pixel-to-pixel neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' pixel-to-centroid pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' centroid-to-centroid pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' anchors anchors Figure 2: The data flow of our proposed multi-level contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (a) feed-forward and feature categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (b) category-wise anchor sampling and computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (c) memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (d) proposed multi-level cross-domain con- trasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Memory Bank & Sampling Strategy Although the number of negative samples is critical for learn- ing the feature representation (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020)), insufficient negative pairs of each batch in the breast MRI segmentation task may occur because of the highly imbalanced ration be- tween foreground and background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To overcome this problem, we utilize two category-wise queues as a memory bank to maintain the pixel and centroid samples from the source images in the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, keeping all pixels in the queue is not feasible because of GPU memory limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Thus, we uniformly sample b pixels from each category in the feature embedding to the pixel queue (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2(b,c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' This under- sampling strategy enables the queue to maintain enough bal- anced, but not redundant, pixel samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The pixel queue Qpixel and the centroid queue Qcentroid can be represented as: Qpixel = {Qk pixel}K−1 k=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Qk pixel = {vk (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='i)} Bp i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (8) Qcentroid = {Qk centroid}K−1 k=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Qk centroid = {ck (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='i)}Bc i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (9) where Qk pixel is the pixel queue of category k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' vk (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='i) is the ith source pixel sample of category k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Qk centroid is the centroid queue of category k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ck (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='i) is the ith source centroid sample of category k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' and Bp and Bc are the size of the queue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Pixel-to-pixel Contrast We perform the pixel-to-pixel (P2P) contrastive loss to align the cross-domain feature representation of the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To resolve this problem, we first sample m anchors from each category of the target feature embedding zt in the student net- work, denoted as set Vk t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Then, for each anchor vk t ∈ Vk t with category label k, we sample a source pixel of the same category from the pixel queue Qpixel to form a positive pair (vk t , vk+ s ), and sample n source pixels of category q ∈ K\\{k} to form n negative pairs (vk t , vq− s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Based on these positive and negative pairs, the InfoNCE loss of a single target anchor is computed by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Overall, the P2P loss is defined as: LP2P ctr = 1 �K−1 k=0 ���Vk t ��� K−1 � k=0 � vk t ∈Vk t Lctr(vk t , vk+ s , Vq− s ), (10) where |·| is the number of elements in a set, and Vq− s is the set of negative source pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Note that the number of pixels labeled as foreground categories might be less than m (or even 0) if the model predicts a few (or no) breast tissue labels in a mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Nevertheless, benefiting from the category-wise memory bank, the contrast loss can still be computed even if all pixels in a mini-batch belong to the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Pixel-to-centroid Contrast Due to the under-sampling strategy in selecting anchors and updating the memory bank, the network may suffer from inade- quate semantic knowledge and thereby be difficult to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' This issue is further addressed by incorporating P2C and C2C contrasts to P2P contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For P2C contrast, we force the pixel representation to learn a more general representation with the guidance of the centroid (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Specifically, a pixel and a centroid from the same category are considered as a positive pair (vk, ck+), while a pixel and a centroid from different cate- gories are considered as a negative pair (vk, cq−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We reuse the anchors in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4 and sample all positive and negative centroids from the centroid queue Qcentroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Similar to P2P loss, the P2C loss is defined as: LP2C ctr = 1 �K−1 k=0 ���Vk t ��� K−1 � k=0 � vk t ∈Vt Lctr(vk t , ck+ s ,Cq− s ), (11) where Cq− s is the set of negative source centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Centroid-to-centroid Contrast For C2C contrast, the ideal situation is that the centroids from the same category are located near to one another, whereas cen- troids from other categories are located far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Unlike P2C contrast, the total number of centroids p (BK ≤ p ≤ 2BK) is much smaller than the pixel number in a mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Be- sides, calculating centroids is computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' There- fore, the centroids of the whole mini-batch can be fully involved as anchors in C2C contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Similar to P2P and P2C contrast, the positive pairs (ck, ck+) and negative pairs (ck, cq−) are de- fined according to whether centroids are from the same cate- gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Thus, the C2C loss is defined as: LC2C ctr = 1 �K−1 k=0 ���Ck t ��� K−1 � k=0 � ck t ∈Ct Lctr(ck t , ck+ s ,Cq− s ), (12) where Ct is the set of target centroid anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6 Table 2: Dataset description and acquisition parameters of dataset 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Subject Number Type Scanner Sequence Acquisition Parameters TR (ms) TE (ms) PS (mm) ST (mm) Dataset 1 11 Healthy volunteers Philips 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5T (Ingenia) T1W 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='36×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='36 2 T2W 2000 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='79×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='79 2 Dataset 2 134 Patients with invasive breast cancer Philips 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5T (Ingenia/Intera) DCE-T1W 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='85×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='85- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='97×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='97 1 T2W 2000 170-259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='65×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='65- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='97×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='97 1 Abbreviations: TR=Repetition time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' TE=Echo time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' PS=Pixel spacing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' ST=Slice thickness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' T1W=T1-weighted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' T2W=T2-weighted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' DCE=Dynamic contrast-enhanced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' T=Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Finally, we combine three above-mentioned contrasts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2(d)) as our proposed multi-level semantic-guided contrastive loss: Lctr = LP2P ctr + LP2C ctr + LC2C ctr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (13) The overall training process of our proposed MSCDA is pre- sented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Datasets Dataset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Dataset 1 consists of test-retest breast T1-weighted (T1W) and T2-weighted (T2W) MRI images and correspond- ing right-breast masks of eleven healthy female volunteers, which is described in Granzier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The images of each subject were collected in two separate sessions (interval<7 days), during which three 3D scans were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Subjects were asked to lay in the prone position and remain still in the MRI scanner while both modalities are sequentially acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All images were acquired with an identical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5T MRI scan- ner (Philips Ingenia, Philips Healthcare, Best, the Netherlands) using a fixed clinical breast protocol without contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The detailed acquisition parameters are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In pre- processing, we first resize all MRI slices and corresponding masks to 256×256 pixels using cubic interpolation and nearest- neighbor interpolation respectively, and then normalize images with z-score transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In total, dataset 1 contains 14520 (11 subjects × 2 sessions × 3 scans × 220 slices) T1W slices and 11220 (11 subjects × 2 sessions × 3 scans × 170 slices) T2W slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Dataset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Dataset 2 consists of the images from 134 subjects with histologically confirmed invasive breast cancer imaged be- tween 2011 and 2017 in Maastricht University Medical Cen- ter+ and collected retrospectively (Granzier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2020, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The images contain dynamic contrast-enhanced breast T1W and T2W MRI and corresponding right-breast masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Simi- lar to Dataset 1, each subject underwent the examinations with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5T MRI scanners (Philips Intera and Philips Ingenia (idem)) in a prone position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In particular, T1W images were acquired before and after the intravenous injection of gadolinium-based contrast Gadobutrol (Gadovist, Bayer Healthcare, Berlin, Ger- many (EU)) with a volume of 15 cc and a flow rate of 2 ml/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The acquisition parameters are also listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We con- duct the same image pre-processing as in Dataset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In total, Dataset 2 contains 21793 T2W and 28540 T1W slices and they are split into three folds with 45, 45 and 44 subjects for the cross-validation depicted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Experiment Setup As shown in Table 2, the subject population, machine vendor and acquisition parameters between the two datasets are het- erogeneous, indicating the common domain shift problem in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We set up the experiment on both Dataset 1 and 2 to transfer the knowledge of breast segmentation from healthy women to patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In particular, the experiment con- sists of two scenarios: (1) Utilizing the T2W images of Dataset 1 as the source domain and the T1W images of Dataset 2 as the target domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2) utilizing the T1W images of Dataset 1 as the source domain and the T2W images of Dataset 2 as the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In each scenario, we establish three tasks with a different number of subjects in the source domain to validate the label- efficient learning ability of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The three tasks con- tain four, eight and eleven (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', the whole dataset) randomly se- lected subjects respectively, and are denoted as S4, S8 and S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To further verify the robustness of UDA performance, we split the target domain into three folds to perform a three-fold cross- validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In each run of the cross-validation, two folds are used as the target domain for training and the remaining fold for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Model Evaluation The DSC is used as the main evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Addition- ally, we use the Jaccard Similarity Coefficient (JSC) as well as precision (PRC) and sensitivity (SEN) as auxiliary evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' These metrics are formulated as follows: DSC = 2 × TP 2 × TP + FP + FN × 100%, (14) 7 Algorithm 1: MSCDA for Breast MRI Input: Source domain image xs and label ys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Target domain image xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 1 Initialize the weights of the student network Θe, Θd with pre-trained weights, Θproj and Θpred via He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Initialize the teacher network by copying weights from the student network and applying stop-gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Initialize the memory bank Qpixel and Qcentroid ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 2 for epoch = 1, Emax do 3 foreach mini-batch do 4 Apply weak and strong data augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5 Forward propagate weak-augmented batch in the student network to get ps, pt and zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6 Forward propagate strong-augmented batch in the teacher network to get p′ t and z′ s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 7 Compute loss Lseg using ps and ys via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 8 Compute loss Lcon using pt and p′ t via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 9 Categorize the feature embedding z′ s and zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 10 foreach category do 11 Sample pixel anchors and compute centroid anchors from zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 12 Sample corresponding positive and negative pairs from Qpixel and Qcentroid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 13 Update Qpixel and Qcentroid using z′ s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 14 end 15 Compute loss LP2P ctr , LP2C ctr and LC2C ctr via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (10)-(12) respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 16 Update the student network via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 17 Update the teacher network by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 18 end 19 end Output: Weights of the student network Θe and Θd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' JSC = TP TP + FP + FN × 100%, (15) PRC = TP TP + FP × 100%, (16) SEN = TP TP + FN × 100%, (17) where TP, FP and FN are the number of true positive, false positive and false negative pixels of the prediction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Note that we show the mean value of each metric of the three- fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Implementation Details 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Architecture Encoder & decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We conduct our experiment by adopting DeepLab-v3+ (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) with ResNet-50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2016)) as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Benefiting from the encoder-decoder architecture, the encoder and decoder of DeepLab-v3+ are adopted in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Specifically, the hidden dimen- sion of ResNet-50 is set to (16, 32, 64, 128), yielding a 512- dimension feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Projection/Prediction Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The projection head fpro j is a shallow network that contains two 1 × 1 convolutional layers with BatchNorm and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' It projects the 512-d feature map into a 128-dimension l2-normalized feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The prediction head fpred shares the same architecture setting with fpro j with the exception that the fpred does not change the di- mension of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The size of the pixel queue and the centroid queue are respectively set to 32768 and 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In each mini- batch, we randomly sample eight pixels per category of each feature embedding to the queue and discard the oldest samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Training Settings To accelerate the training procedure, we pre-train the DeepLab-v3+ on the source domain and then use the weights to initialize the encoder fe and decoder fd of our UDA framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Additionally, the projection and prediction heads are initialized by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' By default, the number of pixel anchors for P2P loss is set to 32 and the number of negative pairs of (P2P, P2C, C2C) loss is set to (32768, 4096, 4096).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The regulariza- tion parameters λ1 and λ2 are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The Adam (Kingma and Ba (2014)) optimizer is used for training the framework for Emax=100 epochs with a fixed learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='01, batch size 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Note that only fe and fd participate in inference, while fpro j, fpred, f ′ e, f ′ d, f ′ pro j and Qp/c are discarded after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All networks are implemented based on Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 and Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1 and are trained on an NVIDIA GeForce GTX 2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Quantitative Comparison with Other Start-of-art Ap- proaches The performance of our proposed MSCDA is depicted in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We compared our proposed method with two state-of- art UDA approaches: CyCADA (Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) using adversarial learning methods and SEDA (Perone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2019)) using self-training methods which are frequently used for med- ical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Additionally, the two selected methods were both trained with two different domain labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' source domain labels (denoted as “Src-Only”) and target domain labels (de- noted as “Supervised”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In summary, we compare MSCDA to four methods and each has two different types of backbones (U- Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2015)) or DeepLab v3+ (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018))), yielding eight combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Note that plain U-Net is not applicable for our method because the very small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', 8 × 8) resolution in latent space leads to the inaccurate classification of embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' MSCDA outperforms the other examined methods under the same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' More specifically, the DSC reaches over 83% in task S4 in both T2W-to-T1W and T1W-to-T2W scenarios (T2W-to- T1W: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2%, T1W-to-T2W: 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4%), while the DSC of other 8 Table 3: The evaluation results of the proposed UDA framework compared with source-only, supervised training and two other UDA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The best performance in each metric is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Method Backbone Task Scenario 1: T2W to T1W Scenario 2: T1W to T2W DSC (%) JSC (%) PRC (%) SEN (%) DSC (%) JSC (%) PRC (%) SEN (%) Src-Only U-Net S11 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 63.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 MSCDA DeepLab v3+ S11 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 88.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 S4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 methods are below 76% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', CyCADA, T2W-to-T1W: 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0%, T1W-to-T2W: 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' SEDA, T2W-to-T1W: 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4%, T1W-to- T2W: 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' This result is supported by other evaluation met- rics, such as JSC and SEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' As it can be seen in the bottom of Table 3, in both scenarios, MSCDA achieved better results in all evaluated metrics except in PRC although it reaches over 92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' For the other two tasks (S8 and S11), the proposed method in general outperforms other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The box plot (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 3) also indicates that MSCDA method not only performs better but also has a smaller interquartile range than Src-Only and the other two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The segmentation results of MSCDA and other methods are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' From Table 3, one can observe that when comparing the performance between different tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', S11, S8 and S4), MSCDA shows high label-efficient learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' More precisely, the DSC of our methods in T2W-to-T1W scenar- ios only drops 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0% from 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2% to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2% while CyCACA and SEDA drop 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3% respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The DSC of our method in T1W-to-T2W scenario remains relatively stable across three tasks with the difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9% across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Compared to our model, the performance of other methods drops significantly as the number of the source sub- jects decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Therefore, the obtained results show that our method is less sensitive to the size of source domain com- pared to other UDA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Notably, the performance of our method is very close to that of supervised learning (MSCDA: DSC=89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2%, JSC=81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0%, PRC=89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3% SEN=89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Super- vised: DSC=95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8%, JSC=92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8%, PRC=98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0%, SEN=94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7%) when training with the eight source subjects (task S8) in T2W- to-T1W scenario, demonstrating the potential of contrastive representation learning and self-training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Figure 3: The box plot comparison of the DSC between our proposed MSCDA and other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All methods are equipped with DeepLab v3+ as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The plots show the distribution of model performance at a sub- ject level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The DSC of each subject is the mean value of all slices containing foreground pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 9 MSCDA MSCDA Figure 4: Segmentation results compared with previous methods on scenario 1/2 task S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All methods are equipped with DeepLab v3+ as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In each scenario, the subplots from left to right indicate the original image, ground truth, Src-Only, SEDA, CyCADA, our proposed MSCDA and supervised train- ing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Ablation Study of Different Loss Function & Augmentation In order to investigate the contribution of augmentation and different loss function, we conduct an ablation experiment by removing/adding each component separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We test the net- work on scenario 1 task S4 fold 1 with combinations of self- training, data augmentation, P2P, P2C and C2C contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All the networks are trained under the same experimental settings as Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' As illustrated in Table 4, adding data augmenta- tion (see case 2) to self-training can increase the DSC by 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3% compared to plain self-training (see case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Combining case 2 with P2P (see case 3) or P2C (see case 4) contrast increase the DSC to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2% and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' However, when adding C2C contrast into case 2 (see case 5), the network per- formance deteriorates to a DSC of 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3%, indicating centroid- level contrastive learning does not benefit feature embeddings in our breast segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Nonetheless, this shortcom- ing is canceled out by adding P2P or P2C contrast, as shown in case 6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' When integrating all contrasts together (see case 9), the DSC reaches highest score of 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2%, an increment of 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9% compared to the simple case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In summary, these results suggest that our proposed contrastive loss can improve the self-training framework to achieve promising segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Ablation Study of Contrast Between Domains As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4, we compute three types of con- trasts between the student and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In particular, only the target feature embeddings in the student network are sampled as anchors, while only the source feature embeddings in the teacher network are sampled to update the memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' To further elaborate our selection, we conduct an additional, complementary ablation study by selecting different domains Table 4: Ablation study of each proposed component on scenario 1 task S4 fold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A check mark indicates that a specific component is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The DSC is utilized to evaluate the performance and the extra points gained compared to the baseline (case 1) are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' case Self- training Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' P2P P2C C2C DSC (%) gain (%) 1 ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 2 ✓ ✓ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6 +21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 3 ✓ ✓ ✓ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 +29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9 4 ✓ ✓ ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 +25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7 5 ✓ ✓ ✓ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 +17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 6 ✓ ✓ ✓ ✓ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1 +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='8 7 ✓ ✓ ✓ ✓ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6 +31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='3 8 ✓ ✓ ✓ ✓ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 +31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 9 ✓ ✓ ✓ ✓ ✓ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 +31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='9 Table 5: Ablation study of contrast between domains on scenario 1 task S4 fold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' A check mark indicates that a specific component is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The DSC is utilized to evaluate the performance and the extra points gained compared to the lowest value (case 1) are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The best performed combination is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' case Student Network (anchor) Teacher Network (queue) DSC (%) gain (%) Source Target Source Target 1 ✓ ✓ ✓ ✓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 2 ✓ ✓ ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 3 ✓ ✓ ✓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 4 ✓ ✓ ✓ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='1 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='6 5 ✓ ✓ ✓ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='0 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='5 6 ✓ ✓ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 7 ✓ ✓ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7 for computing contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Note that all other experimental set- tings remained unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' As shown in Table 5, we observe that the best candidate (see case 7, DSC=82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2%) is the combination of the target samples in the student network and the source sample in the teacher net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' More specifically, we adopt source samples from the teacher network to create the memory bank and to guide the target samples from the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' As expected, when adding target samples to the memory bank (see case 5), the per- formance shows a minor decrease of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='2%, indicating that the pseudo label brings uncertainty to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' It is worth notic- ing that we observe 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='7% of degradation when adopting addi- tional source samples as anchor (see case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' It might be due to the overfitting of the model on the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Visualization of Feature Maps To visualize the effect of our proposed method on domain shift, we plot the learned features from the source and target testing images with t-SNE (Van der Maaten and Hinton (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The learned features are obtained by using DeepLab v3+ (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (2018)) as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' At the pixel level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5), when no domain adaptation method is applied, the breast pixels of Src-Only highly overlap with non-breast pixels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5(a)), making them indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Compared to Src-Only, the 10 (a) Src-Only (b) w/o contrast (self-training) (c) w/ P2P contrast (d) w/ P2P+P2C+C2C contrasts (MSCDA) Figure 5: t-SNE visualization of the pixel representations on scenario 1 task S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Each colored point indicates a categorized pixel representation in the high dimension feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Note that we only partially visualize the testing images of the target domain due to the large dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All methods are equipped with DeepLab v3+ as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' self-training (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5(b)) makes it possible to align part of the breast pixels between domains but fails to separate them from non-breast pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Incorporating P2P contrast (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 5(c)) highly aligns the breast pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' however, a number of breast pixels are contaminated by non-breast pixels which may increase the er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In contrast to the above-mentioned methods, our method nicely aligns the breast pixels and separates them from non- breast pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The visualization of the centroid level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6) further illus- trates the effect of our method on the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Compared to the pixel level, the uneven distribution caused by the imbal- anced dataset is alleviated at the centroid level, making the visu- alization clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We can observe that the learned centroids of different categories in all methods are linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Be- fore self-training, the centroids of the same category are com- pletely separable by domain, as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' When self-training is applied (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6(b)), the non-breast cen- troids are clustered together while the breast centroids are still not aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The P2P contrast (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6(c)) improves the centroid alignment between domains but is still not fully overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In our method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6(d)), the centroids of the same category share a well-aligned tight representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' In summary, the t- SNE visualization demonstrates the effect of domain shift in the feature space, an effect that can be mitigated by applying our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' (a) Src-Only (b) w/o contrast (self-training) (c) w/ P2P contrast (d) w/ P2P+P2C+C2C contrasts (MSCDA) Figure 6: t-SNE visualization of the centroid representations on scenario 1 task S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Each colored point indicates a categorized centroid representation in the high dimension feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All testing images of the target domain are included in the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' All methods are equipped with DeepLab v3+ as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Conclusion In this paper, a novel multi-level semantic-guided con- trastive UDA framework for breast MRI segmentation, named MSCDA, is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' We found that by combining self- training with multi-level contrastive loss, the semantic infor- mation can be further exploited to improve segmentation per- formance on the unlabeled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Furthermore, we built a hybrid memory bank for sample storage and proposed a category-wise cross-domain sampling strategy to balance the contrastive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The proposed model shows a robust and clini- cally relevant performance in a cross-sequence label-sparse sce- nario of breast MRI segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' The code of our MSCDA model is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='com/ShengKuangCN/ MSCDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Acknowledgements The authors disclosed receipt of the following financial sup- port for the research, authorship, and/or publication of this ar- ticle: Authors acknowledge financial support from ERC ad- vanced grant (ERC-ADG-2015 n° 694812 - Hypoximmuno), ERC-2020-PoC: 957565-AUTO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content='DISTINCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Authors also ac- knowledge financial support from the European Union’s Hori- zon 2020 research and innovation programme under grant agreement: ImmunoSABR n° 733008, CHAIMELEON n° 952172, EuCanImage n° 952103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Zhuang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', Shen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Multi-scale patch and multi-modality atlases for whole heart segmentation of mri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' Medical image analysis 31, 77–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQfqgGa/content/2301.02554v1.pdf'} diff --git a/ptAzT4oBgHgl3EQf5v4s/content/tmp_files/load_file.txt b/ptAzT4oBgHgl3EQf5v4s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7f556130a5ec92913c4a917db7ec156032e2370 --- /dev/null +++ b/ptAzT4oBgHgl3EQf5v4s/content/tmp_files/load_file.txt @@ -0,0 +1,772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf,len=771 +page_content='Finite Class 2 Nilpotent and Heisenberg Groups Dávid R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Szabó∗ 6th January 2023 Abstract We present a structural description of finite nilpotent groups of class at most 2 using a specified number of subdirect and central products of 2-generated such groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' As a corollary, we show that all of these groups are isomorphic to a subgroup of a Heisenberg group satisfying certain properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The motivation for these results is of topological nature as they can be used to give lower bounds to the nilpotently Jordan property of the birational automorphism group of varieties and the homeomorphism group of compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1 Introduction A finite non-abelian p-group G is special if its Frattini subgroup Φ(G), derived subgroup G′ and centre Z(G) all coincide and is isomorphic to (Z/pZ)r for some r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' A special group G is extra-special if r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The structure of these groups is described by the following classical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every special p-group is a subdirect product of groups of the form: central product of an extra-special p-group and an abelian group [Suz82, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='16)/(ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every extra- special p-group H of order p2n+1 is the central product of n extra-special p-subgroups of order p3 [Suz82, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every prime p, there are exactly two extra-special groups of order p3 (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We present a generalisation to finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every finite nilpotent group G is a subdirect product of d(Z(G)) groups each with cyclic centre, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every finite ≤2-step nilpotent group G with cyclic commutator subgroup is the internal central product of t many suitable nilpotent 2-generated subgroups of class 2 and an abelian subgroup A satisfying d(G) = 2t + d(A) and some further properties discussed at Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ∗The project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 741420).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The author was supported by the National Research, Development and Innovation Office (NKFIH) Grant K138596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='01863v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='GR] 5 Jan 2023 The p-groups of class 2 with 2 generators are classified in [AMM12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The central product decomposition of Theorem A is a generalisation of [BBC69, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1] where groups with cyclic centre were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The argument presented in the current paper is more structural and gives some invariants needed for a topological application below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' As an application of (the proof of) Theorem A, we embed all of the groups above to matrix groups with a very specific form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that the number of isomorphism classes of groups of order pn is p 2 27 n3+O(n8/3) as n → ∞ [Sim65, Theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 153] of which at least p 2 27 n3− 12 27 n2 is ≤2-step nilpotent [Hig60, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The main statement of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every finite ≤2-step nilpotent group G is isomorphic to a subgroup of a non-degenerate Heisenberg group of the form � 1 A C 0 1 B 0 0 1 � for suitable abelian groups A, B, C whose number of generators and exponents are bounded by concrete invari- ants of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 for the precise statement and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' See [Mag98, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='21] for a weaker statement in a much more general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We remark here that both Theorem A and Theorem B are true in the finitely generated setup, see the thesis of the author [Sza21, Theorems A, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The ideas presented in the current paper are heavily polished and simplified compared to the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Bounding the invariants in both statements is essential for the following topological application (that shall not be proved in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem C ([Sza21, Theorem C]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every natural number d, there exists an algebraic variety Xd, respectively a compact manifold Md, such that every finite ≤2- step nilpotent ≤d-generated group acts faithfully on Xd via birational automorphisms, respectively on Md via diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We briefly discuss the relevance of this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let N be a class of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' A group G is called N -Jordan, if there exists an integer JG such that for every finite subgroup F of G sits in a short exact sequence 1 → N → F → B → 1 with N ∈ N and |B| ≤ JG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' informally if every finite subgroup of G ‘almost’ belong to N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2 (Guld, [Gul20, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The birational automorphism group of any variety over a field of characteristic zero is N -Jordan where N = {≤2-step nilpotent}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 (Csikós, Pyber and Szabó, [CPS22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The homeomorphism group of a compact topological manifold is N -Jordan where N = {nilpotent}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem C shows that N essentially has to contain the ≤2-step nilpotent groups for both Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3, thereby the sharpness of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Structure of the paper In Section 2, we prove the first part of Theorem A using induction and find the smallest number of factors needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In Section 3, we turn our attention to ≤2-step nilpotent groups G with cyclic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The commutator map on these groups induce an alternating bilinear map on the Z-module on G/G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We prove the second part of Theorem A by finding a suitable generating set imitating the Darboux basis of symplectic vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In Section 4, we define a complex structure and isotropic 2 real structure on G/ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This enables to assign a Heisenberg group to G using the same method independently on the prime divisors of |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then we discuss the general idea to modify the previous construction by extending the centre so that G embeds to the resulting Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finally, in Section 5 we prove Theorem B in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' First, we consider the special case when the group has cyclic centre and apply the method of the previous two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Second, we use the reduction of Section 2 to handle the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' G : G′ ⊆ Z(G) � 1 A C 0 1 B 0 0 1 � G1 : G′ 1 ⊆ Z(G1) d(Z(G1)) = 1 Hermitian form with isotropic real structure � 1 A1 C1 0 1 A1 0 0 1 � reduction Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7 embedding Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='10 [−,−] embedding Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='8 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7 � Notation |X| denotes the cardinality of a set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' N+ ⊂ Z is the set of positive integers, N0 = {0}∪N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' n �� k denotes divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' (Note that this symbol is slightly taller than the one denoting the cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=') lcm(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , nk) is the least common multiple of integers n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We apply functions from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For f : X → Y and X0 ⊆ X, we write f|X0 : X0 → Y for restriction and f(X0) := {f(x0) : x0 ∈ X0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For maps fi : Xi → Yi, denote by f1×f2 : X1×X2 → Y1×Y2, (x1, x2) �→ (f1(x1), f2(x2)) their direct product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The arrow indicates a monomorphism (or an injective map), means an epimorphism (or a surjective map), and these arrow notations can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We write for the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In bigger diagrams, we use , or to indicate the ‘chronological order’: the further it is from a solid one, the later it appears in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let G denote a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We denote the identity element of G by 1, or sometimes by 0 when G is an additive abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By abuse of notation, we also write 1 or 0 for the trivial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For a subset S ⊆ G, ⟨S⟩ denotes the subgroup generated by S, and write ⟨g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , gn⟩ := ⟨{g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , gn}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Hom(X, Y ) is the set of morphisms X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' N ◁ G means that N is a normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Write [−, −]: G × G → G′, (g, h) �→ [g, h] for the commutator map where we use the convention [g, h] := g−1h−1gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The commutator subgroup (or derived subgroup) is denoted by G′ := [G, G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Z(G) is the centre of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Syl(G) is the set of Sylow subgroups of G, Sylp(G) consists of Sylow p-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We denote by exp(G) := inf{n ∈ N+ : ∀g ∈ G gn = 1} the exponent of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Write d(G) for the cardinality of the smallest generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We say G is ≤d-generated, if d ≤ d(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' and G is d-generated if d = d(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2 Subdirect product decomposition The goal of this section is to prove the first part of Theorem A from page 1 by passing to abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' To show the existence of such a subdirect product, we recursively factor by the invariant factors of the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' To attain minimal number of factors, we consider intersections with the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let C denote the class of groups with cyclic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' A C- decomposition in a group G is a finite set D of normal subgroups of G such that G/N ∈ C for every N ∈ D and � D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' (Use the convention � D = G if D = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=') Let 3 mC(G) denote the minimal |D| amongst all C-decomposition D in G, or ∞ is no such decomposition exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This is a reformulation of subdirect products, as the associated (central) embedding µD : G ↣ G/D := � N∈D G/N, gK �→ (gN)N∈D makes G a subdirect product of groups from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' There is a C-decomposition in every finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Furthermore, d(Z(G)) ≤ mC(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let l(G) be the maximal length of a strictly increasing subgroup series consisting of normal subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that l(G/N) < l(G) for any non-trivial normal subgroup N of G as a series K0/N < K1/N < · · · < Kn/N of normal subgroups of G/N induces 1 < N < K1 < · · · < Kn in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Write Z(G) = �d i=1 Ci where Ci are non-trivial cyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' If d ≤ 1 (for example when l(G) = 0), then D = {1} is a C-decomposition in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Otherwise, by induction of l(G), there are C-decompositions Di in G/Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lift Di to a set of normal subgroups ¯Di of G containing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Di = {K/Ci : K ∈ ¯Di}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We claim that D := �d i=1 ¯Di is a C-decomposition in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, it is a finite set of normal subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every K ∈ D, we have K/Ci ∈ Di for some i, hence G/K ∼= (G/Ci)/(K/Ci) ∈ C as Di is a C-decomposition in G/Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finally, note that � D = �d i=1 � ¯Di = �d i=1 Ci = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For the second part, suppose D is a C-decomposition in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then µD(Z(G)) ⊆ Z(G/D) = � N∈D Z(G/N), so d(Z(G)) ≤ � N∈D d(Z(G/N)) ≤ |D| since G/N has cyclic centre by assumption and using that d is a monotone function on abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let A be an additive finite abelian p-group, X be a trivially intersecting set of subgroups of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then there exists Y ⊆ X with |Y | ≤ d(A) and � Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We prove this by induction on d(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' If d(A) = 0, then A is trivial, and Y = ∅ works by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Else assume that d(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For any subgroup K ≤ A, define V (K) = {g ∈ K : gp = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that this is an Fp-vector space of dimension d(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Assume by contradiction that V (K) = V (A) for all K ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then V (A) ⊆ K, hence V (A) ⊆ � X = 0, but this contradicts that V (A) has positive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' So we may pick B ∈ X so that d(B) < d(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Now XB := {B ∩K : K ∈ X} is a trivially intersecting set of subgroups of B, so by induction, there is YB ⊆ XB of size at most d(B) with trivial intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lift back YB to Z ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then |Z| = |YB| and YB = {B ∩ K : K ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We show that Y := {B}∪Z ⊆ X satisfies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, |Y | ≤ 1+|YB| ≤ 1+d(B) ≤ d(A) by construction, and � Y = B ∩ � Z = � K∈Z(B ∩ K) = � YB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The next statement is motivated by an idea of Endre Szabó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' mC(P) = d(Z(P)) for any finite p-group P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' There is a C-decomposition D in P by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We claim the existence of a C-decomposition S ⊆ D of size at most d(Z(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This then proves the statement as no smaller C-decomposition may exist by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let A := Z(P) and consider X := {N ∩ A : N ∈ D}, a trivially intersecting set of subgroups of the abelian group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let Y ⊆ X with |Y | ≤ d(A) and � Y = 1 be given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lift Y back to S ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then 1 = � Y = Z(P) ∩ � S, so we must have � S = 1 since in a nilpotent group, there is no non-trivial normal subgroup intersecting the centre trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Also by construction, |S| = |Y | ≤ d(A) = d(Z(P)), so S is indeed a C-decomposition of P with the stated properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 4 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let G be a finite nilpotent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then mC(G) = max{mC(P) : P ∈ Syl(G)} where Syl(G) is the set of Sylow subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let D be a C-decomposition in G and P ∈ Syl(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We claim that DP := {N ∩P : N ∈ D} is a C-decomposition in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, P is a normal subgroup of G because G is finite nilpotent, so N ∩ P is a normal subgroup of G, hence of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' On the other hand � NP = P ∩ � D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This shows that mC(G) ≥ mC(P) for all P ∈ Syl(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For the other direction, let Dp be a C-decomposition for Gp ∈ Sylp(G) for all prime divisors p of |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let D be a partition of � p Dp of size max{|Dp| : p} such that |S∩Dp| ≤ 1 for every S ∈ D and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We claim that D := {� N∈S N : S ∈ D} is a C-decomposition in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, as above, every Np ∈ Dp is normal in G, so every N ∈ D is also a normal subgroup of G being the product of such groups in a finite nilpotent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' On the other hand, Gp ∩ � D = �{Gp ∩ � N∈S N : S ∈ D} = �{K : K ∈ Dp} = � Dp = 1 using the assumption on the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This shows that mC(G) ≤ max{mC(P) : P ∈ Syl(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' mC(G) = d(Z(G)) for any finite nilpotent group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In particular, G is a subdirect product of d(Z(G)) groups each having cyclic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 and the Chinese remainder theorem, we get mC(G) = max{mC(P) : P ∈ Syl(G)} = max{d(Z(P)) : P ∈ Syl(G)} = max{d(Q) : Q ∈ Syl(Z(G))} = d(Z(G)) after noting that G = � P∈Syl(G) P implies Syl(Z(G)) = {Z(P) : P ∈ Syl(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The statement holds when G is finitely generated using a similar reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' See [Sza21, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 3 Alternating modules In this section, we introduce alternating modules to generalise the notion of sym- plectic vector spaces prominently to the abelianisation of ≤2-step nilpotent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We show that, under some conditions, they possess an analogue of the Darboux basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Using this, on one hand we prove the second part of Theorem A from page 1, on the other hand we endow the module with a non-canonical complex structure having an isotropic real structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We start with an elementary statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 (‘Alternating Smith’ normal form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let R be a principal ideal domain, W ∈ Rn×n be an alternating matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' W ⊤ = −W and has 0’s at the main diagonal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then for s = 1 2 rk(W), there exist elements d1 | d2 | · · · | ds ̸= 0 in R (unique up to unit multiples) and B ∈ SLn(R) such that B⊤WB = diag �� 0 d1 −d1 0 � , � 0 d2 −d2 0 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , � 0 ds −ds 0 � , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' (1) 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The idea is similar to the standard proof of Smith normal form [Jac85, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7], but instead of focusing on the main diagonal, we consider the superdiagonal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' At each step, we choose different pivots and apply the base change W �→ X⊤WX (to respect the alternating property) for a series of well-chosen matrices X ∈ SLn(R) until the stated form for W = (wi,j) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Once the existence is verified, the uniqueness statement follows from the fact that for each 1 ≤ k ≤ n, the ideal generated by the k × k minors is unchanged under these transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' See [Sza21, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' A key notion is the following analogue of symplectic vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We call (M, ω, C) an alternating R-module, if M and C are R-modules and ω: M × M → C is a R-bilinear map that is alternating, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ω(m, m) = 0 for every m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' N ≤ M is N isotropic if ω(N N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The orthogonal complement of N is N ⊥ := {m ∈ M : ω(m, N) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Call ω and (M, ω, C) non-degenerate if M ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For simplicity, in this paper we will further require the extra conditions of M being finitely generated, C cyclic and R a principal ideal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 (Darboux-generators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let (M, ω, C) be an alternating R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then there exists a minimal R-module generating set B of M, and a subset {x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , xt, yt} ⊆ B such that ω(M, M) = Rω(x1, y1) ≥ Rω(x2, y2) ≥ · · · ≥ Rω(xt, yt) ̸= 0 and ω(b1, b2) = 0 for all other pairs (b1, b2) ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In any such generating set, the chain of submodules of C above is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' More concretely, t = 1 2d(M/M ⊥) and M/M ⊥ ∼= �t i=1(Rω(xi, yi))⊕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let (M, ω, C) be a Darboux module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Pick a minimal R-module generating set {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , bn} of M, and let c be a fixed generator of ω(M, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Pick wi,j ∈ R such that ω(bi, bj) = wi,jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' (Note that these are not necessarily unique, but wi,j + annR(c) ∈ R/ annR(c) are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=') Without loss of generality, we may assume that W := (wi,j)i,j ∈ Rn×n is an alternating matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let B ∈ SLn(R) given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Since B is an invertible square matrix, B := {�n j=1 Bj,ibj : 1 ≤ i ≤ n} = {x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , xs, ys, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' } is also a (min- imal) generating set of M in which ω can be expressed at the matrix from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then the statement follows by setting t ∈ N0 so that dic ̸= 0 for 1 ≤ i ≤ t, and djc = 0 for t < j ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For the second part, let xi, yi be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We claim that 0 M ⊥ M �t i=1(Rω(xi, yi))⊕2 0 m (ω(m, yi), ω(xi, m))t i=1 ⊆ ω♭ (2) is a short exact sequence of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, ω♭ is well-defined using the bilinearity of ω and the orthogonality elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ker(ω♭) = {m ∈ M : ∀i ω(xi, m) = ω(m, yi)} = {m ∈ M : ∀b ∈ B ω(b, m) = 0} = M ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' On the other hand, using orthogonality once more, ω♭(�t i=1 rixi + siyi) = (ri, yi)i for arbitrary ri, si ∈ R, thus ω♭ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Hence M/M ⊥ ∼= �t i=1(Rω(xi, yi))⊕2, so the isomorphism class of the R-modules Rω(xi, yi) are invariant by the structure theorem of finitely generated modules over a principal ideal domain and t = 1 2d(M/M ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' If annR(c) ̸= 0, then ω(M, M) is a cyclic torsion R-module, so its isomorphic submodules are necessarily equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This means that the submodules Rω(xi, yi) ≤ ω(M, M) are themselves invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Otherwise, ω(M, M) is a free R-module of rank 1 generated by, say, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Thus ω(xi, yi) = dic for some unique di(c) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1, Rdi is independent of the choice the generators of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Hence Rdic = Rω(xi, yi) may depend only on c, but the right-hand side does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 6 Our main resource of alternating Z-modules is the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' A short exact sequence ϵ : 1 → C ι−→ G π−→ M → 1 of groups is called a central-by-abelian extension, if ι(C) ⊆ Z(G) and M is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This extension ϵ is non-degenerate, if ι(C) = Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 (The alternating functor A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every central-by-abelian extension of finitely generated groups ϵ : 1 → C ι−→ G π−→ M → 1 induces an alternating Z-bilinear map ω: M × M → C defined by (m1, m2) �→ ι−1([g1, g2]) for arbitrary gi ∈ π−1(mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In particular, when the extra conditions on M and C from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2 are satisfied, then A(ϵ) := (M, ω, C) is an alternating Z-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We consider M and C as Z-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' First note that G′ ⊆ ι(C) ⊆ Z(G), so G is necessarily ≤2-step nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then the general commutator identities [g1g2, h] = [g1, h][g1, h, g2][g2, h] and [g, h1h2] = [g, h2][g, h1][g, h1, h2] imply that [−, −] : G×G → G′ is a group morphism in both coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Next, we check that ω is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Pick gi, g′ i ∈ π−1(mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then g−1 i g′ i ∈ ker(π) = Im(ι), so there are ci ∈ C with ι(ci) = g−1 i g′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then [g′ 1, g′ 2] = [g1ι(c1), g2ι(c2)] = [g1, g2][g1, ι(c2)][ι(c1), g2][ι(c1), ι(c2)] = [g1, g2] by above as ι(C) ⊆ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finally G′ ⊆ ι(C) implies that we can apply ι−1 to this element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Z-bilinearity of ω follows directly from the previously mentioned fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The alternating property follows as every group element commutes with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 gives the following dictionary between subgroups H, Hi of G and submodules of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The commutator map corresponds to ω ([g, g′] = ι◦ω(π(g), π(g′)) and [H1, H2] = ι ◦ ω(π(H1), π(H2))), commutes to being orthogonal ([H1, H2] = 1 ⇐⇒ π(H1) ⊥ π(H2)), the centraliser to the orthogonal complement (π(CG(H)) = π(H)⊥, in particular π(Z(G)) = M ⊥), abelian to isotropic ([H, H] = 1 ⇐⇒ π(H) ⊥ π(H)), and the notion of non-degeneracy coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The dictionary can be extended to Darboux-generators as the following generalisation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 and [BBC69, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1] shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7 (Central product decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let G be a finitely generated ≤2- step nilpotent group with cyclic commutator subgroup G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then it contains pairwise commuting subgroups A and E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , Et such that G = AE1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Et (a central product) where A ≤ Z(G), Ei are 2-generated and of class exactly 2, d(G) = d(A) + 2t and G′ = E′ 1 ⊋ E′ 2 ⊋ · · · ⊋ E′ t ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In any such case, t = 1 2d(G/ Z(G)) and E′ i ⊆ G′ are invariants given by G/ Z(G) ∼= �t i=1 E′2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let (M, ω, C) = A(1 → G′ ⊆−→ G π−→ G/G′ → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This is an alternating Z-module by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Consider the minimal generating set B = {x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , xt, yt, o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , ok} of M = G/G′ as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every b ∈ B, fix an arbitrary lift ¯b ∈ π−1(b) ⊆ G, and set ¯B := {¯g : g ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We show that the subgroups Ei := ⟨¯xi, ¯yi⟩ ≤ G and A := ⟨¯o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , ¯ok⟩ ≤ G satisfy the statement Indeed, A ⊆ π−1(M ⊥) = Z(G) using Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Moreover, [Ei, Ej] = 1 if and only if (Zxi + Zyi) ⊥ (Zxj + Zyj) if and only if i ̸= j by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6, G′ = [G, G] = ω(M, M) and E′ i = [Ei, Ei] = ω(Zxi + Zyi, Zxi + Zyi) = Zω(xi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then all parts about the derived subgroups follow 7 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In particular, G′ = E′ 1 = Zω(xi, yi) = ⟨[¯x1, ¯y1]⟩, so considering central- by-abelian extension, we see that G = ⟨ ¯B ∪ G′⟩ = ⟨ ¯B ∪ {[¯x1, ¯y1]}⟩ = ⟨ ¯B⟩ = AE1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Et So d(G) ≤ d(A) + �t i=1 d(Ei) ≤ | ¯B| = |B| = d(M) ≤ d(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This forces equality everywhere, so d(Ei) = 2 and 2t + d(A) = d(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finally Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6 shows that M ⊥ = π(Z(G)) = Z(G)/G′, so M/M ⊥ = (G/G′)/(Z(G)/G′) ∼= G/ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The isomorphism class of the subgroups E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , Et are not unique, which is demonstrated by the classical decomposition of extra-special p-groups Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For example, the extra-special p-group G of order p2t+1 of exponent p2 has many different internal central product decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For any 1 ≤ s ≤ t, G = E1E2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Et such that Ei ∼= M for 1 ≤ i ≤ s and Ei ∼= E for s < i ≤ t, where E and M are the non-abelian groups of order p3 and of exponent p and p2, respectively [Suz82, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' If the alternating module is non-degenerate, we can endow it with additional struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For a commutative ring Q, define a ring Q[i] := Q[x]/(x2 + 1) with i := x + (x2 + 1) ∈ Q[i] and maps σ: Q[i] → Q, q + iq′ �→ q − iq′ (conjugation) and ℑ: Q[i] → Q, q + iq �→ q′ (the imaginary part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For a Q[i]-module M, we call a map h: M ×M → Q[i] a Hermitian form on M over Q[i] if h is Q[i]-linear in the first argument and h is σ-conjugate symmetric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' h(m, m′) = σ(h(m′, m))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let (M, ω, C) be a non-degenerate alternating R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Set Q := R/ annR(C), and pick a generator of C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' a Q-module isomorphism ϕ: Q → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then there is a Hermitian form h on M over Q[i] making the following diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' M × M C Q[i] Q ω h ∃ ℑ ϕ ∼ Furthermore, M has an a non-canonical isotropic real structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' an isotropic Q- submodule MQ of M such that M = MQ ⊕ iMQ (as Q-modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The Q[i]-module structure of M is non-canonical, but is compatible with R → Q → Q[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The Q-module iMQ is automatically isotropic as ω(ia, ia′) = ϕ(ℑ(h(ia, ia′))) = ϕ(ℑ(h(a, a′))) = ω(a, a′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' While the structures themselves from the statement depend on the choice of the generators of M, their isomorphism class do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' More concretely, let ¯ M be an arbitrary Q[i]-module structure on M together with a Hermitian form ¯h and an isotropic real structure ¯ M = ¯ MQ⊕i ¯ MQ as at the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 implies the existence of a Q[i]-module isomorphism f : M → ¯ M such that f(MQ) = ¯ MQ and ¯h ◦ (f × f) = h (hence ω ◦ (f × f) = ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' First, we claim that annR(C) ⊆ annR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, for arbitrary r ∈ annR(C) and m ∈ M, ω(rm, m′) = rω(m, m′) = 0 for every m′ ∈ M, hence rm ∈ M ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This then means that M can naturally be considered as a Q-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' To define the Q[i]-module structure, let B = {x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , xt, yt, o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , ok} be a Darboux-generating set as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The isomorphism ω♭ : M ∼= �t j=1(Rω(xj, yj))⊕2 from (2) shows that k = 0 and M = �t j=1(Rxi ⊕ Ryi) = �t j=1(Qxi ⊕ Qyi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In particular, MQ := �t j=1 Qxi and MiQ := �t j=1 Qyi are (non-canonical) isotropic submodules of M giving a Q-module 8 decomposition M = A ⊕ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define a Q-module automorphism ιj of (Qω(xj, yj))⊕2 by ιj : (n, n′) �→ (−n′, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Pulling back �t j=1 ιj along ω♭ gives a non-canonical automorphism ι of M such that ι(xj) = yj and ι(yj) = −xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Thus ι ◦ ι = −idM, ι(MQ) = MiQ and ι(MiQ) = MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Thus defining (q + iq′) · m := qm + ι(q′m) gives the Q[i]-module structure in which MiQ = iMQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Set ωQ := ϕ−1 ◦ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We claim that h: M × M → C[i], (m, m′) �→ ωQ(im, m′) + iωQ(m, m′) (3) is the Hermitian form with the stated properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, Q-linearity in the first argument is inherited from ω, and h(im, m′) = ωQ(−m, m′) + iωQ(im, m′) = i(ωQ(im, m′) + iωQ(m, m′)) = ih(m, m′) then implies Q[i]-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For the conjugate symmetry, first note that ω(ixj, iyj) = ω(yj, −xj) = ω(xj, yj) using the alternating property, and for all other pairs (b1, b2) ∈ B2, we also have ω(ib1, ib2) = 0 = ω(b1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Hence the Q-bilinearity of ωQ implies and ω(im, im′) = ω(m, m′) for every m, m′ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This together with the alternating property of ωQ gives ωQ(im, m′) = −ωQ(m′, im) = ωQ(i2m′, im) = ωQ(im′, m), thus h(m, m′) = ωQ(im, m′) + iωQ(m, m′) = ωQ(im′, m) − iωQ(m′, m) = σ(h(m′, m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Given Hermitian form and the alternating map determine each other uniquely via (3) and ℑ ◦ h = ωQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Furthermore, given the isotropic real structure, these maps are determined by the restriction µ: MQ × iMQ → C, (a, b) �→ ω(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, ω(a + ib, a′ + ib′) = µ(a, ib′) − µ(a′, ib) using the bilinearity of ω and that MQ is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 4 Heisenberg groups In this section, associate a (polarised) Heisenberg group for every Z-bilinear map, in particular to alternating modules, or to finite ≤2-step nilpotent groups G with cyclic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We show that upon extending the centre of the Heisenberg group suitably using an extended polarisation, it will contain a normal subgroup isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 (Heisenberg group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let A, B and C be Z-modules and µ: A × B → C a Z-bilinear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We call µ non-degenerate, if µ(a, B) = 0 implies a = 0 and µ(A, b) = 0 implies b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define the associated Heisenberg group as H(µ) := A ⋉ϕ (B × C) where ϕ: A → Aut(B×C), a �→ ((b, c) �→ (b, µ(a, b)+c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Call H(µ) non-degenerate if Z(H(µ)) = {(0, 0, c) : c ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define a central-by-abelian extension H(µ) : 1 → C ιµ −→ H(µ) πµ −→ A × B → 1 by ι := ιµ : c �→ (0, 0, c) and π := πµ : (a, b, c) �→ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' More explicitly, the group structure on H(µ) is given by (a, b, c) ∗ (a′, b′, c′) = (a + a′, b + b′, c + µ(a, b′) + c′) with (0, 0, 0) being the identity and (a, b, c)−1 = (−a, −b, µ(a, b) − c) the inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' So formally H(µ) ∼= � 1 A C 0 1 B 0 0 1 � with matrix multiplication induced by µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In particular, [(a, b, c), (a′, b′, c′)] = (0, 0, µ(a, b′) − µ(a′, b)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' the commutator coincides with ιµ ◦ ω ◦ (πµ × πµ) using the notation of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that Z(H(µ)) = {(a, b, c) : µ(a, B) = µ(A, b) = 0} ⊇ ιµ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The notion of non-degeneracy for µ, ω, H(µ) and H(µ) all coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 9 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 (Heisenberg group of alternating modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let (M, ω, C) be a non- degenerate alternating R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='10, set A := MQ and B := iMQ considered as Z-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then the restriction µ: A × B → C as at Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='13 is non-degenerate and produces the non-degenerate central-by-abelian extension H(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that while this construction depends on isotropic real structure, the isomorphism class H(µ) does not because f from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='12 induce an isomorphism of short exact sequences, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Sza21, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1] In particular, the isomorphism class of H(µ) is invariant which we call the Heisenberg group of the alternating module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that (M, ω, C), µ, H(µ) and H(µ) basically decode the same information as they mutually determine each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Alternatively, if 2 ∈ R has a multiplicative inverse (for example C is finite of odd order), then there is a canonical way to define this Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' As for symplectic vector spaces, we can define a group on the set H := M × C with binary operation (m, c) · (m′, c′) := (m + m′, c + c′ + 2−1ω(m, m′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then H → H(µ), (a + b, c) �→ (a, b, c − 2−1µ(a, b)) is a group isomorphism to the group from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We shall use the construction of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 to treat all cases uniformly independently whether the group has 2-torsion or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let G be a finite ≤2-step nilpotent group with cyclic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 to the non-degenerate central-by-abelian extension Z(G) 1 Z(G) G G/ Z(G) 1 : ⊆ πZ giving a non-degenerate alternating Z-module (G/ Z(G), ω, Z(G)) := A(Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Hence Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 gives a non-degenerate Z-bilinear map µG : A × B → Z(G) (where actually A ∼= B) and H(µG) 1 Z(G) H(µG) A × B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' : ιµg πµg In this way, we assign a Heisenberg group H(µG) to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' These groups share many proper- ties: the order, isomorphism class of centre and commutator subgroup, nilpotency class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' However, they are non-isomorphic for example if G = Q8 (the quaternion group) or the extra-special p-group of order p3 of exponent p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Thus in general, we cannot even ex- pect to have a morphism between the two short exact sequences, as that would imply G ∼= H(µG) by the 5-lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In fact, this (iso)morphism exists if and only if G ∼= H(µG) since then A ⊕ B gives an isotropic real structure by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Our goal in to establish a monomorphism Z(G) → H(ˆµ) for a suitable (non- degenerate) ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In fact, we will show that ˆµ = ζ ◦ µG works for a suitable ζ : Z(G) ↣ ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For this, we generalise the notion of isotropic real structure from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' An extended polarisation of a central-by-abelian extension ϵ is the pair of the following commutative diagrams (j ∈ {1, 2}) ϵj 1 Cj Gj Lj 1 ϵ 1 C G M 1 ˆC L1 × L2 : ιj κj πj γj ζj λj : ι ζ π ∼ λ:=λ1⊕λ2 (4) 10 such that ϵj is a central-by-abelian extension, λ: L1 × L2 → M, (l1, l2) �→ λ1(l1) + λ2(l2) is an isomorphism and ˆC is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every extended polarisation as in (4) induces a decomposition G = γ2(G2)ι(C)γ1(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Pick g ∈ G arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Set (l1, l2) := λ−1(π(g)) ∈ L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' From the sur- jectivity of πj, pick gj ∈ Gj such that πj(gj) = lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then π(γ2(g2)−1gγ1(g1)−1) = −λ2(π2(g2)) + (λ1(l1) + λ2(l2)) − λ1(π1(g1)) = 0 using the commutativity of the dia- gram (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' So γ2(g2)−1gγ1(g1)−1 ∈ ker(π) = Im(ι), hence there is a c ∈ C such that ι(c) = γ2(g2)−1gγ1(g1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Rearranging this gives the decomposition as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='8 (Key).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Every extended polarisation as at (4) can be completed to a commutative diagram ϵ 1 C G M 1 H(ˆµ) 1 ˆC H(ˆµ) L1 × L2 1 : ι ζ π δ ∼ λ−1 : ιˆµ πˆµ (5) where ˆµ is defined by M × M C L1 × L2 ˆC ω ζ ˆµ λ1×λ2 for the alternating Z-bilinear map ω from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5 when applied to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The 4-lemma implies that δ is injective if and only if ζ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In this case, H(ˆµ) is the external central product of ˆC and G amalgamating C along ι and ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' in particular, G is isomorphic to a normal subgroup of a Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The central-by-abelian extension ϵ is non-degenerate if and only if H(ˆµ) is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' First note that ˆµ is indeed an alternating Z-bilinear map by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5, so H(ˆµ) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We show that δ := (δ1, δ2, δ3) satisfies the statement for j ∈ {1, 2} and δj : G → Lj, g �→ πj(gj), δ3 : G → ˆC, g �→ ζ2(g2)ζ(c)ζ1(g1), for any decomposition g = γ2(g2)ι(c)γ1(g1) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The map δj is actually the natural composition of group morphisms G π−→ M λ−1 −−→ L1 × L2 → Lj, in particular, δj is independent of the choice of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' To show that δ3 is independent of the choice of the decomposition, let γ2(g2)ι(c)γ1(g1) = g = γ2(g′ 2)ι(c′)γ1(g′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then on one hand, πj(gj) = δj(g) = πj(g′ j) by above, hence by the exactness of ϵj, there are cj ∈ Cj such that ι1(c1) = g′ 1g−1 1 and ι2(c2) = g−1 2 g′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' On the other hand, using ι(C) ⊆ Z(G), rearranging the original equation gives ι(cc′−1) = γ2(g−1 2 g′ 2)γ1(g′ 1g−1 1 ) = ι(κ2(c2)κ1(c1)), hence cc′−1 = κ2(c2)κ1(c1) as ι is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Putting these together gives ζ2(g2)ζ(c)ζ1(g1) = ζ2(g′ 2ι2(c2)−1) · ζ(κ2(c2)c′κ1(c1)) · ζ1(ι1(c1)−1g′ 1) 11 = ζ2(g′ 2) · (ζ2(ι2(c2))−1ζ(κ2(c2))) · ζ(c′) · (ζ(κ1(c1))ζ1(ι1(c1))−1) · ζ1(g′ 1) = ζ2(g′ 2)ζ(c′)ζ1(g′ 1) using commutativity of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Thus δ3 is indeed well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that unlike the other maps, δ3 is just a map of sets, not a group morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Its failure to be a group morphism is measured by ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, pick decompositions g = γ2(g2)ι(c)γ1(g1) and g′ = γ2(g′ 2)ι(c′)γ1(g′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Set x := ω(π(γ1(g1)), π(γ2(g′ 2))) ∈ C, so ι(x) = [γ1(g1), γ2(g′ 2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Use this to find a decomposition of the product as gg′ = γ2(g2)ι(c)γ1(g1)γ2(g′ 2)ι(c′)γ1(g′ 1) = γ2(g2)γ2(g′ 2)ι(cc′)[γ1(g1), γ2(g′ 2)]γ1(g1)γ1(g′ 1) = γ2(g2g′ 2)ι(cc′x)γ1(g1g′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then by definitions and using the commutativity of the diagram, δ3(gg′) = ζ2(g2g′ 2)ζ(cc′x)ζ1(g1g′ 1) = ζ2(g2)ζ(c)ζ1(g1) · ζ2(g′ 2)ζ(c′)ζ1(g′ 1) · ζ(ω(π(γ1(g1)), π(γ2(g′ 2)))) = δ3(g)δ3(g′)ˆµ(π1(g1), π2(g′ 2)) = δ3(g)δ3(g′)ˆµ(δ1(g), δ2(g′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This property together with Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2 imply that δ is a group morphism: δ(gg′) = (δ1(gg′), δ2(gg′), δ3(gg′)) = (δ1(g)δ1(g′), δ2(g)δ2(g′), δ3(g)δ3(g′)ˆµ(δ1(g), δ2(g′))) = (δ1(g), δ2(g), δ3(g)) ∗ (δ1(g′), δ2(g′), δ3(g′)) = δ(g) ∗ δ(g′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We check that the diagram (5) is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Indeed, if c ∈ C, then using the decomposition ι(c) = γ2(1)ι(c)γ1(1) gives δ ◦ ι = (c �→ (0, 0, ζ(c))) = ιˆµ ◦ ζ by defin- itions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Similarly, the decomposition g = γ2(g2)ι(c)γ1(g1) ∈ G gives λ−1 ◦ π = (g �→ λ−1(π(γ2(g2)γ1(g1))) = δ1(g) + δ2(g)) = πˆµ ◦ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 5 Heisenberg embeddings In this section, we put together the pieces from earlier sections to prove Theorem B from 2, the main statement of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' First, we handle the cyclic centre case by constructing a suitable extended polarisation using the isotropic real structure from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For the general case, we take the direct product of the resulting Heisenberg groups (which itself is of this type) and use the first part of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We start with an elementary statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The solid arrows of the following diagram can be completed with suitable dashed arrows making a commutative diagram of groups where K is a finite abelian group and l = lcm(m, exp(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Z/nZ K Z/mZ Z/lZ ι κ ϕ θ 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' First, we prove the case K = Z/kZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The map κ is defined by some b ∈ Z such that κ(1+nZ) = m n b+mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Similarly, ι is given by some a ∈ Z with ι(1+nZ) = k na+kZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Since ι is injective, we have gcd(a, n) = 1, hence we may pick x ∈ Z so that ax ≡ b (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define ϕ(i + kZ) := i l kx + lZ and θ(i + mZ) := i l m + lZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Short computation shows that ϕ ◦ ι = θ ◦ κ with these definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In the general case, K = � C∈S C for some suitable set S of cyclic subgroups of K of prime power order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For C ∈ S, let πC : K → C be the natural projection, and write o(C) := | Im(πC ◦ ι)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every prime p dividing n, pick a Cp ∈ S so that o(Cp) = max{o(C) : C ∈ S, p �� |C|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define the composition ¯ι: Z/n ι−→ K ↠ � p|n Cp ∼= Z/dZ → Z/kZ where the second map is � p|n πCp, the isomorphism is given by the Chinese remainder theorem, and the last one is any embedding where d �� k := exp(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This map is injective, because | Im(¯ι)| = lcm{|Cp| : p | n} = lcm{o(C) : C ∈ S} = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' So replacing ι by ¯ι reduces to the special case K = Z/kZ discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Essentially this statement replaces the usage of the second part of Theorem A from 1 and the central product approach of [Sza21] and [Sza19] with a much shorter argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The statement could be extracted when tracking down the behavior of centre the group during taking maximal central products [Sza21, proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='18] and the construction of extended polarisation for 2-generated groups [Sza21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' More concretely, writing G = �t i=1 Ci as a product of cyclic groups, we take iteratively the (so called maximal) central product of Z/mZ, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , Ct at each step amalgamating the largest possible subgroup compatible with the given maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This maximality condition ensures that the resulting groups remain cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every finite ≤2-step nilpotent group G with cyclic centre, there exists a monomorphism Z(G) 1 Z(G) G G/ Z(G) 1 H(ˆµ) 1 ˆC H(ˆµ) A × A 1 : f ⊆ ζ πZ δ ∼ ν : ⊆ πˆµ (6) of non-degenerate central-by-abelian extensions for a suitable ˆµ: A × A → ˆC where ˆC is cyclic and the exp(A) = |G′| �� | Z(G)| �� | ˆC| �� exp(G) divisibility conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Actually, the Heisenberg group from the statement can be replaced with a canonical one as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define a Z-bilinear map ν : Hom(A, ˆC) × A → ˆC by α(a) �→ α(a), and write H(A, ˆV ) := H(ν) for the corresponding Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then the map H(ˆµ) → H(A, ˆC) given by (a, a′, c) �→ (x �→ ˆµ(a, x), a′, c) is an isomorphism because ˆµ is non-degenerate and exp(A) �� |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In particular, every G as above is isomorphic to a normal subgroup of H(A, ˆC) of index at most exp(G)/| Z(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='10 is applicable to (G/ Z(G), ω, Z(G)) := A(Z(G)) giving an isotropic real structure Z(G) = L1 ⊕ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that L2 = iL1 ∼= L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Since Lj is isotropic, π−1 Z (Lj) is abelian by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 13 Consider the inclusion maps from the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Z(G) ∩ π−1 Z (L1) π−1 Z (L1) Z(G) C ˆC Z(G) ∩ π−1 Z (L2) π−1 Z (L2) ⊆ ι1 ⊆ κ1 ϕ1 ζ1 θ1 ζ θ2 ⊆ κ2 ⊆ ι2 ϕ2=ζ2 Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 to the inclusions ι1 and κ1 gives a cyclic group C with a morphisms ϕ1, θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then we can apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 to the inclusion ι2 and the composition κ2 giving another cyclic ˆC with ϕ2, θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By construction, θ1 and θ2 are both injective, hence so is ζ := θ2 ◦θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This diagram is commutative by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Set ζ1 := θ2 ◦ϕ1 and ζ2 := ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The maps above give the following extended polarisation of Z(G) ϵj 1 Z(G) ∩ π−1 Z (Lj) π−1 Z (Lj) Lj 1 Z(G) 1 Z(G) G G/ Z(G) 1 ˆC L1 × L2 : ⊆ ⊆ πj ⊆ ζj ⊆ : ⊆ ζ πZ where πj := πZ|π−1 Z (Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='8 gives the diagram (6) upon setting A := L1 ∼= L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finally, we check that the stated properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Note that the injectivity of ζ together with the 4-lemma implies the injectivity of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' exp(A) = exp(L1 × L2) = exp(G/ Z(G)) = |(G′)| follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Since G is of nilpotency class at most 2, G′ ⊆ Z(G) implies |G′| �� | Z(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1, | ˆC| = lcm(| Z(G)|, exp(π−1 Z (L1)), exp(π−1 Z (L2))) �� exp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every finite ≤2-step nilpotent group G, there exists Z(G) 1 Z(G) G G/ Z(G) 1 H(µ) 1 C H(µ) A × B 1 : f ⊆ ζ πZ δ ν : ⊆ πµ for a suitable non-degenerate µ: A × B → C such that d(G/ Z(G)) ≥ d(A), d(B), d(Z(G)) = d(C), exp(G/ Z(G)) �� exp(A × B) �� exp(G′) �� exp(Z(G)) �� exp(C) �� exp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Furthermore, there is a monomorphism H(µ) ↣ �d(Z(G)) i=1 H(µi) for suitable µi : Ai×Ai → Ci where each H(µi) is non-degenerate, Ci is cyclic and d(Ai) ≤ 1 2d(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 14 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The monomorphism ζ shows that d(C) is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' On the other hand, ν gives d(G/ Z(G)) ≤ d(A × B) ≤ 2d(G/ Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The lower bound is attained in the d(C) ≤ 1 case, see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' It is a natural question to ask for the smallest possible value of d(A×B) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' To give a better upper bound than above, one may need to develop some version of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 for matrices with entries from Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7, write ϕ: G ↣ �n i=1 Gi as a subdirect product where n := d(Z(G)) and each Z(Gi) is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then Gi are all ≤2-step nilpotent as this class is closed under taking quotients, so Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 gives fi = (ζi, δi, νi): Z(Gi) → H(µi) for some µi : Ai × Bi → Ci (where Bi = Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let ¯A := �n i=1 Ai, ¯B := �n i=1 Bi, ¯C := �n i=1 Ci and ¯µ := �n i=1 µi : ¯A × ¯B → ¯C, a non-degenerate Z-bilinear map by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We obtain the following diagram of central-by-abelian extensions where � is a shorthand for �n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Z(G) 1 Z(G) G G/ Z(G) 1 � Z(Gi) 1 � Z(Gi) � Gi � Gi/ Z(Gi) 1 � H(µ) 1 � Ci � H(µi) � Ai × Bi 1 H(¯µ) 1 ¯C H(¯µ) ¯A × ¯B 1 : Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='7 ⊆ ϕ|Z(G) πZ ϕ [ϕ] : � fi Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 ⊆ � ζi � πZ � δi ∼ � νi : ∼ ⊆ � πµi ∼ ∼ : ⊆ π¯µ Denote by ¯f = (¯ζ, ¯δ, ¯ν): Z(G) → H(¯µ) the resulting monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This may have more generators than stated, so we take a suitable subobject of H(¯µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let A ≤ ¯A be the image of G/ Z(G) ¯ν−→ ¯A × ¯B → ¯A, and B ≤ ¯B be that of G/ Z(G) ¯ν−→ ¯A × ¯B → ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then d(A) and d(B) are at most d(G/ Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Let C := ⟨¯ζ(Z(G)), ¯µ(A, B)⟩ ≤ ¯C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then d(Z(G)) = d(¯ζ(Z(G))) ≤ d(C) ≤ d( ¯C) = �n i=1 d(Ci) ≤ n = d(Z(G)), hence comparing the two ends give d(C) = d(Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Define µ: A × B → C, (a, b) �→ ¯µ(a, b), an abelian bihomomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The image of ¯f lies in H(µ) by definition, so restricting the domain to H(µ) gives a map f = (ζ, δ, ν): Z(G) → H(µ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ¯f = (H(µ) ↣ H(¯µ)) ◦ f for the natural inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We show that this f satisfies the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' We check that µ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Pick 0 ̸= a ∈ A and write a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , an) ∈ �n i=1 Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then without loss of generality, a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then by the non-degeneracy of µ1, there is a b′ 1 ∈ B1 such that 0 ̸= µ1(a1, b′ 1) ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By the diagram above, there is g′ 1 ∈ G1 such that ν1(g′ 1 Z(G1)) = (0, b′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' As ϕ is a subdirect product, there is g′ ∈ G such that the 1st factor of [ϕ](g′ Z(G)) is g′ 1 Z(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Write b′ = (b′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , b′ n) for the image of g′ Z(G) under G/ Z(G) ¯ν−→ ¯A × ¯B → ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By construction, b′ 1 coincides with the above choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' By definition, b ∈ B, and µ(a, b) = (¯µ1(a1, b′ 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' , ¯µn(an, b′ n)) ̸= 0 as the first factor is non-trivial by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' This argument remains valid when the roles of A and B are swapped, hence µ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Assume that G is finite and consider the statement on the exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' exp(G/ Z(G)) �� exp(A×B) follows from ν being a monomorphism of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' For every i, exp(Ai× Bi) exp(G′ i) �� exp(G′) using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 and the fact that as Gi is a quotient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Thus exp(A × B) �� exp( ¯A × ¯B) = lcm{exp(Ai × Bi) : 1 ≤ i ≤ n} �� exp(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Since G is ≤2-step nilpotent, we have G′ ⊆ Z(G), so exp(G′) �� exp(Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' The embedding ζ : Z(G) ↣ C shows exp(Z(G)) �� exp(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Once again using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3, we see 15 that exp(Ci) �� exp(Gi) �� exp(G) as Gi is a quotient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Then exp(C) �� exp( ¯C) = lcm{exp(Ci) : 1 ≤ i ≤ n} �� exp(G) as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' References [AMM12] Azhana Ahmad, Arturo Magidin and Robert Fitzgerald Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ‘Two generator p- groups of nilpotency class 2 and their conjugacy classes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In: Publicationes Mathem- aticae Debrecen 81 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 145–166 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [BBC69] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Brady, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Bryce and John Cossey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ‘On certain abelian-by-nilpotent varieties’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In: Bulletin of the Australian Mathematical Society 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='3 (1969), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 403–416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1017/S0004972700042325 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [CPS22] Balázs Csikós, László Pyber and Endre Szabó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finite subgroups of the homeomorph- ism group of a compact topological manifold are almost nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' arXiv: 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 13375 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='GT] (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Gul20] Attila Guld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Finite subgroups of the birational automorphism group are ’almost’ nilpotent of class at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' arXiv: 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='11715 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='AG] (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Hig60] Graham Higman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ‘Enumerating p-Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' I: Inequalities’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In: Proceedings of the London Mathematical Society s3-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1960), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 24–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' issn: 0024-6115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1112/plms/s3-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1112/plms/s3-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='24 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Jac85] Nathan Jacobson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Basic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' New York: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Freeman, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' isbn: 0-7167-1480- 9 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Mag98] Arturo Magidin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Bilinear maps and central extensions of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' arXiv: 9802066 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='GR] (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Sim65] Charles C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Sims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ‘Enumerating p-Groups’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' In: Proceedings of the London Mathemat- ical Society s3-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1965), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 151–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' issn: 0024-6115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1112/plms/ s3-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1112/plms/s3-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='151 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Suz82] Michio Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Group theory II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Grundlehren der mathematischen Wis- senschaften.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Springer-Verlag Berlin Heidelberg, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' isbn: 9783642868870 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 1, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Sza21] Dávid R Szabó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' ‘Jordan Type Problems via Class 2 Nilpotent and Twisted Heisen- berg Groups’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Central European University, 2021 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2, 5, 6, 10, 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' [Sza19] Dávid R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Szabó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Special p-groups acting on compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' arXiv: 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 07319 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='DG] (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' Alfréd Rényi Institute of Mathematics, Reáltanoda u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content=' 13–15, H–1053, Bud- apest, Hungary E-mail address: szabo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='david@renyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} +page_content='hu 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf'} diff --git a/qdFRT4oBgHgl3EQfejci/content/tmp_files/2301.13571v1.pdf.txt b/qdFRT4oBgHgl3EQfejci/content/tmp_files/2301.13571v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..09993d2c91a5c4947367c351f33ea22c0fd0426f --- /dev/null +++ b/qdFRT4oBgHgl3EQfejci/content/tmp_files/2301.13571v1.pdf.txt @@ -0,0 +1,2094 @@ +Exceptional-point-assisted entanglement, squeezing, and reset in a chain of three +superconducting resonators +Wallace S. Teixeira,1 Vasilii Vadimov,1 Timm Mörstedt,1 Suman Kundu,1 and Mikko Möttönen1, 2 +1QCD Labs, QTF Centre of Excellence, Department of Applied Physiscs, +Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland +2VTT Technical Research Centre of Finland Ltd., +QTF Center of Excellence, P.O. Box 1000, FI-02044 VTT, Finland +The interplay between coherent and dissipative dynamics required in various control protocols of +quantum technology has motivated studies of open-system degeneracies, referred to as exceptional +points (EPs). Here, we introduce a scheme for fast quantum-state synthesis using exceptional-point +engineering in a lossy chain of three superconducting resonators. We theoretically find that the rich +physics of EPs can be used to identify regions in the parameter space that favor a fast and quasi- +stable transfer of squeezing and entanglement, or a fast reset of the system. For weakly interacting +resonators with the coupling strength g, the obtained quasi-stabilization time scales are identified as +1/(2 +√ +2g), and reset infidelities below 10−5 are obtained with a waiting time of roughly 6/g in the +case of weakly squeezed resonators. Our results shed light on the role of EPs in multimode Gaussian +systems and pave the way for optimized distribution of squeezing and entanglement between different +nodes of a photonic network using dissipation as a resource. +I. +INTRODUCTION +Quantum mechanics has provided profoundly novel +ways of information processing, communication, and +metrology [1]. Although non-linearity expressed by the +anharmonicity of energy levels is a key metric for phys- +ical realizations of qubits, quantum harmonic systems +have also a broad range of quantum-technological appli- +cations employing, e.g., squeezing and entanglement as +resources [2, 3]. The efficient use of such properties in +experiments typically requires quick transitions from co- +herent to incoherent dynamics for different stages of the +protocols, and hence dissipation engineering using in-situ +tunable components plays an important role towards fast +control and scalability of practical quantum systems [4]. +In circuit quantum electrodynamics (cQED), for ex- +ample, efforts have been made to integrate devices with +in-situ-tunable dissipation to prepare specific quantum +states [5–12], produce fast reset [13–24], and to exploit +the potential benefits of open-system degeneracies, re- +ferred to as exceptional points (EPs) [17, 21, 25–29]. In +contrast to Hermitian degeneracies, EPs induce the coa- +lescence of eigenvalues and eigenvectors of the dynamical +matrix governing the open-system evolution leading to +critical dynamics manifested by polynomial solutions in +time [30, 31]. These features are key elements for op- +timized heat flow [25] and sensitive parameter estima- +tion [30]. +When EPs are dynamically encircled in the +parameter space, counter-intuitive effects not observed +in closed systems appear such as the breakdown of the +adiabatic approximation and topological energy trans- +fer [32–34]. Due to their novelty for the observation of +open-system phenomena and applications, EPs have also +been acknowledged in other physical architectures [35– +37]. +However, the relationship between EPs and the +emergence of non-classical and non-local features in mul- +tipartite continuous-variable (CV) quantum systems has +not been fully explored [38–43]. +Quantum harmonic arrays have a practical appeal +in cQED for the implementation of quantum memo- +ries [44] and for the capability to simulate many-body +physics [45]. Even though the transport of quantum cor- +relations has been extensively theoretically studied in re- +lated setups [46–49], the high dimension of such systems +and their dissipative features render the characterization +of EPs an involved procedure [50–52]. +Motivated by the above-mentioned potential use cases +and issues, in this work, we introduce exceptional-point +engineering for squeezing and entanglement propagation. +We consider a minimal setup for the production of high- +order EPs, consisting of a chain of tree linearly cou- +pled superconducting resonators with independent decay +channels. To some extent, our system can be described +by its first and second moments, so that it can consti- +tute an example of a Gaussian system, i.e., a CV system +represented by a Gaussian Wigner function [3]. To ana- +lytically describe the EP-related phenomena, we employ +the Jordan normal form of the dynamical matrix of the +second moments, allowing for investigations beyond en- +ergy flow. +Interestingly, we observe that even for weakly cou- +pled resonators, the operation in the vicinity of a specific +second-order EP may turn the central resonator into a +fast squeezing splitter and distant-entanglement genera- +tor using only initial squeezing in a single resonator. We +calculate theoretical bounds for the squeezing and en- +tanglement of the quasi-stable states and observe their +rich dependence on the initial squeezing parameter. On +the other hand, operation near a different, third-order +EP branch provides substantial speed up of the decay +towards the ground state. Therefore, the detailed knowl- +edge of its open-system degeneracies render the system a +versatile structure for quantum protocols requiring fast +stabilization or reset of the desired properties. +This article is organized as follows. +In Sec. II, we +arXiv:2301.13571v1 [quant-ph] 31 Jan 2023 + +2 +present the general theory of exceptional points in noisy +Gaussian systems. In Sec. III, we provide the details of +the considered setup, including the characterization of +its EPs. Sections IV and V are dedicated to studies of +different effects arising at or near EPs, with a focus on +the quasi-stabilization and decay of non-classical Gaus- +sian states, respectively. A discussion on the use cases +and limitations of EP engineering is provided in Sec. VI. +The conclusions are drawn in Sec. VII. +II. +EXCEPTIONAL POINTS IN NOISY +GAUSSIAN SYSTEMS +Our general model shown in Fig. 1(a) consists of a sys- +tem of N harmonic modes and of an environment such +that each system mode is interacting with their local +Markovian bath. The j:th mode is described by anni- +hilation and creation operators ˆaj and ˆa† +j, respectively, +with the canonical commutation relations [ˆaj, ˆa† +k] = δjk. +We assume that the modes are linearly coupled to one +another in any desired topology yielding up to quadratic +terms in their coupling Hamiltonian. In Secs. III–V, we +explore a linear network consisting of three lossy super- +conducting resonators as shown in Fig. 1(b). +By defining the quadrature operators of the j:th mode +as ˆqj = (ˆaj + ˆa† +j)/ +√ +2 and ˆpj = −i(ˆaj − ˆa† +j)/ +√ +2 and +their 2N-dimensional vector as ˆx = (ˆq1, ˆp1, ..., ˆqN, ˆpN)⊤, +the total Hermitian Hamiltonian describing the system +classically driven by amplitudes c = (c1, ..., c2N)⊤ can be +cast into the compact quadratic form [53] +ˆH = 1 +2ˆx⊤Hˆx + c⊤Ωˆx, +(1) +where we dropped possible constant energy offsets, in- +troduced the 2N × 2N symmetric matrix H carrying the +internal and mode–mode coupling energies, and utilized +the symplectic matrix +Ω = +N +� +j=1 +� +0 +1 +−1 0 +� +. +(2) +The commutation relations between the elements of ˆx +read [ˆxj, ˆxk] = iΩjk. Note that {ˆqj} and {ˆpj} play the +role of generalized dimensionless position and momen- +tum operators, such that for superconducting LC cir- +cuits they are related to flux and charge operators, re- +spectively [54]. +After tracing out the environmental degrees of free- +dom, the temporal evolution of the reduced density op- +erator of the system, ˆρ, is given by the Lindblad master +equation dˆρ/dt = −i[ ˆH, ˆρ]/ℏ + L↓(ˆρ) + L↑(ˆρ), where +Ll(ˆρ) = 1 +2ℏ +N +� +j=1 +� +2ˆLl +j ˆρ(ˆLl +j)† − +� +(ˆLl +j)† ˆLl +j, ˆρ +�� +, +(3) +... +... +... +... +... +R1 +R2 +R3 +QCRs +mode 1 +mode 2 +mode N +(a) +(b) +Figure 1. (a) Schematic diagram of the general system consid- +ered in this paper consisting of N harmonic quantum modes +linearly coupled to one another (black lines). +In addition, +each mode is coupled to their own Markovian environment +(rounded squares). (b) Particular realization of the system ex- +plored in this work, where three superconducting resonators +are capacitively coupled in a linear-chain configuration. +In +addition, each resonator has their own drive lines (trian- +gles), using which the system can be prepared and measured. +The decay rates of resonators R2 and R3 can be controlled +by quantum-circuit refrigerators (QCRs) placed at the res- +onator input ports. +Each QCR is comprised of a normal- +metal–insulator–superconducting (NIS) junction and can re- +move photons incoherently from the system mediated by elec- +tron tunneling at specific bias-voltage pulses [22, 23]. +describes the incoherent dynamics of the system asso- +ciated to the jump operators {ˆLl +j}, where the labels +l =↓, ↑ refer to thermal emission and absorption, re- +spectively. We restrict to the case where such operators +are local and linear combinations of the elements of ˆx, +i.e., ˆLl +j = (ul +j)⊤Ωˆx, with coefficients given by the 2N- +dimensional vector ul +j that has only a single or two adja- +cent non-zero elements. For example, this corresponds +to jump operators with the form ˆLl +j = nl +jˆaj + ml +jˆa† +j, +thus encompassing N individual squeezed thermal envi- +ronments [55]. The case in which both thermal excitation +and bath squeezing are negligible is thoroughly investi- +gated in Secs. III–V for N = 3. +Under the above conditions and for an initial Gaussian +state of the N oscillators, the dynamics of the system +can be fully characterized by the so-called mean vector +and covariance matrix (CM), the components of which +are ⟨ˆxj⟩ = Tr(ˆxj ˆρ) and Vjk = +1 +2 (⟨ˆxjˆxk⟩ + ⟨ˆxkˆxj⟩) − +⟨ˆxj⟩⟨ˆxk⟩, respectively. +Here, we aim to solve the dy- +namics of the CM, since it captures all squeezing and +non-local properties of the system. By differentiating V +with respect to time and using Eq. (3), we verify that +the CM evolves according to the differential Lyapunov +equation [56] +dV +dt = ΓV + VΓ⊤ + D, +(4) +where we defined the 2N × 2N matrices Γ = Ω(H − +ImΥ)/ℏ, D = ReΥ/ℏ, and Υ = � +l,j[ul +j(ul +j)†]. The CM +is a real, symmetric and positive-definite matrix. As a +compact statement of the uncertainty principle, the CM +must also fulfill the condition V + iΩ/2 ≥ 0 [57]. +Below we focus on the scenario where Γ and D are +independent of time. Given an initial CM V(0) ≡ V0, + +3 +the solution of Eq. (4) in this case is given by [58] +V(t) = eΓt (V0 − Vss) eΓ⊤t + Vss, +(5) +where Vss is the steady-state CM obtained as the solution +of the algebraic Lyapunov equation ΓVss +VssΓ⊤ +D = +0. We observe from Eqs. (4) and (5) that Γ has the role +of a dynamical matrix so that all possible EPs are de- +termined by its structure. Since the entries of Γ are real +numbers with units of angular frequency, its eigenvalues +are the complex-conjugate pairs λ± +sm. +Here, we define +the index sm = (m, µm) to refer to the m:th pair of the +eigenvalues of Γ, each eigenvalue having a multiplicity +µm. Observe that the maximum allowed multiplicity is +thus max(µm) = N. +The matrix Γ admits a Jordan normal form Γ = +PJP−1, where P is a non-singular matrix and J = +diag[J− +s1(λ− +s1), ..., J+ +sk(λ+ +sk)]. The Jordan blocks J± +sm(λsm) +can be decomposed as µm × µm matrices J± +sm(λ± +sm) = +λ± +smIµm + Nµm, with Iµm being the identity matrix and +Nµm having the elements above the diagonal filled with +ones. Naturally, the Jordan blocks for µm = 1 are just +the scalars λ± +sm. With these definitions, Eq. (5) can be +rewritten as +V(t) = PeJtP−1 (V0 − Vss) +� +P−1�⊤ eJ⊤tP⊤ + Vss, +(6) +where eJt = diag(eλ− +s1teNµ1t, ..., eλ+ +sk teNµk t). +The emergence of EPs and the associated critical dy- +namics of the CM correspond to the cases where the dy- +namical matrix Γ becomes non-diagonalizable, i.e., for +any µm > 1. In other words, degeneracies in the spec- +trum of Γ produce nilpotent matrices Nµmt, the expo- +nentials of which yield polynomials in time. Hereafter, +these non-Hermitian degeneracies will be referred to as +EP–µm. Considering the definition of Γ, we remark that +the term ΩH itself does not promote critical dynamics +as it gives rise to unitary evolution of the CM. The pro- +duction of EPs must be accompanied with the incoher- +ent processes caused by the local environments and at- +tributed to the term ΩImΥ. +In summary, Eq. (6) is valid for any time-independent +matrices Γ and D describing the evolution of a system +of coupled quantum harmonic oscillators in noisy Gaus- +sian channels yielding the steady-state CM Vss. At an +EP, Eq. (6) reveals that the solution linked to the crit- +ical dynamics is an exponential function multiplied by +a polynomial, which will be explored below in specific +cases. Alternatively, the description of EPs for quadratic +Liouvillians, such as the one related to Eq. (3), may be +given in terms of annihilation and creation operators as +recently developed in Ref. [59]. +III. +THREE COUPLED RESONATORS UNDER +INDIVIDUAL LOSSES +The system and its environment considered in this +work is depicted in Fig. 1(b). +Three superconducting +resonators, R1, R2, and R3 are capacitively coupled in a +linear-chain configuration through a fixed coupling con- +stant g > 0. We focus on a single electromagnetic mode +for each resonator, which, including the coherent cou- +plings, defines our system. Each mode may dissipate its +energy into its independent linear bath. +Nevertheless, +quantum effects may emerge at low temperatures and +for sufficiently high quality factors and for non-classical +initial states [54], and consequently we need to employ a +quantum-mechanical model. +In the single-mode and rotating-wave approximations, +the Hamiltonian of the system reads +ˆH = ℏ +3 +� +j=1 +ωj +� +ˆa† +jˆaj + 1 +2 +� ++ ℏg(ˆa1ˆa† +2 + ˆa2ˆa† +3 + h.c.), (7) +where ωj is the fundamental angular frequency of the +j:th resonator, {ˆaj} are the corresponding ladder opera- +tors defined as in Sec. II, and h.c. refers to the Hermitian +conjugate. The losses of the system are modeled here as +in Eq. (3), with jump operators ˆL↓ +j = +� +ℏκjˆaj and de- +cay rates κj > 0, for j = 1, 2, 3. Some of the decay rates +can be adjusted experimentally through the QCRs shown +in Fig. 1(b). As we show below, to produce EP–3 with +degenerate resonators, we need asymmetric decay rates, +a scenario which can be realized by the two independent +QCRs shown in Fig. 1(b). In the following analysis, ther- +mal excitations are neglected so that ˆL↑ +j ≈ 0. +By writing the ladder operators in terms of the quadra- +ture operators as ˆaj = (ˆqj + iˆpj)/ +√ +2 and using the nota- +tion of Sec. II, the 6 × 6 dynamical matrix Γ becomes +Γ = +� +� +� +K1 +G +02 +G +K2 +G +02 +G +K3 +� +� +� , +(8) +where 02 is the 2 × 2 null matrix and +Kj = +� +− κj +2 +ωj +−ωj − κj +2 +� +, +G = +� +0 +g +−g 0 +� +. +(9) +By denoting the single-mode CM of the vacuum state as +V(j) +vac = diag (1, 1) /2, one readily obtains +D = +3 +� +j=1 +κjV(j) +vac, Vss = +3 +� +j=1 +V(j) +vac, +(10) +the latter corresponding to the CM of any product of +three coherent states. Since the jump operators here do +not promote incoherent displacements, the steady state +is actually the three-mode vacuum state |0⟩1|0⟩2|0⟩3 as +long as all κj > 0. +A. +Characterization of exceptional points +Finding the EPs directly from the spectrum of Γ may +be challenging as one needs to solve a 2N:th degree poly- +nomial equation, or in the studied case, a sextic equation. + +4 +However, owing to the absence of counter-rotating terms +in the form of ˆH here, the characterization of EPs can be +simplified to the study of the dynamical equation for the +3×3 vector a = (⟨ˆa1⟩, ⟨ˆa2⟩, ⟨ˆa3⟩)⊤. By moving Eq. (3) to +a frame rotating with ω1, one can obtain ˙a = −iHa, with +H having the role of an effective non-Hermitian Hamil- +tonian. Explicitly, we have +H = +� +� +� +−i κ1 +2 +g +0 +g +δ2 − i κ2 +2 +g +0 +g +δ3 − i κ3 +2 +� +� +� , +(11) +where δ2 = ω2 − ω1 and δ3 = ω3 − ω1 are frequency +detunings. +Without loss of generality, we assume that the param- +eters g, ω1, and κ1 are fixed. Thus it is convenient to +express the parameters of R2 and R3 with respect to +those of R1. We proceed with this parametrization using +complex-valued parameters {εk} such that for k = 2, 3, +we have +δk(εk) = +√ +2gIm(εk), +κk(εk) = κ1 + 2 +√ +2gRe(εk). (12) +As detailed in Appendix A, degeneracies in the spectrum +of H appear provided that +f(ε) = 1 +2 +� +�ε ± +� +ε4 + 10ε2 − 2 ± 2 (1 + 2ε2) +3 +2 +ε2 +� +� , +(13) +where ε = ε3 and f(ε) = ε2. Note that the complex- +valued function f(ε) presents four branches indicated by +the signs ‘±’ as shown for a purely real ε in Fig. 2a. +At the degeneracies of H, such a matrix has at most +two distinct eigenvalues, from which the effective detun- +ings and decay rates of the normal modes are extracted as +δeff +j (ε) = +√ +2gIm[hj(ε)] and κeff +j (ε) = κ1 + 2 +√ +2gRe[hj(ε)] +(Appendix A), where +h1(ε) = f 3 − εf 2 − (ε2 + 4)f + ε3 + ε/2 +f 2 − εf + ε2 − 3 +, +h2(ε) = h3(ε) = 1 +4 +�2εf 2 + 2(ε2 + 1)f − 7ε +f 2 − εf + ε2 − 3 +� +, +(14) +and we write f = f(ε) for brevity. Consequently, the +degenerate eigenvalues of Γ are given by the pairs (Ap- +pendix A) +λ± +sj(ε) = −κeff +j (ε) +2 +± i +� +ω1 + δeff +j (ε) +� +, +(15) +which coincide at an EP–3. +The rich structure of the +decay rates and frequencies of the normal modes is shown +in Fig. 2b for a purely real ε. +Without imposing further restrictions, the considered +open system presents six EP–3, two of which are ob- +tained for ε = 2f(ε) = ±2, so that all modes are de- +generate, κ2 = κ1 ± 2 +√ +2g, and κ3 = κ1 ± 4 +√ +2g. These +cases correspond to the square-root singularity of f(ε) +- 4 +- 2 +0 +2 +4 +- 4 +- 2 +0 +2 +4 +- 4 +- 2 +0 +2 +4 +(a) +- 4 +- 2 +0 +2 +4 +- 4 +- 2 +0 +2 +4 +- 4 +- 2 +0 +2 +4 +(b) +Figure 2. +Exceptional-point engineering for a linear chain +of three lossy resonators with degenerate angular frequencies +ω1 = ω3, expressed by a purely real parameter ε = ε3, see +Eqs. (12)–(14). +(a) Decay rate (top panel) and frequency +(bottom panel) offsets of resonator R2 as functions of the +decay rate offset of resonator R3, expressed by the complex- +valued function f(ε) = ε2 defined in Eq. (13). (b) Effective +decay rate (top) and effective frequency (bottom) offsets of +the eigenmodes of the system as functions of the decay rate +offset of resonator R3, expressed by the complex-valued func- +tions hj(ε) defined in Eq. (14). All offsets are given with re- +spect to the parameters of resonator R1. In (b), solid (dashed) +curves represent the single (double) roots of the characteristic +polynomial of H. In all cases, the labels ++, −+, +−, and +−− indicate the four branches of f(ε) obtained from the cor- +responding selection of signs in Eq. (13). The vertical dashed +lines in all panels highlight the values of ε producing EP–3. +The shaded area in (a) indicates the relevant region of the +Re(ε3)–Re(ε2) parameter space for this work. +and are highlighted in Fig. 2. The remaining four EP– +3 are obtained with f(ε) = (±3 +√ +3 ± i)/(2 +√ +2), and +ε = 2i Im[f(ε)] = ±i/ +√ +2, thus requiring equal decay +rates for R1 and R3, κ2 = κ1 ± 3 +√ +3g, in addition to +the detunings δ2 = ±g/2 and δ3 = ±g. The degeneracy +map for such cases is shown in Fig. 6 of Appendix A for +completeness. +All other cases expressed through Eqs. (13) and (14) +are associated to EP–2. Our numerical tests show the +coalescence of eigenvectors of H following the branches +f(ε), indeed indicating open-system degeneracies. The +Jordan decompositions of Γ yielding polynomial-in-time +features of the dynamics are shown in Appendix B for +relevant EPs in this work. +We emphasize that the experimental feasibility of EP +engineering in the present model is strongly dependent +on the physical limitations of the setup. For instance, to +obtain the four instances of EP–3 with non-degenerate +frequencies, one needs frequency detunings of the order +of g/(2π), which are typically much smaller than the fre- +quency of superconducting resonators themselves [54]. +Hereafter, we restrict our discussion to degenerate res- + +5 +onators, i.e., Im(ε) = Im[f(ε)] = 0. By also considering +κ1 as the smallest decay rate, another restriction for ob- +taining EPs is imposed, such that both Re(ε) ≥ 0 and +Re[f(ε)] ≥ 0. +In this case, the only allowed branches +of f(ε) are ‘+−’ and ‘−−’ for ε ≥ 2, and ‘++’ for +ε ≥ 0, see the shaded region in Fig. 2a. In particular, +the branch ‘++’ at ε = 0 yields weakly dissipative nor- +mal modes, with one of them decaying according to κ1, +see Fig. 2b and Eq. (14). This behavior suggests that a +quasi-stabilization of some properties of the system can +be obtained with the combination of a small κ1 and a +proper choice of the EP, as explored in detail in Sec. IV. +B. +Single-mode squeezing and bipartite +entanglement +Below, we specifically investigate single-mode squeez- +ing and bipartite entanglement for the three-resonator +system. For Gaussian evolution, these quantities can be +addressed directly from the specific partitions of the total +CM +V = +� +� +� +V(1) +C(12) +C(13) +C(12)⊤ +V(2) +C(23) +C(13)⊤ C(23)⊤ +V(3) +� +� +� , +(16) +where V(j) is the reduced CM of resonator Rj and C(jk) +is the intermodal correlation matrix between resonators +Rj and Rk [53]. +Since all single-mode Gaussian states can be written as +squeezed thermal states apart from local displacements, +the components of the reduced CM of resonator Rj can +be cast into the form [60] +V(j) +11 = ( ¯Nj + 1/2)[cosh(2rj) + sinh(2rj) cos φj], +V(j) +22 = ( ¯Nj + 1/2)[cosh(2rj) − sinh(2rj) cos φj], +V(j) +12 = ( ¯Nj + 1/2) sinh(2rj) sin φj, +(17) +where rj and φj are real-valued quantities defining the +squeezing parameter ξj = rjeiφj and ¯Nj is the effective +thermal occupation number of resonator Rj. As a con- +sequence, one can extract rj and ¯Nj as +rj = 1 +2 sinh−1 +� +� +� +(V(j) +11 − V(j) +22 )2 + 4V(j)2 +12 +2( ¯Nj + 1/2) +� +� , +¯Nj = +� +det V(j) − 1 +2, +(18) +and the single-mode purity is readily given by Pj = +(2 ¯Nj + 1)−1. +While bipartite entanglement can be quantified by the +reduced von Neuman entropy given a pure state of the +complete system [61], an entanglement measure for mixed +states is not uniquely defined [62]. Here, we focus on the +concept of logarithmic negativity [63], which is based on +the Peres–Horodecki separability criterion [64, 65] and +fulfills the conditions for an entanglement monotone [66]. +Given Eq. (16) and considering the subsystems Rj and +Rk (j < k), one can write their joint CM as +V(jk) = +� +V(j) +C(jk) +C(jk)⊤ +V(k) +� +. +(19) +For Gaussian states, the logarithmic negativity, Ejk, can +then be computed as [63, 67] +Ejk = max[0, − log2(2˜ν− +jk)], +(20) +where ˜ν− +jk = { ˜∆jk − [ ˜∆2 +jk − 4 det V(jk)] +1 +2 } +1 +2 / +√ +2 being +the smallest symplectic eigenvalue of ˜V(jk), which cor- +responds to the two-mode CM obtained after the Peres– +Horodecki partial transposition of the associated bipar- +tite density matrix, and ˜∆jk = det V(j) + det V(k) − +2 det C(jk) [65, 67]. The inequality ˜ν− +jk ≥ 1/2 is a neces- +sary and sufficient condition for separability of bipartite +Gaussian systems of two modes [65, 67]. +IV. +QUASI-STABILIZATION OF SQUEEZING +AND ENTANGLEMENT +In this section, we study the propagation of single- +mode squeezing and bipartite entanglement in the open +quantum system of Fig. 1b. The initial state is chosen +as |0⟩1|0⟩2 ˆS3(r)|0⟩3, where ˆS3(r) = exp +� +r(ˆa†2 +3 − ˆa2 +3)/2 +� +is the single-mode squeezing operator of R3 and r ≥ 0. +Such a state has the CM +V0 = 1 +2diag +� +1, 1, 1, 1, e2r, e−2r� +, +(21) +which indicates that the variances of R3 are initially mod- +ified by the factors e±2r. We employ Eq. (5) to numeri- +cally obtain the 6 × 6 time-evolved CM V(t) at different +points of the parameter space. Here, we set κ1 = κ3 as +the smallest decay rates of the system and test different +κ2 = κ1 + 2 +√ +2g Re(ε2) with Re(ε2) ≥ 0 and Im(ε2) = 0. +Within these conditions, the only allowed EP–branch is +‘++’ so that an EP–2 is produced at f(ε = 0) = 2, see +Eq. (13) and Fig. 2a. +In Figure 3a, we observe the emergence of squeezed +thermal states for resonator R1 and bipartite quantum +correlations expressed through the logarithmic negativ- +ity E13, with a clear passage from underdamped to over- +damped dynamics with increasing κ2. The squeezing de- +gree of R2 along with the logarithmic negativities E12 +and E23 (data not shown) is rapidly suppressed for large +ratios κ2/g. On the other hand, the small values of κj/g, +j = 1, 3, help to delay the decay of the system towards the +three-mode vacuum state, and this quasi-stability tends +to be achieved faster near the critical-damping regime +produced by the EP–2. Such a behavior is not present at +the EP–2 if R1 is directly connected to R3, which reduces + +6 +0.0 +0.2 +0.4 +0.6 +0.0 +0.1 +0.2 +0.0 +0.2 +0.4 +0 +4 +8 +12 +16 20 +0.0 +0.2 +0.4 +0.6 +0.0 +0.4 +0.8 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +(a) +(b) +Figure 3. (a) Dynamics of the squeezing parameter r1 and +effective thermal occupation ¯ +N1 of resonator R1 and the loga- +rithmic negativity between R1 and R3, E13, for the indicated +values of the damping rate of R2, κ2/g. +The shown data +correspond to a crossover from underdamped to overdamped +dynamics, with critical damping at κ2/g = 5.658. The fre- +quencies of the resonator modes are chosen as ω1/g = ω2/g = +ω3/g = 5000, the other damping rates as κ1/g = κ3/g = +10−3, and the initial squeezing parameter of R3 as r = 1. +The corresponding values of ε2 as defined in Eq. (12) are +ε2 = 0.2 (gray curves), ε2 = 2.0 (blue curves) and ε2 = 4.0 +(red curves). +In the chosen parameter regime, the results +are essentially independent of the resonator–resonator cou- +pling strength g. (b) Maximum achieved quantities in tem- +poral evolutions corresponding to (a) at the critical damping +as functions of the initial squeezing parameter r for selected +values of κ1/g. In all panels, dashed lines correspond to long- +time values in the limit κ1/g → 0, see Eq. (22). +the dimension of the system to N = 2. In such a case, +the only two normal modes of the system decay at equal +rates [25]. +The maximum achieved values of r1, ¯N1, and E13 as +functions of the initial squeezing parameter r for the sys- +tem dynamics at the EP–2 are shown in Figure 3b. Their +values in the limit κj → 0, j = 1, 3, and t → ∞, can be +estimated directly from Eqs. (18) and (20) with the help +of the Jordan decomposition of Γ shown in Appendix B. +One readily obtains r⋆ +2 = ¯N ⋆ +2 = E⋆ +12 = E⋆ +23 = 0, whereas +r⋆ +1 = r⋆ +3 = 1 +2 log +� +3 + e2r +� +10 + 6 cosh(2r) +� +, +¯N ⋆ +1 = ¯N ⋆ +3 = 1 +8 +�� +10 + 6 cosh(2r) − 4 +� +, +E⋆ +13 = 1 +2 +� +1 − log2(1 + e−2r) +� +. +(22) +The superscripts ‘⋆’ in Eqs. (22) indicate that such quan- +tities are bounds for the quasi-stabilized states, shown as +dashed lines in Fig. 3. Interestingly, we can generate en- +tanglement between resonators R1 and R3 although the +entanglement with resonator R2 is rapidly suppressed. +From Fig. 3b and Eqs. (22), we observe that the +squeezing splitting increases linearly with r for r ≪ 1, +where thermal occupancy is insignificant. The squeezing- +splitting capacity r⋆ +1/r and the degree of entanglement +between R1 and R3 tend to saturate to 1/2 in the limit +r → ∞ with the expense of also thermally populat- +ing these resonators. Using the decibel scale defined by +r = 10 log10(e2r) dB [68], an initial amount of squeezing +r ≈ 3 dB is roughly converted into squeezed states with +r⋆ +1 = r⋆ +3 ≈ 0.772 dB and purities P⋆ +1 = P⋆ +3 ≈ 0.997, with +E⋆ +13 ≈ 0.207. Despite producing a faster decay towards +the actual steady state of the system, an increase of two +orders of magnitude in κ1/g does not provide significant +differences in the maximum quantities for small r. +To further address the quasi-stabilization of entangle- +ment and squeezing transferred to R1 for different κ2, +we diagonalize Eq. (11) to obtain the effective frequency +detunings and decay rates of the system as shown in +Figs. 4a and 4b, respectively. +For κ1 ≪ κ2, we ob- +tain two eigenmodes with frequency detunings δeff +± +≈ +± Im( +� +κ2 +2 − 32g2)/4 and dissipation rates κeff +± ≈ κ2/2 ± +Re( +� +κ2 +2 − 32g2)/2, which coalesce at κ2 ≈ 4 +√ +2g. The +frequency detuning δeff +0 = 0 and dissipation rate κeff +0 = κ1 +are preserved, thus indicating that one of the eigenmodes +remains hidden from the dissipation of resonator R2. +Since clearly the speed of quasi-stabilization for the +squeezing and entanglement of resonator R1 depend on +κ2 (Fig. 3a) and since κeff ++ ≥ κeff +− , we conclude that the +time scale for this quasi-stabilization is roughly given by +1/κeff +− ≈ 2/[κ2 − Re( +� +κ2 +2 − 32g2)]. To arrive at a more +accurate expression for the quasi-stabilization time, we +first fit functions of the form +rfit +1 (t) = r⋆ +1 +2 e−yr1κ1t � +e−κeff +− t/2 � +1 − 3 cos +� +δeff +− t +�� ++ 2 +� +, +Efit +13(t) = E⋆ +13e−yE13κ1t � +1 − e−κeff +− t cos2(δeff +− t) +� +, +(23) +to time traces similar to those in Fig. 3a and find yr1 ≈ +0.75 and yE13 ≈ 1.3. Although these functions neglect the +polynomial-in-time solution at the EP–2, they capture +the main features of the over and underdamped dynam- +ics, and hence are accurate enough from our following +analysis. +Next, we define the quasi-stabilization time tα as the +earliest time instant after which the quantity α = r1, E13 +stays within an uncertainty σα from the ideal value +α⋆e−yακ1tα, where we take into account also the slow +decay of the maximum attainable value owing to finite +κ1. More precicely, +tα = min{t|α⋆e−yακ1t − ˜α(t) ≤ σα}, +(24) +where ˜α(t) is the lower envelope of the possibly oscillating +α(t). +Note that by this definition, ˜α(t) = α(t) in the +critically and overdamped dynamics. + +7 +- 1.5 +- 1.0 +- 0.5 +0.0 +0.5 +1.0 +1.5 +0 +2 +4 +6 +8 +2 +4 +6 +8 +5 +10 +15 +20 +25 +30 +35 +40 +(a) +(b) +(c) +Figure 4. +(a) Effective frequency detunings and (b) effec- +tive decay rates of the eigenmodes of the coupled system as +functions of the decay rate of resonator R2, κ2, in units of +the resonator–resonator coupling strength g. (c) Time tα to +yield quasi-stable squeezing (filled circles, α = r1) and en- +tanglement (filled squares, α = E13) within an uncertainty +σα = 10−5, see main text. The dashed lines represent cor- +responding results from the fit functions of Eq. (23). In all +panels, the parameters are chosen as in Fig. 3a and the colored +regions separate the underdamped from the overdamped dy- +namics, with critical damping at κ2/g = 5.658, corresponding +to an EP–2. +In Fig. 4c, +we show the behavior of the quasi- +stabilitation time tα on the dissipation rates κ2 for an +error σα = 10−5 as obtained from the solutions of the +temporal evolution of the system similar to those in +Fig. 3a. The shortest quasi-stabilization times are ob- +tained in the vicinity of the EP–2 owing to the peak in +κeff +− illustrated in Fiq. 4b. Using the lower envelopes of +the fitting functions (23) in Eq. (24), one can estimate +the quasi-stabilization time as +tα ≈ +log +� +α⋆ +σα +� +yακ1 + zακeff +− +, +(25) +with zr1 ≈ 0.5 and zE13 ≈ 1. Therefore, tα tends to scale +logarithmically with the desired error. +V. +FAST RESET NEAR EXCEPTIONAL +POINTS +As the final application of EPs, we discuss the reset +of the resonator chain to its ground state |0⟩1|0⟩2|0⟩3. +Typically, stronger dissipation leads to faster decay, but +of course in our system where the coupling between the +different resonators is weak compared with the excitation +frequencies of the bare resonators, the critical dynamics +plays an important role. Similar features are prone to +arise in a quantum register of several coupled qubits. +To quantitatively study the accuracy of the reset, we +define the infidelity +Iss(ˆρ) = 1 − Fss(ˆρ), +(26) +where Fss(ˆρ) = ⟨0|1⟨0|2⟨0|3ˆρ|0⟩1|0⟩2|0⟩3 is the overlap +probability between an arbitrary three-mode state ˆρ and +the ground state. For multimode Gaussian states with +null mean vector ⟨ˆx⟩, Fss can be directly computed from +the covariance matrix V, which for the present case be- +comes [3] +Fss = +1 +� +det (V + Vss) +, +(27) +where Vss given in Eq. (10). +An optimized reset is +achieved with the set of free parameters producing the +fastest decay to the ground state, i.e., the minimal Iss in +a given time. +Figure 5 shows the reset infidelity for different param- +eter values and for an initial state which is obtained by +waiting for a preparation time τs at EP–2 after squeezing +the vacuum at resonator R3 by a finite r. Note that if +τs = 0, one has the initial squeezed state with the covari- +ance matrix given by Eq. (21), and with τs = 8/g, one +prepares an initial state with entanglement and squeez- +ing split between R1 and R3, see Fig. 3a. In Fig. 5a, we +show the dependence of Iss on the decay rates κ2 and κ3 +in the region corresponding to the shaded area in Fig. 2a +for the above-mentioned preparation times and immedi- +ately following reset times τr. Although the regions of +low infidelity are relatively broad if all squeezing is con- +centrated in R3, so that no entanglement is present, we +observe a narrowing of such regions if τs = 8/g. These +regions tend to cover the EP–3 and follow the real com- +ponents of the ‘−−’ branch of f(ε3) as ε3 is increased. +Such a feature is even more prominent for long reset times +naturally leading to lower reset infidelities. Note from +Fig. 2b that this branch tends to produce highly dissi- +pative normal modes for ε3 > 2. In contrast, at least +one decay rate produced by the ‘+−’ and ‘++’ branches +is slow even with increasing ε3, rendering such branches +less favorable for the reset. +Figure 5b shows the reset infidelity Iss as a function +of the reset times τr at the EP–3 for different initial +states. In all displayed cases, low infidelities Iss are in- +deed achieved beyond τr ∼ 6/g, owing to the exponential +dependence on τr. For such reset times, the distribution + +8 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +2 +4 +6 +8 +10 +10- 7 +0.0 +0.5 +1.0 +1.5 +2.0 +10- 8 +10- 6 +10- 4 +10- 5 +10- 3 +10- 1 +10- 9 +10- 2 +(a) +EP-3 reset +10- 1 +10- 2 +10- 3 +10- 4 +10- 5 +(b) +(c) +Figure 5. +(a) Reset infidelity Iss of degenerate resonators +as a function of the dimensionless decay rate offsets Re(ε3) +and Re(ε2) for selected choices of preparation times τs (top +and bottom panels) and reset times τr (left and right panels). +During the time-interval τs, the system is set at the EP–2 +with Re(ε3) = 0 and Re(ε2) = 2. +Solid curves on top of +the contour plots show the components of the EP branches +‘++’ (blue), ‘+−’ (gray), and ‘−−’ (green) in the Re(ε3)– +Re(ε2) parameter space as in the shaded region of Fig. 2a, +with EP–3 indicated by dashed circles. The other parameters +are ωj/g = 5000, j = 1, 2, 3, κ1/g = 10−3, and r = 1. (b) +Reset infidelity Iss at the EP–3 as a function of reset times τr +(in units of g−1) for different preparation times τs and decay +rate κ1/g = 10−3. Solid (dashed) curves show data for r = 1.0 +(r = 2.0). (c) Reset infidelity Iss at the EP–3 as a function +of squeezing parameter r for different reset times τr and for +preparation time τs = 8/g. Solid (dashed) curves show data +for κ1/g = 10−3 (κ1/g = 10−1). The remaining parameters +are chosen as in (a). +of squeezing and entanglement tends to have a minor rel- +ative effect on the reset performance. This is in stark +contrast with the short-reset-time cases, where the decay +towards the ground state tends to significantly acceler- +ate if all initial squeezing is poorly distributed, remaining +mostly in R3. We observe that the reset performance is +degraded for small ratios of κ1/g and for increasing ini- +tial squeezing parameters as displayed in Fig. 5c. In such +scenarios, for a finite reset time, the infidelity tends to +grow asymptotically to unity in the limit r → ∞. +VI. +DISCUSSION +We observed that fast generation of entanglement and +propagation of squeezing in a linear chain of three su- +perconducting resonators may benefit from the detailed +understanding of critical damping in the system. Here, +the highly dissipative resonator R2 acts as an incoherent +entanglement generator and squeezing splitter with the +cost of reducing the purity of the local states through +the increase of their effective temperatures. The role of +critical damping towards stabilization has also been ac- +knowledged recently in an autonomous quantum thermal +machine with two qubits [69]. +The stabilization of squeezed states through reservoir +engineering in superconducting circuits has been recently +reported in +[12]. We highlight that the scheme in our +paper differs from typical two-mode squeezing opera- +tions, since it arises from the combination of dissipa- +tion and only a single-mode squeezing source available +in the beginning of the dynamics, thus being also dis- +tinct from conventional reservoir-engineering protocols. +On the other hand, we do not need continuous driving +terms since the structure of couplings and dissipation of +the system promote a separation of time scales for the +decay of the normal modes. +We explicitly show that +this can be beneficial if fine-tuning κ2 near a particular +EP–2 instead of only roughly assuming the conditions +κj ≪ κ2, g, for j = 1, 3. +The results shown in Figs. 3 and 4 also suggest that +concatenating similar structures can be used for fast and +stable distribution of entanglement to every other node +in a photonic network. Although, spoiling Gaussian fea- +tures of the system [70, 71], entanglement distillation +protocols [72] may be used in such cases to increase the +amount of entanglement shared by the nodes. Particu- +lar low-order EPs of high-dimensional systems may be +used to speed up the generation of quasi-stable states, +and hence they may have potential use cases in quantum +protocols, although the open-system-degeneracy map in +such cases becomes more intricate. +Regarding the unconditional dissipative reset of the +system, the role of critical damping becomes more evi- +dent. Here, the region near the EP–3 and also following +a particular EP–2 branch is a reasonable choice of param- +eters to produce a substantial performance enhancement +of the reset. Since the covariance matrices of the vacuum +state and a product of coherent states are identical, such +regions in the parameter space could also be used to pro- +mote unconditional fast stabilization of coherent states +with a proper inclusion of driving terms in the system +Hamiltonian. +Let us present typical experimental parameters of the +circuit 1b that could reproduce the findings of this work. + +9 +For a resonance frequency of ω/(2π) = 5.0 GHz, the sim- +ulated values of coupling strength and lowest decay fre- +quencies are g/(2π) = 1.0 MHz and κ1/(2π) = 1.0 kHz, +respectively. +Such resonance frequency and coupling +strength has been conveniently experimentally achiev- +able for longer than a decade, and the quality factor +of five million implied by the lowest decay rate can +be achieved with state-of-the-art fabrication techniques. +The EP–2 used for stabilization is thus achieved with +κ2/(2π) ≈ 5.66 MHz and κ3/(2π) = 1.0 kHz, while the +EP–3 with κ2/(2π) ≈ 2.83 MHz and κ3/(2π) ≈ 5.66 +MHz. Even though the almost four-orders-of-magnitude +tunability required to interchange between this particu- +lar EP–2 and the EP–3 may be technically challenging, +the maximum achievable decay rates with the QCR are +beyond the ones considered here and their demonstrated +on/off ratios are close to these requirements [23]. +VII. +CONCLUSIONS +We demonstrated the theory of exceptional-point- +related phenomena for continuous-variable systems de- +scribed entirely by their second moments, consequently +capturing different non-classical features and non-locality +largely neglected in previous work. +For a linear chain +of three lossy superconducting resonators, we analyti- +cally obtained its open-system-degeneracy map and ob- +served that different parameter sets yielding different ex- +ceptional points can be used to identify sweet spots for +the optimization of squeezing propagation, entanglement +generation, and reset. +More precisely, we assessed the role of critical dynamics +for dissipative state synthesis by numerically simulating +the temporal evolution of the covariance matrix of the +system. The region of the parameter space considered in +the simulations is physically motivated by recent experi- +mental advances in dissipation-tunable devices embedded +to superconducting circuits. +We found that the quasi-stabilization into mixed bipar- +tite entangled states generated from an initially squeezed +resonator R3 is optimized in the vicinities of a particu- +lar low-dissipative EP–2 produced with symmetric de- +cay rates of resonators R1 and R3 [see the ‘++’ branch +of f(ε) in Eq. (13)]. +In such scenarios, one observed +that the time scale for this quasi-stabilization is mini- +mum for κ2 ≈ 4 +√ +2g and κ1, κ3 ≪ κ2. Using the Jor- +dan decomposition of the dynamical matrix, we obtained +analytical bounds for the maximum achievable quasi- +stable squeezing-splitting capacity and logarithmic neg- +ativity. +Remarkably, all residual squeezing of the cen- +tral resonator is removed within the quasi-stabilitization +time-scales, and consequently, the choice of EP–2 also +quickly removes the entanglement of R2 with the other +resonators. +Furthermore, we investigated the dissipative reset of +such non-classical states to the ground state. The region +in the parameter space producing the lowest reset infi- +delities at given reset times τr requires asymmetric res- +onator decay rates and tend to follow a particular high- +dissipative EP branch, which includes the physically at- +tainable EP–3 [see the ‘−−’ branch of f(ε) in Eq. (13)]. +In this EP–3 case, the distribution of the initial squeezing +into the different resonators tends to become irrelevant +for the reset performance beyond τr ∼ 6/g. +In conclusion, this work paves the way for a deep un- +derstanding of the role of exceptional points in multimode +continuous-variable systems, with potential applications +in quantum technology such as in using dissipation as +an ingredient for fast transfer of desired quantum prop- +erties. +For example, heat engines [73] operating with +nonequilibrium reservoirs [74] and presenting quantum +resources [75] arise as systems with promising near-term +opportunities. Moreover, the investigation of exceptional +points in such superconducting systems through involved +models, see e.g. [76], is also a potential future line of re- +search. 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Research 3, +033126 (2021). +Appendix A: Explicit determination of EPs +Here, we show the characterization of EPs for the open +quantum system presented in Sec. III, namely, a linear +chain of three resonators. Given fixed coupling constants +g and parameters of resonator R1, we aim at finding the +parameters of resonators R2 and R3 that produce EPs. +Assuming resonance frequencies ωj > 0 and decay +rates κj > 0, for j = 1, 2, 3, we move the Lindblad mas- +ter equation (3) to a frame rotating with ω1 so that we +obtain Eq. (11) of the main text +H = +� +� +� +−i κ1 +2 +g +0 +g +δ2 − i κ2 +2 +g +0 +g +δ3 − i κ3 +2 +� +� +� , +(A1) +where δ2 = ω2 − ω1 and δ3 = ω3 − ω1. Below, we use the +parametrizations +δ2 = +√ +2gIm(ε2), +δ3 = +√ +2gIm(ε3), +κ2 = κ1 + 2 +√ +2gRe(ε2), +κ3 = κ1 + 2 +√ +2gRe(ε3), (A2) +so that the effective offsets from ω1 and κ1 are given +according to the imaginary and real parts of the com- +plex parameters ε2 and ε3. +This allows one to write +the characteristic polynomial associated to H, P(x) = +ax3 + bx2 + cx + d, with the coefficients +a =1, +b = i +2[3κ1 + 2 +√ +2g(ε2 + ε3)], +c = − 3κ2 +1 +4 +− +√ +2gκ1(ε2 + ε3) − 2g2(1 + ε2ε3), +d = − i +8[κ3 +1 + 8 +√ +2g3ε3 + 2 +√ +2gκ2 +1(ε2 + ε3) ++ 8g2κ1(1 + ε2ε3)]. +(A3) +Given the cubic discriminant +∆ = −(4v3 + 27w2), +(A4) +where +v = 3ac − b2 +3a2 +, +w = 2b3 − 9abc + 27a2d +27a3 +, +(A5) +we search for degeneracies in the spectrum of H, which +occur when ∆ = 0. Interestingly, the appearance of EPs +depends only on the relationship between ε2 and ε3 given +by the condition +4ε4 +2ε2 +3 − 8ε3 +2ε3 +3 + 4ε2 +2(ε4 +3 − 5ε2 +3 + 1) ++ 4ε2(5ε3 +3 − ε3) − 8ε4 +3 + 13ε2 +3 − 16 = 0. +(A6) +Solving Eq. (A6) for ε2 yields the four branches +ε2 = 1 +2 +� +�ε3 ± +� +ε4 +3 + 10ε2 +3 − 2 ± 2 (1 + 2ε2 +3) +3 +2 +ε2 +3 +� +� , +(A7) +where the signs ‘±’ can be chosen independently. +To identify the order of the EPs, we inspect Eq. (A5) +and (A7) more carefully. +All EP–3 correspond to the +triple root of P(x) = 0, which is obtained when v = w = +0. First setting v = 0 reduces ε2 in Eq. (A7) to +ε2 = 1 +2 +� +ε3 ± +� +12 − 3ε2 +3 +� +, +(A8) +and imposing w = 0 yields ε3 ∈ {±2, ±i/ +√ +2}. Hence, +the studied open system presents six distinct EP–3, two +of which are produced when all resonators are degen- +erate such that ε3 = 2ε2 = ±2. +The remaining four +EP–3 are obtained with ε2 = (±3 +√ +3 ± i)/(2 +√ +2) and +ε3 = 2iIm(ε2) = ±i/ +√ +2, thus requiring frequency shifts +from the resonance and equal decay rates for R1 and R3. +By defining ε = ε3, the parameters of R2 produce EPs +provided that they are chosen as ε2 ≡ f(ε), with f(ε) +defined in Eq. (13) of the main text. When degeneracies +of H are present, we express the complex roots of P(x) = +0 as +x1 = 4abc − 9a2d − b3 +a(b2 − 3ac) +, +x2 = x3 = +9ad − bc +2(b2 − 3ac), (A9) + +12 +- 4 +- 2 +0 +2 +4 +- 4i - 2i +0i +2i +4i +- 4 +- 2 +0 +2 +4 +(a) +- 4 +- 2 +0 +2 +4 +- 4i - 2i +0i +2i +4i +- 4 +- 2 +0 +2 +4 +(b) +Figure 6. Open-system-degeneracy map for a linear chain of +three coupled resonators with degenerate decay rates κ1 = κ3, +which are expressed by a pure imaginary parameter ε. De- +pendence of (a) f(ε) and (b) hj(ε) on ε. In (b), solid (dashed) +curves represent the single (double) root of the characteristic +polynomial of H. In all cases, the labels ++, −+, +−, and +−− indicate the four branches of f(ε) obtained from the cor- +responding selection of signs in Eq. (13). The vertical dashed +lines in all plots highlight the values of ε yielding EP–3. +to extract the effective detunings and decay rates of the +normal modes as given in Eq. (14). In Fig. 6, we show +the rich structure of the branches yielding the EPs for a +purely imaginary ε. +Let us comment on the relationship between the spec- +trum of the non-Hermitian Hamiltonian H defined in +Eq. (11) and the dynamical matrix Γ defined in Eq. (8). +First, we note that Γ also determines the temporal evo- +lution of the mean vector x = ⟨ˆx⟩ introduced in Sec. II. +We define the 6 × 6 vector of ladder operators, ˆA = +[ˆa, (ˆa†)⊤]⊤, where ˆa = (ˆa1, ˆa2, ˆa3)⊤, ˆa† = (ˆa† +1, ˆa† +2, ˆa† +3). +In the Schrodinger picture, the vector of expectation +values A = ⟨ˆA⟩ is obtained from the dynamical equa- +tion ˙A = diag[−i(H + H1), i(H∗ + H∗ +1)]A, where H1 = +ω1I3. +By introducing the vector x′ = (q, p)⊤, where +q = (⟨ˆa⟩ + ⟨ˆa⟩∗)/ +√ +2 and p = −i(⟨ˆa⟩ − ⟨ˆa⟩∗)/ +√ +2, one +verifies that x′ is related to A through a unitary trans- +formation so that x′ = ΛA, where +Λ = +1 +√ +2 +� +I3 +I3 +−iI3 iI3 +� +. +(A10) +Consequently, the dynamical equation for x′ becomes +˙x′ = Γ′x′, where +Γ′ = Λ +� +−i(H + H1) +03 +03 +i(H∗ + H∗ +1) +� +Λ†. +(A11) +Since the vectors x′ and x are equivalent except for the +different orderings, the spectrum of Γ′ coincides with that +of Γ, which in the case of open-system degeneracies is +given by the eigenvalues defined in Eq. (15) of the main +text. +Appendix B: Jordan normal form +In this appendix, we present the explicit Jordan nor- +mal form of Γ defined in Eq. (8) at some relevant EPs +considered in this work. +EP–3. +We start with the EP–3 used for the reset of +the system, i.e., the one produced with degenerate res- +onators (ω1 = ω2 = ω3 = ω), κ2 = κ1 + 2 +√ +2g, and κ3 = +κ1 + 4 +√ +2g. Using the notation introduced in Sec. II, the +Jordan blocks in the matrix J = diag +� +J− +s1(λ− +s1), J+ +s1(λ+ +s1) +� +, +and the non-singular matrix P read +J± +s1(λ± +s1) = +� +� +� +λ± +s1 +1 +0 +0 +λ± +s1 +1 +0 +0 +λ± +s1 +� +� +� , +s1 = (1, 3), +(B1) +P = +� +� +� +� +� +� +� +� +� +� +−i +− i +√ +2 +g +− i +g2 +i +i +√ +2 +g +i +g2 +−1 +− +√ +2 +g +− 1 +g2 +−1 +− +√ +2 +g +− 1 +g2 +− +√ +2 +− 1 +g +0 +− +√ +2 +− 1 +g +0 +i +√ +2 +i +g +0 +−i +√ +2 +− i +g +0 +i +0 +0 +−i +0 +0 +1 +0 +0 +1 +0 +0 +� +� +� +� +� +� +� +� +� +� +, +(B2) +where λ± +s1 are given as in Eq. (15) of the main text with +ε = 2 and f(ε) = 1. Consequently, one obtains +eJt = +� +m=∓ +eλm +s1t +� +� +� +1 t t2/2 +0 1 +t +0 0 +1 +� +� +� . +(B3) +EP–2. +Let us consider the EP–2 used for the quasi- +stabilization of squeezing and entanglement, i.e., the one +obtained with degenerate resonators, κ2 = κ1 + 4 +√ +2g, +and κ3 = κ1. The Jordan blocks here defining the matrix +J = diag +� +J− +s1(λ− +s1), J− +s2(λ− +s2), J+ +s1(λ+ +s1), J+ +s2(λ+ +s2) +� +are +J± +s1(λ± +s1) = λ± +s1, +s1 = (1, 1), +J± +s2(λ± +s2) = +� +λ± +s2 +1 +0 +λ± +s2 +� +, +s2 = (2, 2), +(B4) +and the singular matrix P reads +P = +� +� +� +� +� +� +� +� +� +� +−i +i +0 +i +−i +0 +−1 +1 +0 +−1 +1 +0 +0 +√ +2 +− 1 +g +0 +√ +2 − 1 +g +0 +−i +√ +2 +i +g +0 +i +√ +2 − i +g +i +i +0 +−i +−i +0 +1 +1 +0 +1 +1 +0 +� +� +� +� +� +� +� +� +� +� +, +(B5) + +13 +where λ± +sj are given as in Eq. (15) of the main text with +ε = 0 and f(ε) = 2. In this case, the Jordan decomposi- +tion of Γ gives rise to +eJt = +� +m=∓ +� +� +� +eλm +s1t +0 +0 +0 +eλm +s2t eλm +s2tt +0 +0 +eλm +s2t +� +� +� . +(B6) +This decomposition is employed in Eq. (6) to analytically +obtain the time-evolved covariance matrix V and, conse- +quently, the theoretical bounds presented in Sec. IV. + diff --git a/qdFRT4oBgHgl3EQfejci/content/tmp_files/load_file.txt b/qdFRT4oBgHgl3EQfejci/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa8d51ae037443665355253d7ffa19ef9f4db467 --- /dev/null +++ b/qdFRT4oBgHgl3EQfejci/content/tmp_files/load_file.txt @@ -0,0 +1,1251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf,len=1250 +page_content='Exceptional-point-assisted entanglement, squeezing, and reset in a chain of three superconducting resonators Wallace S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Teixeira,1 Vasilii Vadimov,1 Timm Mörstedt,1 Suman Kundu,1 and Mikko Möttönen1, 2 1QCD Labs, QTF Centre of Excellence, Department of Applied Physiscs, Aalto University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Box 15100, FI-00076 Aalto, Finland 2VTT Technical Research Centre of Finland Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', QTF Center of Excellence, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Box 1000, FI-02044 VTT, Finland The interplay between coherent and dissipative dynamics required in various control protocols of quantum technology has motivated studies of open-system degeneracies, referred to as exceptional points (EPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, we introduce a scheme for fast quantum-state synthesis using exceptional-point engineering in a lossy chain of three superconducting resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We theoretically find that the rich physics of EPs can be used to identify regions in the parameter space that favor a fast and quasi- stable transfer of squeezing and entanglement, or a fast reset of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For weakly interacting resonators with the coupling strength g, the obtained quasi-stabilization time scales are identified as 1/(2 √ 2g), and reset infidelities below 10−5 are obtained with a waiting time of roughly 6/g in the case of weakly squeezed resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Our results shed light on the role of EPs in multimode Gaussian systems and pave the way for optimized distribution of squeezing and entanglement between different nodes of a photonic network using dissipation as a resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' INTRODUCTION Quantum mechanics has provided profoundly novel ways of information processing, communication, and metrology [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Although non-linearity expressed by the anharmonicity of energy levels is a key metric for phys- ical realizations of qubits, quantum harmonic systems have also a broad range of quantum-technological appli- cations employing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', squeezing and entanglement as resources [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The efficient use of such properties in experiments typically requires quick transitions from co- herent to incoherent dynamics for different stages of the protocols, and hence dissipation engineering using in-situ tunable components plays an important role towards fast control and scalability of practical quantum systems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In circuit quantum electrodynamics (cQED), for ex- ample, efforts have been made to integrate devices with in-situ-tunable dissipation to prepare specific quantum states [5–12], produce fast reset [13–24], and to exploit the potential benefits of open-system degeneracies, re- ferred to as exceptional points (EPs) [17, 21, 25–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In contrast to Hermitian degeneracies, EPs induce the coa- lescence of eigenvalues and eigenvectors of the dynamical matrix governing the open-system evolution leading to critical dynamics manifested by polynomial solutions in time [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' These features are key elements for op- timized heat flow [25] and sensitive parameter estima- tion [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' When EPs are dynamically encircled in the parameter space, counter-intuitive effects not observed in closed systems appear such as the breakdown of the adiabatic approximation and topological energy trans- fer [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Due to their novelty for the observation of open-system phenomena and applications, EPs have also been acknowledged in other physical architectures [35– 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' However, the relationship between EPs and the emergence of non-classical and non-local features in mul- tipartite continuous-variable (CV) quantum systems has not been fully explored [38–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Quantum harmonic arrays have a practical appeal in cQED for the implementation of quantum memo- ries [44] and for the capability to simulate many-body physics [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Even though the transport of quantum cor- relations has been extensively theoretically studied in re- lated setups [46–49], the high dimension of such systems and their dissipative features render the characterization of EPs an involved procedure [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Motivated by the above-mentioned potential use cases and issues, in this work, we introduce exceptional-point engineering for squeezing and entanglement propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We consider a minimal setup for the production of high- order EPs, consisting of a chain of tree linearly cou- pled superconducting resonators with independent decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' To some extent, our system can be described by its first and second moments, so that it can consti- tute an example of a Gaussian system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', a CV system represented by a Gaussian Wigner function [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' To ana- lytically describe the EP-related phenomena, we employ the Jordan normal form of the dynamical matrix of the second moments, allowing for investigations beyond en- ergy flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Interestingly, we observe that even for weakly cou- pled resonators, the operation in the vicinity of a specific second-order EP may turn the central resonator into a fast squeezing splitter and distant-entanglement genera- tor using only initial squeezing in a single resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We calculate theoretical bounds for the squeezing and en- tanglement of the quasi-stable states and observe their rich dependence on the initial squeezing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' On the other hand, operation near a different, third-order EP branch provides substantial speed up of the decay towards the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Therefore, the detailed knowl- edge of its open-system degeneracies render the system a versatile structure for quantum protocols requiring fast stabilization or reset of the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II, we arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='13571v1 [quant-ph] 31 Jan 2023 2 present the general theory of exceptional points in noisy Gaussian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' III, we provide the details of the considered setup, including the characterization of its EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Sections IV and V are dedicated to studies of different effects arising at or near EPs, with a focus on the quasi-stabilization and decay of non-classical Gaus- sian states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A discussion on the use cases and limitations of EP engineering is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The conclusions are drawn in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' EXCEPTIONAL POINTS IN NOISY GAUSSIAN SYSTEMS Our general model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1(a) consists of a sys- tem of N harmonic modes and of an environment such that each system mode is interacting with their local Markovian bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The j:th mode is described by anni- hilation and creation operators ˆaj and ˆa† j, respectively, with the canonical commutation relations [ˆaj, ˆa† k] = δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We assume that the modes are linearly coupled to one another in any desired topology yielding up to quadratic terms in their coupling Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' III–V, we explore a linear network consisting of three lossy super- conducting resonators as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By defining the quadrature operators of the j:th mode as ˆqj = (ˆaj + ˆa† j)/ √ 2 and ˆpj = −i(ˆaj − ˆa† j)/ √ 2 and their 2N-dimensional vector as ˆx = (ˆq1, ˆp1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', ˆqN, ˆpN)⊤, the total Hermitian Hamiltonian describing the system classically driven by amplitudes c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', c2N)⊤ can be cast into the compact quadratic form [53] ˆH = 1 2ˆx⊤Hˆx + c⊤Ωˆx, (1) where we dropped possible constant energy offsets, in- troduced the 2N × 2N symmetric matrix H carrying the internal and mode–mode coupling energies, and utilized the symplectic matrix Ω = N � j=1 � 0 1 −1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (2) The commutation relations between the elements of ˆx read [ˆxj, ˆxk] = iΩjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Note that {ˆqj} and {ˆpj} play the role of generalized dimensionless position and momen- tum operators, such that for superconducting LC cir- cuits they are related to flux and charge operators, re- spectively [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' After tracing out the environmental degrees of free- dom, the temporal evolution of the reduced density op- erator of the system, ˆρ, is given by the Lindblad master equation dˆρ/dt = −i[ ˆH, ˆρ]/ℏ + L↓(ˆρ) + L↑(ˆρ), where Ll(ˆρ) = 1 2ℏ N � j=1 � 2ˆLl j ˆρ(ˆLl j)† − � (ˆLl j)† ˆLl j, ˆρ �� , (3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' R1 R2 R3 QCRs mode 1 mode 2 mode N (a) (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (a) Schematic diagram of the general system consid- ered in this paper consisting of N harmonic quantum modes linearly coupled to one another (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In addition, each mode is coupled to their own Markovian environment (rounded squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (b) Particular realization of the system ex- plored in this work, where three superconducting resonators are capacitively coupled in a linear-chain configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In addition, each resonator has their own drive lines (trian- gles), using which the system can be prepared and measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The decay rates of resonators R2 and R3 can be controlled by quantum-circuit refrigerators (QCRs) placed at the res- onator input ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Each QCR is comprised of a normal- metal–insulator–superconducting (NIS) junction and can re- move photons incoherently from the system mediated by elec- tron tunneling at specific bias-voltage pulses [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' describes the incoherent dynamics of the system asso- ciated to the jump operators {ˆLl j}, where the labels l =↓, ↑ refer to thermal emission and absorption, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We restrict to the case where such operators are local and linear combinations of the elements of ˆx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', ˆLl j = (ul j)⊤Ωˆx, with coefficients given by the 2N- dimensional vector ul j that has only a single or two adja- cent non-zero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For example, this corresponds to jump operators with the form ˆLl j = nl jˆaj + ml jˆa† j, thus encompassing N individual squeezed thermal envi- ronments [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The case in which both thermal excitation and bath squeezing are negligible is thoroughly investi- gated in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' III–V for N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Under the above conditions and for an initial Gaussian state of the N oscillators, the dynamics of the system can be fully characterized by the so-called mean vector and covariance matrix (CM), the components of which are ⟨ˆxj⟩ = Tr(ˆxj ˆρ) and Vjk = 1 2 (⟨ˆxjˆxk⟩ + ⟨ˆxkˆxj⟩) − ⟨ˆxj⟩⟨ˆxk⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, we aim to solve the dy- namics of the CM, since it captures all squeezing and non-local properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By differentiating V with respect to time and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (3), we verify that the CM evolves according to the differential Lyapunov equation [56] dV dt = ΓV + VΓ⊤ + D, (4) where we defined the 2N × 2N matrices Γ = Ω(H − ImΥ)/ℏ, D = ReΥ/ℏ, and Υ = � l,j[ul j(ul j)†].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The CM is a real, symmetric and positive-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' As a compact statement of the uncertainty principle, the CM must also fulfill the condition V + iΩ/2 ≥ 0 [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Below we focus on the scenario where Γ and D are independent of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Given an initial CM V(0) ≡ V0, 3 the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (4) in this case is given by [58] V(t) = eΓt (V0 − Vss) eΓ⊤t + Vss, (5) where Vss is the steady-state CM obtained as the solution of the algebraic Lyapunov equation ΓVss +VssΓ⊤ +D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We observe from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (4) and (5) that Γ has the role of a dynamical matrix so that all possible EPs are de- termined by its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Since the entries of Γ are real numbers with units of angular frequency, its eigenvalues are the complex-conjugate pairs λ± sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, we define the index sm = (m, µm) to refer to the m:th pair of the eigenvalues of Γ, each eigenvalue having a multiplicity µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Observe that the maximum allowed multiplicity is thus max(µm) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The matrix Γ admits a Jordan normal form Γ = PJP−1, where P is a non-singular matrix and J = diag[J− s1(λ− s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', J+ sk(λ+ sk)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The Jordan blocks J± sm(λsm) can be decomposed as µm × µm matrices J± sm(λ± sm) = λ± smIµm + Nµm, with Iµm being the identity matrix and Nµm having the elements above the diagonal filled with ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Naturally, the Jordan blocks for µm = 1 are just the scalars λ± sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' With these definitions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (5) can be rewritten as V(t) = PeJtP−1 (V0 − Vss) � P−1�⊤ eJ⊤tP⊤ + Vss, (6) where eJt = diag(eλ− s1teNµ1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', eλ+ sk teNµk t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The emergence of EPs and the associated critical dy- namics of the CM correspond to the cases where the dy- namical matrix Γ becomes non-diagonalizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', for any µm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In other words, degeneracies in the spec- trum of Γ produce nilpotent matrices Nµmt, the expo- nentials of which yield polynomials in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hereafter, these non-Hermitian degeneracies will be referred to as EP–µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Considering the definition of Γ, we remark that the term ΩH itself does not promote critical dynamics as it gives rise to unitary evolution of the CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The pro- duction of EPs must be accompanied with the incoher- ent processes caused by the local environments and at- tributed to the term ΩImΥ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In summary, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (6) is valid for any time-independent matrices Γ and D describing the evolution of a system of coupled quantum harmonic oscillators in noisy Gaus- sian channels yielding the steady-state CM Vss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' At an EP, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (6) reveals that the solution linked to the crit- ical dynamics is an exponential function multiplied by a polynomial, which will be explored below in specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Alternatively, the description of EPs for quadratic Liouvillians, such as the one related to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (3), may be given in terms of annihilation and creation operators as recently developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' THREE COUPLED RESONATORS UNDER INDIVIDUAL LOSSES The system and its environment considered in this work is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Three superconducting resonators, R1, R2, and R3 are capacitively coupled in a linear-chain configuration through a fixed coupling con- stant g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We focus on a single electromagnetic mode for each resonator, which, including the coherent cou- plings, defines our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Each mode may dissipate its energy into its independent linear bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Nevertheless, quantum effects may emerge at low temperatures and for sufficiently high quality factors and for non-classical initial states [54], and consequently we need to employ a quantum-mechanical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In the single-mode and rotating-wave approximations, the Hamiltonian of the system reads ˆH = ℏ 3 � j=1 ωj � ˆa† jˆaj + 1 2 � + ℏg(ˆa1ˆa† 2 + ˆa2ˆa† 3 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='), (7) where ωj is the fundamental angular frequency of the j:th resonator, {ˆaj} are the corresponding ladder opera- tors defined as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II, and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' refers to the Hermitian conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The losses of the system are modeled here as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (3), with jump operators ˆL↓ j = � ℏκjˆaj and de- cay rates κj > 0, for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Some of the decay rates can be adjusted experimentally through the QCRs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' As we show below, to produce EP–3 with degenerate resonators, we need asymmetric decay rates, a scenario which can be realized by the two independent QCRs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In the following analysis, ther- mal excitations are neglected so that ˆL↑ j ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By writing the ladder operators in terms of the quadra- ture operators as ˆaj = (ˆqj + iˆpj)/ √ 2 and using the nota- tion of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II, the 6 × 6 dynamical matrix Γ becomes Γ = � � � K1 G 02 G K2 G 02 G K3 � � � , (8) where 02 is the 2 × 2 null matrix and Kj = � − κj 2 ωj −ωj − κj 2 � , G = � 0 g −g 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (9) By denoting the single-mode CM of the vacuum state as V(j) vac = diag (1, 1) /2, one readily obtains D = 3 � j=1 κjV(j) vac, Vss = 3 � j=1 V(j) vac, (10) the latter corresponding to the CM of any product of three coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Since the jump operators here do not promote incoherent displacements, the steady state is actually the three-mode vacuum state |0⟩1|0⟩2|0⟩3 as long as all κj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Characterization of exceptional points Finding the EPs directly from the spectrum of Γ may be challenging as one needs to solve a 2N:th degree poly- nomial equation, or in the studied case, a sextic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 4 However, owing to the absence of counter-rotating terms in the form of ˆH here, the characterization of EPs can be simplified to the study of the dynamical equation for the 3×3 vector a = (⟨ˆa1⟩, ⟨ˆa2⟩, ⟨ˆa3⟩)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By moving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (3) to a frame rotating with ω1, one can obtain ˙a = −iHa, with H having the role of an effective non-Hermitian Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Explicitly, we have H = � � � −i κ1 2 g 0 g δ2 − i κ2 2 g 0 g δ3 − i κ3 2 � � � , (11) where δ2 = ω2 − ω1 and δ3 = ω3 − ω1 are frequency detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Without loss of generality, we assume that the param- eters g, ω1, and κ1 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Thus it is convenient to express the parameters of R2 and R3 with respect to those of R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We proceed with this parametrization using complex-valued parameters {εk} such that for k = 2, 3, we have δk(εk) = √ 2gIm(εk), κk(εk) = κ1 + 2 √ 2gRe(εk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (12) As detailed in Appendix A, degeneracies in the spectrum of H appear provided that f(ε) = 1 2 � �ε ± � ε4 + 10ε2 − 2 ± 2 (1 + 2ε2) 3 2 ε2 � � , (13) where ε = ε3 and f(ε) = ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Note that the complex- valued function f(ε) presents four branches indicated by the signs ‘±’ as shown for a purely real ε in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' At the degeneracies of H, such a matrix has at most two distinct eigenvalues, from which the effective detun- ings and decay rates of the normal modes are extracted as δeff j (ε) = √ 2gIm[hj(ε)] and κeff j (ε) = κ1 + 2 √ 2gRe[hj(ε)] (Appendix A), where h1(ε) = f 3 − εf 2 − (ε2 + 4)f + ε3 + ε/2 f 2 − εf + ε2 − 3 , h2(ε) = h3(ε) = 1 4 �2εf 2 + 2(ε2 + 1)f − 7ε f 2 − εf + ε2 − 3 � , (14) and we write f = f(ε) for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Consequently, the degenerate eigenvalues of Γ are given by the pairs (Ap- pendix A) λ± sj(ε) = −κeff j (ε) 2 ± i � ω1 + δeff j (ε) � , (15) which coincide at an EP–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The rich structure of the decay rates and frequencies of the normal modes is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2b for a purely real ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Without imposing further restrictions, the considered open system presents six EP–3, two of which are ob- tained for ε = 2f(ε) = ±2, so that all modes are de- generate, κ2 = κ1 ± 2 √ 2g, and κ3 = κ1 ± 4 √ 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' These cases correspond to the square-root singularity of f(ε) 4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 (a) 4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Exceptional-point engineering for a linear chain of three lossy resonators with degenerate angular frequencies ω1 = ω3, expressed by a purely real parameter ε = ε3, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (12)–(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (a) Decay rate (top panel) and frequency (bottom panel) offsets of resonator R2 as functions of the decay rate offset of resonator R3, expressed by the complex- valued function f(ε) = ε2 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (b) Effective decay rate (top) and effective frequency (bottom) offsets of the eigenmodes of the system as functions of the decay rate offset of resonator R3, expressed by the complex-valued func- tions hj(ε) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' All offsets are given with re- spect to the parameters of resonator R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In (b), solid (dashed) curves represent the single (double) roots of the characteristic polynomial of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In all cases, the labels ++, −+, +−, and −− indicate the four branches of f(ε) obtained from the cor- responding selection of signs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The vertical dashed lines in all panels highlight the values of ε producing EP–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The shaded area in (a) indicates the relevant region of the Re(ε3)–Re(ε2) parameter space for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' and are highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The remaining four EP– 3 are obtained with f(ε) = (±3 √ 3 ± i)/(2 √ 2), and ε = 2i Im[f(ε)] = ±i/ √ 2, thus requiring equal decay rates for R1 and R3, κ2 = κ1 ± 3 √ 3g, in addition to the detunings δ2 = ±g/2 and δ3 = ±g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The degeneracy map for such cases is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 6 of Appendix A for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' All other cases expressed through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13) and (14) are associated to EP–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Our numerical tests show the coalescence of eigenvectors of H following the branches f(ε), indeed indicating open-system degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The Jordan decompositions of Γ yielding polynomial-in-time features of the dynamics are shown in Appendix B for relevant EPs in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We emphasize that the experimental feasibility of EP engineering in the present model is strongly dependent on the physical limitations of the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For instance, to obtain the four instances of EP–3 with non-degenerate frequencies, one needs frequency detunings of the order of g/(2π), which are typically much smaller than the fre- quency of superconducting resonators themselves [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hereafter, we restrict our discussion to degenerate res- 5 onators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', Im(ε) = Im[f(ε)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By also considering κ1 as the smallest decay rate, another restriction for ob- taining EPs is imposed, such that both Re(ε) ≥ 0 and Re[f(ε)] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In this case, the only allowed branches of f(ε) are ‘+−’ and ‘−−’ for ε ≥ 2, and ‘++’ for ε ≥ 0, see the shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In particular, the branch ‘++’ at ε = 0 yields weakly dissipative nor- mal modes, with one of them decaying according to κ1, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2b and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' This behavior suggests that a quasi-stabilization of some properties of the system can be obtained with the combination of a small κ1 and a proper choice of the EP, as explored in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Single-mode squeezing and bipartite entanglement Below, we specifically investigate single-mode squeez- ing and bipartite entanglement for the three-resonator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For Gaussian evolution, these quantities can be addressed directly from the specific partitions of the total CM V = � � � V(1) C(12) C(13) C(12)⊤ V(2) C(23) C(13)⊤ C(23)⊤ V(3) � � � , (16) where V(j) is the reduced CM of resonator Rj and C(jk) is the intermodal correlation matrix between resonators Rj and Rk [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Since all single-mode Gaussian states can be written as squeezed thermal states apart from local displacements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' the components of the reduced CM of resonator Rj can be cast into the form [60] V(j) 11 = ( ¯Nj + 1/2)[cosh(2rj) + sinh(2rj) cos φj],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' V(j) 22 = ( ¯Nj + 1/2)[cosh(2rj) − sinh(2rj) cos φj],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' V(j) 12 = ( ¯Nj + 1/2) sinh(2rj) sin φj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (17) where rj and φj are real-valued quantities defining the squeezing parameter ξj = rjeiφj and ¯Nj is the effective thermal occupation number of resonator Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' As a con- sequence, one can extract rj and ¯Nj as rj = 1 2 sinh−1 � � � (V(j) 11 − V(j) 22 )2 + 4V(j)2 12 2( ¯Nj + 1/2) � � , ¯Nj = � det V(j) − 1 2, (18) and the single-mode purity is readily given by Pj = (2 ¯Nj + 1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' While bipartite entanglement can be quantified by the reduced von Neuman entropy given a pure state of the complete system [61], an entanglement measure for mixed states is not uniquely defined [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, we focus on the concept of logarithmic negativity [63], which is based on the Peres–Horodecki separability criterion [64, 65] and fulfills the conditions for an entanglement monotone [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Given Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (16) and considering the subsystems Rj and Rk (j < k), one can write their joint CM as V(jk) = � V(j) C(jk) C(jk)⊤ V(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (19) For Gaussian states, the logarithmic negativity, Ejk, can then be computed as [63, 67] Ejk = max[0, − log2(2˜ν− jk)], (20) where ˜ν− jk = { ˜∆jk − [ ˜∆2 jk − 4 det V(jk)] 1 2 } 1 2 / √ 2 being the smallest symplectic eigenvalue of ˜V(jk), which cor- responds to the two-mode CM obtained after the Peres– Horodecki partial transposition of the associated bipar- tite density matrix, and ˜∆jk = det V(j) + det V(k) − 2 det C(jk) [65, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The inequality ˜ν− jk ≥ 1/2 is a neces- sary and sufficient condition for separability of bipartite Gaussian systems of two modes [65, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' QUASI-STABILIZATION OF SQUEEZING AND ENTANGLEMENT In this section, we study the propagation of single- mode squeezing and bipartite entanglement in the open quantum system of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The initial state is chosen as |0⟩1|0⟩2 ˆS3(r)|0⟩3, where ˆS3(r) = exp � r(ˆa†2 3 − ˆa2 3)/2 � is the single-mode squeezing operator of R3 and r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Such a state has the CM V0 = 1 2diag � 1, 1, 1, 1, e2r, e−2r� , (21) which indicates that the variances of R3 are initially mod- ified by the factors e±2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We employ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (5) to numeri- cally obtain the 6 × 6 time-evolved CM V(t) at different points of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, we set κ1 = κ3 as the smallest decay rates of the system and test different κ2 = κ1 + 2 √ 2g Re(ε2) with Re(ε2) ≥ 0 and Im(ε2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Within these conditions, the only allowed EP–branch is ‘++’ so that an EP–2 is produced at f(ε = 0) = 2, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Figure 3a, we observe the emergence of squeezed thermal states for resonator R1 and bipartite quantum correlations expressed through the logarithmic negativ- ity E13, with a clear passage from underdamped to over- damped dynamics with increasing κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The squeezing de- gree of R2 along with the logarithmic negativities E12 and E23 (data not shown) is rapidly suppressed for large ratios κ2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' On the other hand, the small values of κj/g, j = 1, 3, help to delay the decay of the system towards the three-mode vacuum state, and this quasi-stability tends to be achieved faster near the critical-damping regime produced by the EP–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Such a behavior is not present at the EP–2 if R1 is directly connected to R3, which reduces 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='4 0 4 8 12 16 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='4 (a) (b) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (a) Dynamics of the squeezing parameter r1 and effective thermal occupation ¯ N1 of resonator R1 and the loga- rithmic negativity between R1 and R3, E13, for the indicated values of the damping rate of R2, κ2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The shown data correspond to a crossover from underdamped to overdamped dynamics, with critical damping at κ2/g = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The fre- quencies of the resonator modes are chosen as ω1/g = ω2/g = ω3/g = 5000, the other damping rates as κ1/g = κ3/g = 10−3, and the initial squeezing parameter of R3 as r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The corresponding values of ε2 as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (12) are ε2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='2 (gray curves), ε2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 (blue curves) and ε2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In the chosen parameter regime, the results are essentially independent of the resonator–resonator cou- pling strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (b) Maximum achieved quantities in tem- poral evolutions corresponding to (a) at the critical damping as functions of the initial squeezing parameter r for selected values of κ1/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In all panels, dashed lines correspond to long- time values in the limit κ1/g → 0, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' the dimension of the system to N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In such a case, the only two normal modes of the system decay at equal rates [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The maximum achieved values of r1, ¯N1, and E13 as functions of the initial squeezing parameter r for the sys- tem dynamics at the EP–2 are shown in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Their values in the limit κj → 0, j = 1, 3, and t → ∞, can be estimated directly from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (18) and (20) with the help of the Jordan decomposition of Γ shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' One readily obtains r⋆ 2 = ¯N ⋆ 2 = E⋆ 12 = E⋆ 23 = 0, whereas r⋆ 1 = r⋆ 3 = 1 2 log � 3 + e2r � 10 + 6 cosh(2r) � , ¯N ⋆ 1 = ¯N ⋆ 3 = 1 8 �� 10 + 6 cosh(2r) − 4 � , E⋆ 13 = 1 2 � 1 − log2(1 + e−2r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (22) The superscripts ‘⋆’ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (22) indicate that such quan- tities are bounds for the quasi-stabilized states, shown as dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Interestingly, we can generate en- tanglement between resonators R1 and R3 although the entanglement with resonator R2 is rapidly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3b and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (22), we observe that the squeezing splitting increases linearly with r for r ≪ 1, where thermal occupancy is insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The squeezing- splitting capacity r⋆ 1/r and the degree of entanglement between R1 and R3 tend to saturate to 1/2 in the limit r → ∞ with the expense of also thermally populat- ing these resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Using the decibel scale defined by r = 10 log10(e2r) dB [68], an initial amount of squeezing r ≈ 3 dB is roughly converted into squeezed states with r⋆ 1 = r⋆ 3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='772 dB and purities P⋆ 1 = P⋆ 3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='997, with E⋆ 13 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Despite producing a faster decay towards the actual steady state of the system, an increase of two orders of magnitude in κ1/g does not provide significant differences in the maximum quantities for small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' To further address the quasi-stabilization of entangle- ment and squeezing transferred to R1 for different κ2, we diagonalize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (11) to obtain the effective frequency detunings and decay rates of the system as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 4a and 4b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For κ1 ≪ κ2, we ob- tain two eigenmodes with frequency detunings δeff ± ≈ ± Im( � κ2 2 − 32g2)/4 and dissipation rates κeff ± ≈ κ2/2 ± Re( � κ2 2 − 32g2)/2, which coalesce at κ2 ≈ 4 √ 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The frequency detuning δeff 0 = 0 and dissipation rate κeff 0 = κ1 are preserved, thus indicating that one of the eigenmodes remains hidden from the dissipation of resonator R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Since clearly the speed of quasi-stabilization for the squeezing and entanglement of resonator R1 depend on κ2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3a) and since κeff + ≥ κeff − , we conclude that the time scale for this quasi-stabilization is roughly given by 1/κeff − ≈ 2/[κ2 − Re( � κ2 2 − 32g2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' To arrive at a more accurate expression for the quasi-stabilization time, we first fit functions of the form rfit 1 (t) = r⋆ 1 2 e−yr1κ1t � e−κeff − t/2 � 1 − 3 cos � δeff − t �� + 2 � , Efit 13(t) = E⋆ 13e−yE13κ1t � 1 − e−κeff − t cos2(δeff − t) � , (23) to time traces similar to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3a and find yr1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='75 and yE13 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Although these functions neglect the polynomial-in-time solution at the EP–2, they capture the main features of the over and underdamped dynam- ics, and hence are accurate enough from our following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Next, we define the quasi-stabilization time tα as the earliest time instant after which the quantity α = r1, E13 stays within an uncertainty σα from the ideal value α⋆e−yακ1tα, where we take into account also the slow decay of the maximum attainable value owing to finite κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' More precicely, tα = min{t|α⋆e−yακ1t − ˜α(t) ≤ σα}, (24) where ˜α(t) is the lower envelope of the possibly oscillating α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Note that by this definition, ˜α(t) = α(t) in the critically and overdamped dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 0 2 4 6 8 2 4 6 8 5 10 15 20 25 30 35 40 (a) (b) (c) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (a) Effective frequency detunings and (b) effec- tive decay rates of the eigenmodes of the coupled system as functions of the decay rate of resonator R2, κ2, in units of the resonator–resonator coupling strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (c) Time tα to yield quasi-stable squeezing (filled circles, α = r1) and en- tanglement (filled squares, α = E13) within an uncertainty σα = 10−5, see main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The dashed lines represent cor- responding results from the fit functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In all panels, the parameters are chosen as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3a and the colored regions separate the underdamped from the overdamped dy- namics, with critical damping at κ2/g = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='658, corresponding to an EP–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 4c, we show the behavior of the quasi- stabilitation time tα on the dissipation rates κ2 for an error σα = 10−5 as obtained from the solutions of the temporal evolution of the system similar to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The shortest quasi-stabilization times are ob- tained in the vicinity of the EP–2 owing to the peak in κeff − illustrated in Fiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Using the lower envelopes of the fitting functions (23) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (24), one can estimate the quasi-stabilization time as tα ≈ log � α⋆ σα � yακ1 + zακeff − , (25) with zr1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 and zE13 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Therefore, tα tends to scale logarithmically with the desired error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' FAST RESET NEAR EXCEPTIONAL POINTS As the final application of EPs, we discuss the reset of the resonator chain to its ground state |0⟩1|0⟩2|0⟩3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Typically, stronger dissipation leads to faster decay, but of course in our system where the coupling between the different resonators is weak compared with the excitation frequencies of the bare resonators, the critical dynamics plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Similar features are prone to arise in a quantum register of several coupled qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' To quantitatively study the accuracy of the reset, we define the infidelity Iss(ˆρ) = 1 − Fss(ˆρ), (26) where Fss(ˆρ) = ⟨0|1⟨0|2⟨0|3ˆρ|0⟩1|0⟩2|0⟩3 is the overlap probability between an arbitrary three-mode state ˆρ and the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For multimode Gaussian states with null mean vector ⟨ˆx⟩, Fss can be directly computed from the covariance matrix V, which for the present case be- comes [3] Fss = 1 � det (V + Vss) , (27) where Vss given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' An optimized reset is achieved with the set of free parameters producing the fastest decay to the ground state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', the minimal Iss in a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Figure 5 shows the reset infidelity for different param- eter values and for an initial state which is obtained by waiting for a preparation time τs at EP–2 after squeezing the vacuum at resonator R3 by a finite r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Note that if τs = 0, one has the initial squeezed state with the covari- ance matrix given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (21), and with τs = 8/g, one prepares an initial state with entanglement and squeez- ing split between R1 and R3, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 5a, we show the dependence of Iss on the decay rates κ2 and κ3 in the region corresponding to the shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2a for the above-mentioned preparation times and immedi- ately following reset times τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Although the regions of low infidelity are relatively broad if all squeezing is con- centrated in R3, so that no entanglement is present, we observe a narrowing of such regions if τs = 8/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' These regions tend to cover the EP–3 and follow the real com- ponents of the ‘−−’ branch of f(ε3) as ε3 is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Such a feature is even more prominent for long reset times naturally leading to lower reset infidelities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Note from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2b that this branch tends to produce highly dissi- pative normal modes for ε3 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In contrast, at least one decay rate produced by the ‘+−’ and ‘++’ branches is slow even with increasing ε3, rendering such branches less favorable for the reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Figure 5b shows the reset infidelity Iss as a function of the reset times τr at the EP–3 for different initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In all displayed cases, low infidelities Iss are in- deed achieved beyond τr ∼ 6/g, owing to the exponential dependence on τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For such reset times, the distribution 8 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 2 4 6 8 10 10- 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 10- 8 10- 6 10- 4 10- 5 10- 3 10- 1 10- 9 10- 2 (a) EP-3 reset 10- 1 10- 2 10- 3 10- 4 10- 5 (b) (c) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (a) Reset infidelity Iss of degenerate resonators as a function of the dimensionless decay rate offsets Re(ε3) and Re(ε2) for selected choices of preparation times τs (top and bottom panels) and reset times τr (left and right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' During the time-interval τs, the system is set at the EP–2 with Re(ε3) = 0 and Re(ε2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Solid curves on top of the contour plots show the components of the EP branches ‘++’ (blue), ‘+−’ (gray), and ‘−−’ (green) in the Re(ε3)– Re(ε2) parameter space as in the shaded region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2a, with EP–3 indicated by dashed circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The other parameters are ωj/g = 5000, j = 1, 2, 3, κ1/g = 10−3, and r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (b) Reset infidelity Iss at the EP–3 as a function of reset times τr (in units of g−1) for different preparation times τs and decay rate κ1/g = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Solid (dashed) curves show data for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 (r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (c) Reset infidelity Iss at the EP–3 as a function of squeezing parameter r for different reset times τr and for preparation time τs = 8/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Solid (dashed) curves show data for κ1/g = 10−3 (κ1/g = 10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The remaining parameters are chosen as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' of squeezing and entanglement tends to have a minor rel- ative effect on the reset performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' This is in stark contrast with the short-reset-time cases, where the decay towards the ground state tends to significantly acceler- ate if all initial squeezing is poorly distributed, remaining mostly in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We observe that the reset performance is degraded for small ratios of κ1/g and for increasing ini- tial squeezing parameters as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In such scenarios, for a finite reset time, the infidelity tends to grow asymptotically to unity in the limit r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' DISCUSSION We observed that fast generation of entanglement and propagation of squeezing in a linear chain of three su- perconducting resonators may benefit from the detailed understanding of critical damping in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, the highly dissipative resonator R2 acts as an incoherent entanglement generator and squeezing splitter with the cost of reducing the purity of the local states through the increase of their effective temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The role of critical damping towards stabilization has also been ac- knowledged recently in an autonomous quantum thermal machine with two qubits [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The stabilization of squeezed states through reservoir engineering in superconducting circuits has been recently reported in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We highlight that the scheme in our paper differs from typical two-mode squeezing opera- tions, since it arises from the combination of dissipa- tion and only a single-mode squeezing source available in the beginning of the dynamics, thus being also dis- tinct from conventional reservoir-engineering protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' On the other hand, we do not need continuous driving terms since the structure of couplings and dissipation of the system promote a separation of time scales for the decay of the normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We explicitly show that this can be beneficial if fine-tuning κ2 near a particular EP–2 instead of only roughly assuming the conditions κj ≪ κ2, g, for j = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3 and 4 also suggest that concatenating similar structures can be used for fast and stable distribution of entanglement to every other node in a photonic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Although, spoiling Gaussian fea- tures of the system [70, 71], entanglement distillation protocols [72] may be used in such cases to increase the amount of entanglement shared by the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Particu- lar low-order EPs of high-dimensional systems may be used to speed up the generation of quasi-stable states, and hence they may have potential use cases in quantum protocols, although the open-system-degeneracy map in such cases becomes more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Regarding the unconditional dissipative reset of the system, the role of critical damping becomes more evi- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Here, the region near the EP–3 and also following a particular EP–2 branch is a reasonable choice of param- eters to produce a substantial performance enhancement of the reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Since the covariance matrices of the vacuum state and a product of coherent states are identical, such regions in the parameter space could also be used to pro- mote unconditional fast stabilization of coherent states with a proper inclusion of driving terms in the system Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Let us present typical experimental parameters of the circuit 1b that could reproduce the findings of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 9 For a resonance frequency of ω/(2π) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 GHz, the sim- ulated values of coupling strength and lowest decay fre- quencies are g/(2π) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 MHz and κ1/(2π) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 kHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Such resonance frequency and coupling strength has been conveniently experimentally achiev- able for longer than a decade, and the quality factor of five million implied by the lowest decay rate can be achieved with state-of-the-art fabrication techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The EP–2 used for stabilization is thus achieved with κ2/(2π) ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='66 MHz and κ3/(2π) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='0 kHz, while the EP–3 with κ2/(2π) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='83 MHz and κ3/(2π) ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='66 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Even though the almost four-orders-of-magnitude tunability required to interchange between this particu- lar EP–2 and the EP–3 may be technically challenging, the maximum achievable decay rates with the QCR are beyond the ones considered here and their demonstrated on/off ratios are close to these requirements [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' CONCLUSIONS We demonstrated the theory of exceptional-point- related phenomena for continuous-variable systems de- scribed entirely by their second moments, consequently capturing different non-classical features and non-locality largely neglected in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For a linear chain of three lossy superconducting resonators, we analyti- cally obtained its open-system-degeneracy map and ob- served that different parameter sets yielding different ex- ceptional points can be used to identify sweet spots for the optimization of squeezing propagation, entanglement generation, and reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' More precisely, we assessed the role of critical dynamics for dissipative state synthesis by numerically simulating the temporal evolution of the covariance matrix of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The region of the parameter space considered in the simulations is physically motivated by recent experi- mental advances in dissipation-tunable devices embedded to superconducting circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We found that the quasi-stabilization into mixed bipar- tite entangled states generated from an initially squeezed resonator R3 is optimized in the vicinities of a particu- lar low-dissipative EP–2 produced with symmetric de- cay rates of resonators R1 and R3 [see the ‘++’ branch of f(ε) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In such scenarios, one observed that the time scale for this quasi-stabilization is mini- mum for κ2 ≈ 4 √ 2g and κ1, κ3 ≪ κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Using the Jor- dan decomposition of the dynamical matrix, we obtained analytical bounds for the maximum achievable quasi- stable squeezing-splitting capacity and logarithmic neg- ativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Remarkably, all residual squeezing of the cen- tral resonator is removed within the quasi-stabilitization time-scales, and consequently, the choice of EP–2 also quickly removes the entanglement of R2 with the other resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Furthermore, we investigated the dissipative reset of such non-classical states to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The region in the parameter space producing the lowest reset infi- delities at given reset times τr requires asymmetric res- onator decay rates and tend to follow a particular high- dissipative EP branch, which includes the physically at- tainable EP–3 [see the ‘−−’ branch of f(ε) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In this EP–3 case, the distribution of the initial squeezing into the different resonators tends to become irrelevant for the reset performance beyond τr ∼ 6/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In conclusion, this work paves the way for a deep un- derstanding of the role of exceptional points in multimode continuous-variable systems, with potential applications in quantum technology such as in using dissipation as an ingredient for fast transfer of desired quantum prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' For example, heat engines [73] operating with nonequilibrium reservoirs [74] and presenting quantum resources [75] arise as systems with promising near-term opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Moreover, the investigation of exceptional points in such superconducting systems through involved models, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [76], is also a potential future line of re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' As a final remark, we note that the role of the counter-rotating terms in the system Hamiltonian and of squeezed reservoirs on the exceptional points may also be addressed with the tools presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors acknowledge the Academy of Finland Cen- tre of Excellence program (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 336810), Eu- ropean Research Council under Consolidator Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 681311 (QUESS) and Advanced Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 101053801 (ConceptQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Dowling and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Milburn, Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A 361, 1655 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Weedbrook, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Pirandola, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Garcia-Patron, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Cerf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Ralph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Shapiro, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lloyd, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 84, 621 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Serafini, Quantum Continuous Variables (CRC Press, Boca Raton, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [4] Q.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Heeres, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Reagor, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Vool, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Leghtas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Frunzio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Kirchmair, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Devoret, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mirrahimi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Schoelkopf, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} 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rahimi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Devoret, Science 347, 853 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Kimchi-Schwartz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Martin, E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 116, 240503 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Premaratne, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 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Bienfait, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Huard, PRX Quantum 2, 020323 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Valenzuela, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Oliver, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Silveri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Par- tanen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Jenei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Masuda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Goetz, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Vesterinen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Grönberg, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Möttönen, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 115, 082601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Yoshioka and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Tsai, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 119, 124003 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Yin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Huai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Gu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Allcock, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Xi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Song, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' An, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zheng, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Zhang, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 12, 5924 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [22] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Vadimov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Viitanen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mörstedt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Ala-Nissila, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Möttönen, AIP Advances 12, 075005 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mörstedt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Viitanen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Vadimov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Sevriuk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Partanen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hyyppä, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Catelani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Silveri, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Tan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Möttönen, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 534, 2100543 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Sevriuk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rönkkö, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hsu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Marxer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mörstedt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Partanen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Räbinä, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Venkatesh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hotari, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Grönberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Heinsoo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Tuorila, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Chan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hassel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Tan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Möttönen, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 121, 234002 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Partanen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Goetz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Tan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Kohvakka, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Sevriuk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lake, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Kokkoniemi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Ikonen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hazra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mäkinen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hyyppä, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Grönberg, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Vesterinen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Silveri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Möttönen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' B 100, 134505 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 15, 1232 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [27] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Abbasi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Joglekar, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Murch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 127, 140504 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Abbasi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Ha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Erdamar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Joglekar, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Murch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 128, 110402 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Abbasi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A 100, 062131 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Uzdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Mailybaev, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Moiseyev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Landi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Brunelli, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Ciccarello, Quantum Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 6, 025005 (2021).' metadata={'source': 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P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Groszkowski, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Earnest, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' McKay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Koch, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Takeoka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hayasaka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Furusawa, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Sasaki, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Photonics 4, 178 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [73] N.' metadata={'source': 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Klaers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Faelt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Imamoglu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Togan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' X 7, 031044 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [75] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Camati, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Santos, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Serra, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' A 99, 062103 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' [76] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Viitanen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Silveri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Jenei, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Sevriuk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Partanen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Goetz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Grönberg, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Vadimov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Lahtinen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Möttönen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Research 3, 033126 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Appendix A: Explicit determination of EPs Here, we show the characterization of EPs for the open quantum system presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' III, namely, a linear chain of three resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Given fixed coupling constants g and parameters of resonator R1, we aim at finding the parameters of resonators R2 and R3 that produce EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Assuming resonance frequencies ωj > 0 and decay rates κj > 0, for j = 1, 2, 3, we move the Lindblad mas- ter equation (3) to a frame rotating with ω1 so that we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (11) of the main text H = � � � −i κ1 2 g 0 g δ2 − i κ2 2 g 0 g δ3 − i κ3 2 � � � , (A1) where δ2 = ω2 − ω1 and δ3 = ω3 − ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Below, we use the parametrizations δ2 = √ 2gIm(ε2), δ3 = √ 2gIm(ε3), κ2 = κ1 + 2 √ 2gRe(ε2), κ3 = κ1 + 2 √ 2gRe(ε3), (A2) so that the effective offsets from ω1 and κ1 are given according to the imaginary and real parts of the com- plex parameters ε2 and ε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' This allows one to write the characteristic polynomial associated to H, P(x) = ax3 + bx2 + cx + d, with the coefficients a =1, b = i 2[3κ1 + 2 √ 2g(ε2 + ε3)], c = − 3κ2 1 4 − √ 2gκ1(ε2 + ε3) − 2g2(1 + ε2ε3), d = − i 8[κ3 1 + 8 √ 2g3ε3 + 2 √ 2gκ2 1(ε2 + ε3) + 8g2κ1(1 + ε2ε3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A3) Given the cubic discriminant ∆ = −(4v3 + 27w2), (A4) where v = 3ac − b2 3a2 , w = 2b3 − 9abc + 27a2d 27a3 , (A5) we search for degeneracies in the spectrum of H, which occur when ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Interestingly, the appearance of EPs depends only on the relationship between ε2 and ε3 given by the condition 4ε4 2ε2 3 − 8ε3 2ε3 3 + 4ε2 2(ε4 3 − 5ε2 3 + 1) + 4ε2(5ε3 3 − ε3) − 8ε4 3 + 13ε2 3 − 16 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A6) Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A6) for ε2 yields the four branches ε2 = 1 2 � �ε3 ± � ε4 3 + 10ε2 3 − 2 ± 2 (1 + 2ε2 3) 3 2 ε2 3 � � , (A7) where the signs ‘±’ can be chosen independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' To identify the order of the EPs, we inspect Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A5) and (A7) more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' All EP–3 correspond to the triple root of P(x) = 0, which is obtained when v = w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' First setting v = 0 reduces ε2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A7) to ε2 = 1 2 � ε3 ± � 12 − 3ε2 3 � , (A8) and imposing w = 0 yields ε3 ∈ {±2, ±i/ √ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Hence, the studied open system presents six distinct EP–3, two of which are produced when all resonators are degen- erate such that ε3 = 2ε2 = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The remaining four EP–3 are obtained with ε2 = (±3 √ 3 ± i)/(2 √ 2) and ε3 = 2iIm(ε2) = ±i/ √ 2, thus requiring frequency shifts from the resonance and equal decay rates for R1 and R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By defining ε = ε3, the parameters of R2 produce EPs provided that they are chosen as ε2 ≡ f(ε), with f(ε) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' When degeneracies of H are present, we express the complex roots of P(x) = 0 as x1 = 4abc − 9a2d − b3 a(b2 − 3ac) , x2 = x3 = 9ad − bc 2(b2 − 3ac), (A9) 12 4 2 0 2 4 4i - 2i 0i 2i 4i 4 2 0 2 4 (a) 4 2 0 2 4 4i - 2i 0i 2i 4i 4 2 0 2 4 (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Open-system-degeneracy map for a linear chain of three coupled resonators with degenerate decay rates κ1 = κ3, which are expressed by a pure imaginary parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' De- pendence of (a) f(ε) and (b) hj(ε) on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In (b), solid (dashed) curves represent the single (double) root of the characteristic polynomial of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In all cases, the labels ++, −+, +−, and −− indicate the four branches of f(ε) obtained from the cor- responding selection of signs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The vertical dashed lines in all plots highlight the values of ε yielding EP–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' to extract the effective detunings and decay rates of the normal modes as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 6, we show the rich structure of the branches yielding the EPs for a purely imaginary ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Let us comment on the relationship between the spec- trum of the non-Hermitian Hamiltonian H defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (11) and the dynamical matrix Γ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' First, we note that Γ also determines the temporal evo- lution of the mean vector x = ⟨ˆx⟩ introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We define the 6 × 6 vector of ladder operators, ˆA = [ˆa, (ˆa†)⊤]⊤, where ˆa = (ˆa1, ˆa2, ˆa3)⊤, ˆa† = (ˆa† 1, ˆa† 2, ˆa† 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In the Schrodinger picture, the vector of expectation values A = ⟨ˆA⟩ is obtained from the dynamical equa- tion ˙A = diag[−i(H + H1), i(H∗ + H∗ 1)]A, where H1 = ω1I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' By introducing the vector x′ = (q, p)⊤, where q = (⟨ˆa⟩ + ⟨ˆa⟩∗)/ √ 2 and p = −i(⟨ˆa⟩ − ⟨ˆa⟩∗)/ √ 2, one verifies that x′ is related to A through a unitary trans- formation so that x′ = ΛA, where Λ = 1 √ 2 � I3 I3 −iI3 iI3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A10) Consequently, the dynamical equation for x′ becomes ˙x′ = Γ′x′, where Γ′ = Λ � −i(H + H1) 03 03 i(H∗ + H∗ 1) � Λ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (A11) Since the vectors x′ and x are equivalent except for the different orderings, the spectrum of Γ′ coincides with that of Γ, which in the case of open-system degeneracies is given by the eigenvalues defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (15) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Appendix B: Jordan normal form In this appendix, we present the explicit Jordan nor- mal form of Γ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (8) at some relevant EPs considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' EP–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' We start with the EP–3 used for the reset of the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', the one produced with degenerate res- onators (ω1 = ω2 = ω3 = ω), κ2 = κ1 + 2 √ 2g, and κ3 = κ1 + 4 √ 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Using the notation introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' the Jordan blocks in the matrix J = diag � J− s1(λ− s1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J+ s1(λ+ s1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' and the non-singular matrix P read J± s1(λ± s1) = � � � λ± s1 1 0 0 λ± s1 1 0 0 λ± s1 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' s1 = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (B1) P = � � � � � � � � � � −i − i √ 2 g − i g2 i i √ 2 g i g2 −1 − √ 2 g − 1 g2 −1 − √ 2 g − 1 g2 − √ 2 − 1 g 0 − √ 2 − 1 g 0 i √ 2 i g 0 −i √ 2 − i g 0 i 0 0 −i 0 0 1 0 0 1 0 0 � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (B2) where λ± s1 are given as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (15) of the main text with ε = 2 and f(ε) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Consequently, one obtains eJt = � m=∓ eλm s1t � � � 1 t t2/2 0 1 t 0 0 1 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (B3) EP–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' Let us consider the EP–2 used for the quasi- stabilization of squeezing and entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=', the one obtained with degenerate resonators, κ2 = κ1 + 4 √ 2g, and κ3 = κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' The Jordan blocks here defining the matrix J = diag � J− s1(λ− s1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J− s2(λ− s2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J+ s1(λ+ s1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J+ s2(λ+ s2) � are J± s1(λ± s1) = λ± s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' s1 = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' J± s2(λ± s2) = � λ± s2 1 0 λ± s2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' s2 = (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (B4) and the singular matrix P reads P = � � � � � � � � � � −i i 0 i −i 0 −1 1 0 −1 1 0 0 √ 2 − 1 g 0 √ 2 − 1 g 0 −i √ 2 i g 0 i √ 2 − i g i i 0 −i −i 0 1 1 0 1 1 0 � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (B5) 13 where λ± sj are given as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (15) of the main text with ε = 0 and f(ε) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' In this case, the Jordan decomposi- tion of Γ gives rise to eJt = � m=∓ � � � eλm s1t 0 0 0 eλm s2t eλm s2tt 0 0 eλm s2t � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (B6) This decomposition is employed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' (6) to analytically obtain the time-evolved covariance matrix V and, conse- quently, the theoretical bounds presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFRT4oBgHgl3EQfejci/content/2301.13571v1.pdf'} diff --git a/rtE1T4oBgHgl3EQf3AU7/content/tmp_files/2301.03484v1.pdf.txt b/rtE1T4oBgHgl3EQf3AU7/content/tmp_files/2301.03484v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3b270133af33395992e76963d1a15ab88104731 --- /dev/null +++ b/rtE1T4oBgHgl3EQf3AU7/content/tmp_files/2301.03484v1.pdf.txt @@ -0,0 +1,5706 @@ +arXiv:2301.03484v1 [math.PR] 9 Jan 2023 +A Lyapunov approach to stability of positive +semigroups: An overview with illustrations +Marc Arnaudon1, Pierre Del Moral2 & El Maati Ouhabaz1 +1University of Bordeaux, Institut de Math´ematiques de Bordeaux, France. E-Mail: +marc.arnaudon@math.u-bordeaux.fr, Elmaati.Ouhabaz@math.u-bordeaux.fr +2Centre de Recherche Inria Bordeaux Sud-Ouest, Talence, 33405, FR. E-Mail: +pierre.del-moral@inria.fr +January 10, 2023 +Abstract +The stability analysis of possibly time varying positive semigroups on non +necessarily compact state spaces, including Neumann and Dirichlet boundary +conditions is a notoriously difficult subject. These crucial questions arise in +a variety of areas of applied mathematics, including nonlinear filtering, rare +event analysis, branching processes, physics and molecular chemistry. This ar- +ticle presents an overview of some recent Lyapunov-based approaches, focusing +principally on practical and powerful tools for designing Lyapunov functions. +These techniques include semigroup comparisons as well as conjugacy princi- +ples on non necessarily bounded manifolds with locally Lipschitz boundaries. +All the Lyapunov methodologies discussed in the article are illustrated in a +variety of situations, ranging from conventional Markov semigroups on general +state spaces to more sophisticated conditional stochastic processes possibly re- +stricted to some non necessarily bounded domains, including locally Lipschitz +and smooth hypersurface boundaries, Langevin diffusions as well as coupled +harmonic oscillators. +Keywords: Integral operators, semigroups, Markov and Sub-Markov semi- +groups, harmonic oscillators, Langevin diffusions, Lyapunov function, hyper- +surfaces, shape matrices, boundary problems. +Mathematics Subject Classification: Primary 47D08, 60J25, 47D06, 47D07, +47H07; secondary 47B65, 37A30, 37M25. +1 + +Contents +1 +Introduction +3 +1.1 +Description of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.2 +Some basic notation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.3 +Regularity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2 +A brief review on semigroups +8 +2.1 +A V -norm contraction theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.2 +Normalized semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.3 +Time homogenous models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +2.4 +Markov diffusion semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +2.5 +Sub-Markov semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +3 +Lyapunov design principles +25 +3.1 +Foster-Lyapunov conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +3.2 +Semigroup domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +3.3 +Some conjugacy principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4 +Boundary problems +30 +4.1 +Bounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +4.2 +Unbounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.3 +Smooth boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5 +Riccati type processes +39 +5.1 +Positive diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +5.2 +Matrix valued diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +5.3 +Logistic birth and death process +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +5.4 +Multivariate birth and death processes . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +6 +Some conditional diffusions +43 +6.1 +Coupled harmonic oscillators +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +6.2 +Half-harmonic linear diffusions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +6.3 +Linear diffusions in some domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +6.4 +Langevin diffusions in some domains . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +6.5 +Coupled oscillators in some domains . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +7 +Some hypersurface boundaries +51 +7.1 +Defining functions and charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +7.2 +The shape matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +7.3 +Surface and volume forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +7.4 +Boundary decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +2 + +1 +Introduction +1.1 +Description of the models +Let BpEq be the algebra of locally bounded measurable functions on a locally compact +Polish space E. We denote by BbpEq Ă BpEq the sub-algebra of bounded measurable +functions endowed with the supremum norm }.}. Let Qs,t, be a semigroup of positive +integral operators on BbpEq indexed by a continuous time indices s, t P T “ R` :“ +r0, 8r or by a discrete time index set T “ N, with s ď t. +For a given uniformly positive function V P BpEq, we let BV pEq Ă BpEq be the +sub-space of functions f P BpEq equipped with the norm }f}V :“ }f{V }. +We also let B8pEq Ă BpEq be the subalgebra of locally bounded and uniformly +positive functions V that grow at infinity; that is, supK V ă 8 for any compact set +K Ă E, and for any r ě V‹ :“ infE V ą 0 the r-sub-level set Vprq :“ tV ď ru Ă E is +a non empty compact subset. We denote by B0pEq :“ t1{V +: V P B8u Ă BbpEq the +sub-algebra of positive functions, locally lower bounded and that vanish at infinity. +For a given V P B8pEq, consider the subspace +B0,V pEq :“ tf P BpEq : |f|{V P B0pEqu . +We say that Qs,t is a V -positive semigroup on BV pEq for some Lyapunov function +V P B8pEq as soon as there exists some τ ą 0 and some function Θτ P B0pEq such +that for any 0 ă f P BV pEq and s ă t we have 0 ă Qs,tpfq P B0,V pEq as well as +Qs,s`τpV q{V ď Θτ +and +sup +|t´s|ďτ +` +|||Qs,t||| _ |||Qs,t|||V +˘ +ă 8. +(1) +The growth conditions stated above are discussed in some details in Section 1.3. +As shown in Section 3.1, the l.h.s. criterion in (1) can be seen as a uniform Foster- +Lyapunov condition (a.k.a. drift condition). +Foster-Lyapunov criterion dates back to the 1950s with the seminal articles [32, 38]. +These criteria are nowadays an essential tool to analyze the stability properties of +Markov semigroups on general state spaces [7, 26, 37, 39, 50, 51]. Their use in the +context of positive semigroup arising in discrete time nonlinear filtering goes back to +the pioneering articles [27, 64], based on coupling techniques developed in [41, 42]. +The extension of Foster-Lyapunov criterion to discrete or continuous time varying +positive semigroups and their normalized versions on general state spaces were further +developed in [21], extending Dobrushin’s ergodic coefficient techniques introduced +in [11, 12] and further developed in [15, 16, 17, 14, 22] to unbounded state space +models. +Recall that the Dobrushin’s ergodic coefficient of a Markov semigroup is the oper- +ator norm of the Markov integral operator acting on probability measures equipped +with the total variation norm (see for instance [16] and references therein). In the +same vein, the V -Dobrushin’s ergodic coefficient of a Markov transition is defined as +the operator norm of the Markov integral operator acting on probability measures +3 + +equipped with the V -norm. In this operator theoretical framework, the contraction +w.r.t. V -norms is deduced by coupling the Foster-Lyapunov criterion with a local +contraction on a sufficiently large compact sub-level set of the Lyapunov function. A +brief overview on this subject is provided in Section 2. +The local contraction on the compact sub-level sets of the Lyapunov function +is generally an easily verifiable condition. +This property is often deduced from a +Doeblin type local minorization property of integral operators on the compact sub- +level sets of the Lyapunov function. For instance, this local minorization condition +is satisfied as soon as the semigroup is lower bounded by an absolutely continuous +integral operator (a.k.a. transition kernel operator). This class of models includes +hypo-elliptic diffusion semigroups as well as some regular jump processes on non +necessarily bounded domains. +Even for diffusion semigroups with smooth densities on bounded manifolds with +entrance boundaries (i.e. boundary states that cannot be reached from the inside), the +existence of a sufficiently strong Lyapunov function is essential to ensure the stability +of the semigroup. In this context, the transition densities are null on entrance bound- +ary states so that the local minorization condition alone applied to some exhausting +sequence of compact subsets is not sufficient to ensure the stability of the process. +The exhausting sequence of compact subsets needs to be equivalent to the sub-level +sets of some sufficiently strong Lyapunov function near entrance boundaries. For a +more thorough discussion on this subject we refer to Section 2 and the article [21], +see also the series of Riccati-type diffusions discussed in Section 5. +The general problem of constructing Lyapunov functions for positive semigroups, +including for Markov semigroups often requires to have some good intuition about a +candidate for a Lyapunov function on some particular class of model. As for determin- +istic dynamical systems, the design of Lyapunov functions for sub-Markov semigroups +associated with a non-absorbed stochastic process requires to use some physical in- +sight on the stability and the behavior of the free evolution stochastic process near +possible absorbing boundaries. +Constructing Lyapunov functions for general classes of positive semigroups is well +known as a very hard problem in system theory as well as in applied probability +literature. The main subject of this article is to find practical ways to design these +Lyapunov functions for various classes of positive semigroups that have been discussed +in the literature, including conditional diffusions on manifolds with Neumann and +Dirichlet boundaries. We did our best to cover the subject as broadly as possible, we +also refer to the article [21] for additional historical and reference pointers. Due to the +vast literature on this subject we apologize for possible omissions of some important +contributions due to the lack of knowledge. +The remainder of this article is structured as follows: +In Section 2, we begin with a brief review of V -norm contraction theorems and +semigroup stability properties stemming from an assumed Lyapunov structure. Sec- +tion 2.1 is dedicated to time varying Markov semigroups. The extension of these re- +sults to time varying positive semigroups are discussed in Section 2.2. In Section 2.3, +4 + +we present some consequences of these results in the context of time homogenous +models, including existence of ground states and quasi-invariant measures. Section 2.4 +and Section 2.5 present different tools to design Lyapunov functions for continuous +time Markov semigroups and sub-Markov semigroups. We also illustrate these results +through different examples of semigroups arising in physics and applied probability, +including overdamped Langevin diffusions, Langevin and hypo-elliptic diffusions, as +well as typical examples of solvable one-dimensional sub-Markov semigroups such as +the harmonic oscillator, the half-harmonic oscillator and the Dirichlet heat kernel. +General comparison and conjugacy principles to construct Lyapunov functions for +positive semigroups are provided in Section 3. Boundary problems are discussed in +some details in Section 4. We then turn in Section 5 to the design of Lyapunov func- +tions for Riccati type processes, including positive definite matrix valued diffusions, +logistic and multivariate birth and death processes arising respectively in Ensemble +Kalman-Bucy filter theory and population dynamic analysis. +In Section 6 we illustrate the power of the Lyapunov approach in the context +of multivariate conditional diffusions. +Section 7 is dedicated to illustrations with +explicit computations of geometrical objects for the Lyapunov functions discussed in +Section 4.3 in the context of hypersurface Dirichlet boundaries. +1.2 +Some basic notation +We denote by CpEq Ă BpEq the sub-algebra of continuous functions and by CbpEq Ă +CpEq the sub-algebra of bounded continuous functions. +We also set CV pEq :“ BV pEq X CpEq, C0pEq :“ B0pEq X CpEq and C8pEq :“ +B8pEq X CpEq and C0,V pEq :“ B0,V pEq X CpEq. Note that none of the sub-algebras +B0pEq and B8pEq have an unit unless E is compact, the null function 0 R B0pEq but +the unit function 1 P C0,V pEq as soon as V P B8pEq. +Let MbpEq be the set of bounded signed measures µ on E equipped with the +total variation norm }µ}tv :“ |µ|pEq{2, where |µ| :“ µ` ` µ´. It stands for the total +variation measure associated with a Hahn-Jordan decomposition µ “ µ` ´ µ´ of the +measure. Also let PpEq Ă MbpEq be the subset of probability measures on E. +With a slight abuse of notation, we denote by 0 and 1 the null and unit scalars as +well as the null and unit function on E. +The action of Qs,t on BbpEq is given for any f P BbpEq by the formulae +Qs,tpfqpxq :“ +ż +Qs,tpx, dyq fpyq. +(2) +The left action of Qs,t on MbpEq is given for any η P MbpEq by the formulae +pη Qs,tqpdyq :“ +ż +ηpdxq Qs,tpx, dyq. +(3) +In this notation, the semigroup property takes the following form +Qs,uQu,t “ Qs,t +with +Qs,s “ I, +the identity operator. +(4) +5 + +In the above display, Qs,uQu,t is a shorthand notation for the composition Qs,u˝Qu,t of +the left or right-action operators. Unless otherwise stated, all the semigroups discussed +in this article are indexed by conformal indices s ď t in the set T . To avoid repetition, +we often write Qs,t without specifying the order s ď t of the indices s, t P T . +We denote by MV pEq be the space of measures µ P MbpEq equipped with the +operator V -norm |||µ|||V :“ |µ|pV q, and by PV pEq Ă MV pEq be the convex set of +probability measures. We associate with a function h P B0,V pEq the Boltzmann-Gibbs +transformation +Ψh : µ P PV pEq ÞÑ Ψhpµq P PV hpEq +(5) +with the probability measure +Ψhpµqpdxq :“ hpxq +µphq µpdxq +and +V h :“ V {h P B8pEq. +We also denote by |||Q|||V the operator norm of a bounded linear operator Q : f P +BV pEq ÞÑ Qpfq P BV pEq; that is +|||Q|||V :“ supt}Qpfq}V +: f P BV pEq +such that +}f}V ď 1u. +(6) +In terms of the V -conjugate semigroup +f P BbpEq ÞÑ QV pfq :“ QpV fq{V P BbpEq +we have +|||Q|||V “ }QV p1q} “ +ˇˇˇˇˇˇQV ˇˇˇˇˇˇ :“ supt}QV pfq} : f P BbpEq +such that +}f} ď 1u. +For a given measurable function f and a given measurable subset, we use the short- +hand notation +´8 ď inf +A f :“ inf +xPA fpxq ď sup +A +f :“ sup +xPA +fpxq ď `8. +For a given s P T and τ P T with τ ą 0, we consider the time mesh +rs, 8rτ:“ ts ` nτ P rs, 8r : n P Nu. +Throughout, unless otherwise is stated we write c for some positive constants +whose values may vary from line to line, and we write cα, as well as cpβq and cαpβq +when their values may depend on some parameters α, β defined on some parameter +sets. We also set a ^ b “ minpa, bq, a _ b “ maxpa, bq, and a` “ a _ 0 for a, b P R. +6 + +1.3 +Regularity conditions +The irreducibility condition f ą 0 ùñ Qs,tpfq ą 0 is satisfied if and only if we have +Qs,tp1q ą 0. We check this claim by contradiction. Assume that Qs,tp1q ą 0 and +consider a function f ą 0 and some x P E such that Qs,tpfqpxq “ 0. In this case, for +any ǫ ą 0 we would have +0 “ ǫ Qs,t p1fěǫq pxq ď Qs,tpfqpxq +by Fatou’s lemma we would find the contraction +lim inf +ǫÑ0 +Qs,t p1fěǫqp xq “ 0 ě Qs,tp1qpxq ùñ Qs,tp1qpxq “ 0. +Without further mention, all semigroups Qs,t considered in this article are assumed to +be semigroups of positive integral operators Qs,t on BbpEq satisfying the irreducibility +condition Qs,tp1q ą 0 for any s ď t. Notice that the condition +0 ă f P BV pEq +ùñ +@s ă t +0 ă Qs,tpfq P B0,V pEq +is met as soon as Qs,t is a strong V -Feller semigroup, in the sense that for any s ă t +we have Qs,tpBV pEqq Ă CV pEq and when we have Qs,tpV q{V P B0pEq. +To check +this claim, observe that for any positive function f P BV pEq and s ă t the function +Qs,tpfq is positive and continuous; and thus locally lower bounded. In this situation, +whenever }f}V ď 1, for any s ă t we have the comparison property +Qs,tpfq{V ď Qs,tpV q{V P B0pEq ùñ Qs,tpfq{V P B0pEq ðñ Qs,tpfq P C0,V pEq. +In summary, a strong V -Feller semigroup Qs,t is V -positive on BV pEq as soon as there +exists some τ ą 0 and some function Θτ P B0pEq such that the l.h.s. condition in (1) +is met and for any s ă t we have +Qs,tpV q{V P B0pEq +and +Qs,s`τpV q{V ď Θτ P B0pEq. +When V P C8pEq, we say that Qs,t is a V -positive semigroup on CV pEq as soon as +Qs,tpCV pEqq Ă C0,V pEq for any s ă t and condition (1) is met. +A V -Feller semigroup Qs,t for some V P C8pEq, in the sense that for any s ă t we +have Qs,tpCV pEqq Ă CV pEq, is also said to be V -positive on CV pEq as soon as there +exists some τ ą 0 and some function Θτ P B0pEq such that the l.h.s. condition in (1) +is met and for any s ă t we have +Qs,tpV q{V P C0pEq +and +Qs,s`τpV q{V ď Θτ P B0pEq. +Last but not least, observe that positive semigroups Qs,t with continuous time +indices s ď t P R` can be turned into discrete time models by setting Qp,n “ Qpτ,nτ +for any p ď n P N and some parameter τ ą 0. Up to a time rescaling, the parameter +τ ą 0 arising in the definition of a discrete time V -positive semigroups Qp,n can be +chosen as the unit time parameter. In this context, the r.h.s. condition in (1) is +automatically satisfied. +7 + +2 +A brief review on semigroups +2.1 +A V -norm contraction theorem +The aim of this section is to present some stability theorems for uniform V -positive +semigroups. We first examine the situation where Qs,t “ Ps,t is a semigroup of Markov +integral operators Ps,t on BbpEq. Note that the Lyapunov condition stated in the l.h.s. +of p1q ensures the following geometric drift condition +Ps,s`τpV q ď ǫτ V ` cτ +(7) +some parameter ǫτ P r0, 1r and some finite constant cτ ă 8. The geometric drift +condition (7) ensures that the sequence |||Ps,s`nτ|||V indexed by s ě 0 and n ě 1 is +uniformly bounded. In this context, the r.h.s. condition in (1) applied to Qs,t “ Ps,t +ensures that the operator norms of Ps,t are uniformly bounded w.r.t. any time horizon. +More precisely, whenever (7) is met we have the equivalence +sup +sě0 sup +těs |||Ps,t|||V ă 8 ðñ sup +|t´s|ďτ +|||Ps,t|||V ă 8. +(8) +Note that (8) is automatically satisfied whenever (7) is met for any τ ą 0 with +supτPr0,1s cτ ă 8. For instance, consider the Markov transition semigroup Ps,t of a +continuous time stochastic flow Xs,tpxq on some locally compact normed vector space +pE, }.}q with generator Lt defined on some common domain DpLq Ă BpEq. In this +context, for any non negative function V P DpLq and any parameters a ą 0, c ă 8 +and τ ą 0 we have +LτpV q ď ´aV ` c ùñ p7q and p8q +with +ǫτ “ p1 ` aτq´1 ă 1 +and +cτ “ cτ. (9) +The above estimate is rather well known, a detailed proof is provided in the appendix +on page 63. Further examples of Markov diffusion semigroups on Rn satisfying (7) +are discussed in Section 2.4. We further assume there exists some r0 ě 1 and some +function ατ : r P rr0, 8r ÞÑ ατprq P s0, 1s, such that for any r ě r0 we have +sup +px,yqPVprq2 }δxPs,s`τ ´ δyPs,s`τ}tv ď 1 ´ ατprq +with +Vprq :“ tV ď ru. +(10) +Consider the V -norm operator βV pPs,tq (a.k.a. the V -Dobrushin coefficient) of Ps,t +defined by +βV pPs,tq :“ +sup +µ,ηPPV pEq +|||pµ ´ ηqPs,t|||V {|||µ ´ η|||V . +(11) +In this notation, conditions (7), (8) and (10) ensure the existence of some parameter +τ ą 0 such that +sup +|t´s|ďτ +βV pPs,tq ă 8 +and +sup +sě0 βV pPs,s`τq ă 1. +(12) +8 + +The proof of the above assertion can be found in [21] (see also [22] in the context +of time homogeneous models). The next exponential contraction theorem is a direct +consequence of the operator norm estimates (12) and it is valid on abstract measurable +spaces as well as for any function V ě 1. +Theorem 2.1. Let Ps,t be a semigroup of Markov integral operators Ps,t on some +measurable state space E satisfying condition (12) for some function V ě 1 and some +parameter τ ą 0. In this situation, there exists a parameter b ą 0 and some finite +constant c ă 8 such that for any s ď t and µ, η P PV pEq we have the exponential +estimate +|||pµ ´ ηqPs,t|||V ď c e´bpt´sq |||µ ´ η|||V . +(13) +In particular, the above exponential Lipschitz estimates are met as soon as conditions +(7), (8) and (10) are satisfied. +The estimates (13) also hold for any s ě 0 and +t P rs, 8rτ as soon as (7) and (10) are satisfied for some τ ą 0. +The proof of Theorem 2.1 is based on discrete time type V -norm operator contrac- +tion techniques combining the geometric drift condition (7) with the total variation +estimates (10). The r.h.s. condition in (8) is a technical condition only made for con- +tinuous time semigroups to ensure that (13) also holds for continuous time indices. +For time homogeneous semigroups Pt :“ Ps,s`t the contraction estimate (13) en- +sures the existence of a single invariant probability measure µ8 “ µ8Pt P PV pEq. +In this context, similar approaches are presented in the article [37], simplifying the +Foster-Lyapunov methodologies and the small-sets return times estimation techniques +developed in [50]. Theorem 2.1 can be seen as an extension of Harris’ theorem to time +varying Markov semigroup. The operator-theoretic framework discussed above pro- +vides a very direct proof based on the V -Dobrushin coefficient (11). +For a more +thorough discussion on this subject we refer to [21, 22]. +Note that the strength of conditions (7) and (10) depends on the strength of the +function V : when the function V is bounded, the geometric drift condition (7) and +the uniform norm condition (8) are trivially met but in this case condition (10) is a +uniform contraction condition on the state E. In the reverse angle, when V P B8pEq +is a function with compact sub-level sets, the geometric drift condition (7) combined +with (8) ensures that µPs,t is a tight collection of probability measures indexed by +s ď t. In this context, the local contraction condition (10) is met if and only if for +any s ě 0 and any px, yq P Vprq2 there exists some probability measure µ on E (that +may depends on the parameters pτ, r, s, x, yq) such that +@z P tx, yu +δzPs,s`τpdyq ě ατprq µpdyq. +For instance, the above condition is met as soon as +Ps,s`τpx, dyq ě ps,s`τpx, yq ντpdyq +(14) +9 + +for some Radon positive measure ντ on E and some density function ps,s`τ, satisfying +for any r ě r0 the local minorization condition +0 ă inf +sPT inf +Vprq2 ps,s`τ +and +0 ă ντpVprqq ă 8. +(15) +For locally compact Polish spaces condition 0 ă ντpVprqq ă 8 is met as soon as V has +compact sub-levels sets Vprq with non empty interior and ντ is a Radon measure of full +support; that is ντ is finite on compact sets and strictly positive on non-empty open +sets. For time homogeneous models, also note that the l.h.s. minorization condition +(15) is satisfied as soon as px, yq P pE˝q2 ÞÑ pτpx, yq is a continuous positive function +on the interior E˝ of the set E. +Several illustrations of Theorem 2.1 are discussed in Section 2.4 in the context of +diffusion processes on Euclidean spaces as well as in Section 5 in the context of Riccati- +type diffusion on positive definite matrix spaces and multivariate birth and death jump +type processes on countable state spaces. The stability of Markov semigroups on man- +ifolds with entrance boundaries can also be analyzed using the Lyapunov techniques +developed in Section 4. For instance, as shown in Section 4.1, any absolutely con- +tinuous Markov semigroup Ps,t on a bounded connected subset E Ă Rn with locally +Lipschitz boundary BE satisfies the conditions of Theorem 2.1 with the (non unique) +Lyapunov function V pxq “ 1{ +a +dpx, BEq and the distance to the boundary defined +for any x P E by +dpx, BEq :“ inf t}x ´ y} : y P BEu. +We illustrate the above discussion with some elementary one dimensional examples. +Example 2.2. Consider a one dimensional Brownian on the compact interval E “ +r0, 1s with reflected boundaries. In this situation, Pt :“ P0,t coincides with the Neu- +mann heat semigroup on r0, 1s. In this context, recalling that the Neumann heat kernel +is smooth and strictly positive on the compact interval r0, 1s, the conditions of Theo- +rem 2.1 are satisfied with the unit Lyapunov function V pxq “ 1, as well as for any of +the Lyapunov functions V pxq “ 1{?x, V pxq “ 1{ +? +1 ´ x or V pxq “ 1{?x`1{ +? +1 ´ x. +The same reasoning applies to the one dimensional positive Riccati-type diffusions +with an entrance boundary at the origin discussed in Section 5. Reflecting this class of +positive diffusions at x “ 1, the conditions of Theorem 2.1 are satisfied on E “s0, 1s +with the Lyapunov functions V pxq “ 1{?x as well as for V pxq “ 1{?x ` 1{?1 ´ x. +2.2 +Normalized semigroups +For non necessarily Markov V -positive semigroups Qs,t one natural idea is to normalise +the semigroups. For any probability measure η P PV pEq we let Φs,tpηq P PV pEq be +the normalised distribution defined for any f P BV pEq by the formula +Φs,tpηqpfq :“ ηQs,tpfq +ηQs,tp1q +and we set +Qs,tpfqpxq :“ Qs,tpfqpxq +Qs,tp1qpxq “ Φs,tpδxqpfq. +(16) +10 + +The mapping Φs,t is a well defined semigroup on PV pEq. The denormalisation formula +connecting these semigroups is given for any t P rs, `8rτ by +µQs,tpfq “ Φs,tpµqpfq +ź +uPrs,trτ +Φs,upµqpQu,u`τp1qq. +(17) +with +rs, trτ:“ ts ` nτ P rs, tr : n P Nu. +To check this claim, observe that for any t :“ s ` nτ we have +Φs,s`pτpµqpQs`pτ,s`pp`1qτp1qq “ µQs,s`pp`1qτp1q{µQs,s`pτp1q +and therefore +ś +0ďpăn Φs,s`pτpµqpQs`pτ,s`pp`1qτp1qq “ µQs,s`nτp1q +The above formula coincides with the product formula relating the unnormalised +operators Qs,t with the normalised semigroup Φs,t discussed in [14, Section 1.3.2], see +also [17, Proposition 2.3.1] and [19, Section 12.2.1]. +We strengthen (14) and assume that for any s ě 0 and τ ą 0, the integral operator +Qt,t`τ has a density qs,s`τ with respect to some Radon positive measure ντ on E; that +is we have that +Qs,s`τpx, dyq “ qs,s`τpx, yq ντpdyq. +(18) +We also assume there exists some r0 ą 1 such that for any r ě r0 we have +0 ă ιrpτq :“ inf +sPT inf +Vprq2 qs,s`τ ď sup +sPT +sup +Vprq2 qs,s`τ ă 8 +and +ντpVprqq ą 0. +(19) +In this situation, for any r ě r0 and r ě r we have the uniform estimate +inf +Vprq Qs,s`τp1q ě r,rpτq :“ inf +Vprq Qs,s`τp1Vprqq ě ιrpτq ντpVprqq ą 0. +We associate with a given µ P PV pEq and some function H P B0,V pEq the finite rank +(and hence compact) operator +f P BV pEq ÞÑ T µ,H +s,t pfq :“ +Qs,tpHq +µspQs,tp1qq µtpfq P CV pEq +with the flow of measures µt “ Φs,tpµsq starting at µ0 “ µ. With this notation at +hand, one has the following theorem. +Theorem 2.3 ([21]). Consider a V -positive semigroups Qs,t with a density (18) sat- +isfying (19) for some parameter τ ą 0 and some r0 ą 1. In this situation, there exists +a parameter b ą 0 such that for any µ, η P PV pEq and any s ě 0 and t P rs, 8rτ we +have the local Lipschitz estimate +|||Φs,tpµq ´ Φs,tpηq|||V ď cpµ, ηq e´bpt´sq |||µ ´ η|||V . +(20) +11 + +For any pµ, Hq P pPV pEqˆB0,V pEqq there exists some finite constant cHpµq ă 8 such +that for any s ě 0 and t P rs, 8rτ we have +ˇˇˇˇ +ˇˇˇˇ +ˇˇˇˇ +Qs,t +µsQs,tp1q ´ T µ,H +s,t +ˇˇˇˇ +ˇˇˇˇ +ˇˇˇˇ +V +ď cHpµq e´bpt´sq. +(21) +For continuous time semigroups, the above estimates also hold for any continuous +time indices s ď t as soon as for any r ě r0 there exists some r ě r such that +infδPr0,τs r,rpδq ą 0. +The proof of Theorem 2.3 is based on discrete time type V -norm operator contrac- +tion techniques combining the geometric drift condition stated in the l.h.s. of (1) with +the local minorization condition stated in (19). The condition infδPr0,τs r,rpδq ą 0 is +a technical condition only made for continuous time semigroups to ensure that (20) +and (21) also hold for continuous time indices. +Theses regularity conditions are rather flexible as we will now explain. +Absolutely continuous integral operators arise in a natural way in discrete time +settings [17, 14, 27, 64] and in the analysis of continuous time elliptic diffusion absorp- +tion models [2, 29, 30, 61]. In connection to this, two-sided estimates for stable-like +processes are provided in [6, 43, 59, 63]. Two sided Gaussian estimates can also be +obtained for some classes of degenerate diffusion processes of rank 2, that is when the +Poisson brackets of the first order span the whole space [44]. This class of diffusions +includes frictionless Hamiltonian kinetic models. +Diffusion density estimates can be extended to sub-Markovian semigroups using +the multiplicative functional methodology developed in [15]. Whenever the trajecto- +ries of these diffusion flows, say t ÞÑ Xtpxq, where x P E is the initial position, are +absorbed on the smooth boundary BE of a open connected domain E, for any τ ą 0 +the densities qτpx, yq of the sub-Markovian semigroup Qτ (with respect to the trace of +the Lebesgue measure on E) associated with the non absorption event are null at the +boundary. Nevertheless, whenever these densities are positive and continuous on the +open set E2 for some τ ą 0, they are uniformly positive and bounded on any compact +subset of E; thus condition (19) is satisfied. +In this context, whenever Tpxq stands for first exit time from E and Trpxq the +first exit time from the compact level set Vprq Ă E starting from x P Vprq, for any +δ P r0, τs and r` ą r we have the estimate +Qδp1Vpr`qqpxq +:“ +E +` +1Vpr`qpXδpxqq 1Tpxqąδ +˘ +ě P +` +Tr`pxq ą δ +˘ +ě P +` +Tr`pxq ą τ +˘ +. +In this context, we have +inf +xPVprq P +` +Tr`pxq ą τ +˘ +ą 0 ùñ inf +δPr0,τs inf +Vprq Qδp1q ě inf +δPr0,τs r,r`pδq ą 0. +(22) +Whenever the interior E` :“ Vpr`q˝ is a connected domain, the l.h.s. estimate in (22) +is met as soon as the sub-Markovian semigroup Q` +τ associated with the non absorption +12 + +event at the boundary BE` has a continuous density px, yq P E2 +` ÞÑ q` +τ px, yq. To check +this claim, observe that for any x P Vprq we have +P +` +Tr`pxq ą τ +˘ +“ Q` +τ p1qpxq ě Q` +τ p1Vprqqpxq +“ +ż +q` +τ px, yq 1Vprqpyq ντpdyq +ě +ντpVprqq inf +Vprq2 q` +τ ą 0. +It is out of the scope of this article to review the different classes of absolutely con- +tinuous operators and related two-sided Gaussian estimates arising in the analysis of +continuous time elliptic diffusion and particle absorption models. For a more thorough +discussion on this topic we refer to the series of reference pointers presented above. +Needless to say that the design of Lyapunov functions is a crucial and challenging +problem in the stability analysis of positive semigroups. We have chosen to concen- +trate our review on presenting practical and general principles for designing Lyapunov +functions. +2.3 +Time homogenous models +For time homogeneous models we use the notation +pΦt, Qt, Qtq :“ pΦ0,t, Q0,t, Q0,tq. +As expected for time homogeneous semigroups a variety of results follow almost im- +mediately from the estimates obtained in Theorem 2.3. Following [21], these results +include the existence of an unique leading eigen-triple +pρ, η8, hq P pR ˆ PV pEq ˆ B0,V pEqq +with +η8phq “ 1 +(23) +in the sense that for any t P T we have +Qtphq “ eρt h +and +η8Qt “ eρt η8 +or equivalently +Φtpη8q “ η8. +(24) +The eigenfunction h is sometimes called the ground state and the fixed point measure +η8 the quasi-invariant measure. +For any x P E we also have the product series +formulation +0 ă hpxq :“ +ź +ně0 +␣ +1 ` e´ρτ rΦnτpδxqpQτp1qq ´ Φnτpη8qpQτp1qqs +( +. +In this context, choosing pµ, Hq “ pη8, hq in (21), we readily check that +T η8,h +s,s`tpfq “ Tpfq :“ +h +η8phq η8pfq +and +ˇˇˇˇˇˇe´ρt Qt ´ T +ˇˇˇˇˇˇ +V ď chpη8q e´bt. +For any η P PV pEq we have the conjugate formulae +ΨhpΦtpηqq “ ΨhpηqP h +t +(25) +13 + +with the Doob h-transform of Qt defined by the Markov semigroup +P h +t +: f P BV hpEq ÞÑ P h +t pfq :“ e´ρt 1 +h Qtphfq P BV hpEq. +Observe that +η8 “ Φtpη8q ðñ +ηh +8 :“ Ψhpη8q “ ηh +8P h +t . +The Markov semigroup P h +t is sometimes called the transition semigroup of the h- +process, a.k.a. the process evolving in the ground state. +We further assume that Qt is a sub-Markov semigroup of self-adjoint operators on +L2pνq with respect to some locally finite measure ν on E. In addition, there exists +an orthonormal basis pϕnqně1 associated with a decreasing sequence of eigenvalues +ρn ď 0 such that +Qtpx, dyq “ +ÿ +ně1 +eρnt ϕnpxq ϕnpyq νpdyq. +(26) +In this context, the formulae (24) are satisfied with the parameters +pρ, hq “ pρ1, ϕ1q +and +η8pdxq “ Ψhpνqpdxq :“ +1 +νphq hpxq νpdxq. +Note that in this case h has unit norm νph2q “ 1. The spectral resolution (26) yields +for any t ě 0 and f P L2pνq the following decomposition +e´ρtQtpfqpxq ´ hpxq +η8phq η8pfq “ +ÿ +ně2 +eρh +nt ϕnpxq νpϕnfq +with +ρh +n “ ρn ´ ρ1. +(27) +This yields the following result. +Proposition 2.4. For any time horizon t ě 0 and any f P L2pνq we have the expo- +nential estimates +››››e´ρtQtpfq ´ +h +η8phq η8pfq +›››› +L2pνq +ď eρh +2 t ` +νpf 2q ´ νphfq2˘1{2 . +(28) +Whenever Qt is a positive semigroup of self-adjoint operators on L2pνq the Doob +h-transform P h +t is a semigroup of self-adjoint operators on L2pηh +8q and we have the +following spectral decomposition +Lemma 2.5. For any t ě 0 and f P L2pηh +8q we have +P h +t px, dyq “ ηh +8pdyq ` +ÿ +ně2 +eλnt hnpxq hnpyq ηh +8pdyq +(29) +with the L2pηh +8q orthonormal basis phnqně2 defined for any n ě 2 by +hn :“ ϕn{h +and +λn “ ρn ´ ρ1 ă 0 +and +ηh +8 “ Ψh2pνq. +14 + +Note that the density of the integral operator P h +t px, dyq w.r.t. ηh +8pdyq is given by +ph +t px, yq “ e´ρ1t +qtpx, yq +hpxqhpyq “ 1 ` +ÿ +ně2 +eλnt hnpxq hnpyq. +(30) +We further assume that h P B0pEq and P h +t is ultra contractive, in the sense that +for any t ą 0 we have +ˇˇˇˇˇˇP h +t +ˇˇˇˇˇˇ +L2pηh8qÞÑL8pηh8q “ e´ρ1t +sup +px,yqPE2 +qtpx, yq +hpxqhpyq “ +sup +px,yqPE2 ph +t px, yq ă 8. +(31) +Proposition 2.6. Assume that νpEq ă 8 and h P B0pEq. +In addition, for any +t ą 0 (31) holds and the mapping x ÞÑ +ş +ph +t px, yq νpdyq is u.s.c. and locally lower +bounded. In this situation, the function V :“ 1{h P B8pEq and for any t ą 0 we have +QtpV q{V P B0pEq. In addition, for any t ą 0 we have +QtpV q{V ď ct{V 2 P B0pEq. +(32) +2.4 +Markov diffusion semigroups +This section is mainly concerned with the design of Lyapunov functions for continuous +time Markov semigroups. To simplify notation, we only consider time homogeneous +models. +All the semigroups discussed in this section satisfy condition (8). +Thus, +by (14) the contraction theorem, Theorem 2.1 applies to all the Markov semigroups +discussed in this section as soon as the transition semigroups have a continuous density +with respect to the Lebesgue measure. +Section 2.4.1 presents some elementary principles based on spectral conditions +on the drift function and a simple way to design Lyapunov functions in terms of +the generator of diffusion process. These generator-type techniques are illustrated +in Section 2.4.2 in the context of overdamped Langevin diffusions. +The design of +Lyapunov functions for hypo-elliptic diffusions and Langevin diffusions are discussed +respectively in Section 2.4.3 and Section 2.4.4. +2.4.1 +Some general principles +Consider the Markov semigroup Pt of a diffusion flow Xtpxq on E “ Rn defined by +dXtpxq “ bpXtpxqq dt ` σpXtpxqq dBt. +(33) +In the above display, Bt is a n1-dimensional Brownian motion starting at the origin for +some n ě 1, b is a differentiable drift function from Rn into itself with gradient-matrix +∇b “ pBxibjq1ďi,jďn, and σ stands for some diffusion function from Rn into Rnˆn1. We +set Σ2 :“ σσ1, where σ1pxq :“ σpxq1 stands for the transposition of the matrix σpxq, +so that Σ2pxq :“ σpxqσ1pxq. The absolutely continuity of the transition semigroup +Ptpx, dyq “ PpXtpxq P dyq “ ptpx, yqνpdyq for some continuous transition densities +15 + +ptpx, yq (w.r.t. the Lebesgue measure νpdyq) is ensured as soon as pb, σq are globally +Lipschitz continuous and the diffusion matrix is invertible or more generally satisfying +a parabolic H¨ormander condition (see for instance [53, 58, 56] and references therein). +The generator L of the diffusion flow Xtpxq and its carr´e du champ operator ΓL are +given respectively by the formula +Lpfq :“ b1∇f ` 1 +2 Tr +` +Σ2∇2f +˘ +and +ΓLpf, gq :“ p∇fq1Σ2∇g. +(34) +The next proposition provides a rather elementary way to design a Lyapunov +function. +Proposition 2.7. Assume that σpxq “ σ0 for some σ0 P Rnˆn1 and we have +∇b ` p∇bq1 ď ´2λ I +for some +λ ą 0. +(35) +Then for any v ą 0 and t ą 0 there exists some δt ą 0 such that +V pxq :“ exp pv}x}q ùñ PtpV q{V ď ct{V δt. +(36) +The proof of the above proposition is rather technical, thus it is provided in the +appendix on page 63. +The next proposition is a slight extension of Theorem 2.6 [49] on reversible semi- +groups to stochastic flows in Euclidean spaces. It provides a rather simple way to +design Lyapunov functions in terms of generators. +Proposition 2.8. Assume there exists some α ą 0, β P R and 0 ă ǫ ă 1 such that +α W ` β ` LpWq ď ´ǫ ΓLpW, Wq. +(37) +In this situation, for any t ą 0 we have +V :“ exp p2ǫWq ùñ Pt pV q {V ď vt{V δt +(38) +with the parameters +vt “ exp +` +´2βǫ p1 ´ e´αtq{α +˘ +and +δt :“ p1 ´ e´αtq. +The proof of the above proposition follows word-for-word the proof of Theorem +2.6 in [49], thus it is provided in the appendix on page 64. +We further assume that Pt satisfies for any t ą 0 the sub-Gaussian estimate +Ptpx, dyq ď ct exp +ˆ +´ 1 +2σ2 +t +}y ´ mtpxq}2 +˙ +dy +(39) +for some parameters σt ą 0 and some some function mt on Rn such that +}mtpxq} ď ct p1 ` }x}q. +16 + +In this situation, for any n ě 1 and t ě 0 we have +V pxq :“ 1 ` }x}n ùñ }PtpV q{V } ă 8. +More refined estimates can be found when the function mt is such that +|mtpxq| ď ǫt |x| +with +ǫt Ps0, 1r +(40) +for some norm |.| on Rn. In this situation, observe that any v ě 0 and any centered +Gaussian random variable Y on Rn with identity covariance matrix In we have +e´v|x| E +´ +ev|mtpxq`σ2 +t Y |¯ +ď ct e´vp1´ǫtq|x|. +This yields the following lemma. +Lemma 2.9. Consider a Markov semigroup Pt satisfying the sub-Gaussian estimate +(39) as well as (40) for some norm |.| on Rn. Then for any v ě 0 and t ą 0 there +also exists some finite constant δt ą 0 such that +V pxq :“ exp pv|x|q ùñ PtpV q{V ď ct{V δt. +2.4.2 +Overdamped Langevin diffusion +Let Wpxq be some twice differentiable potential function from Rn into R. The over- +damped Langevin diffusion is defined by choosing in (33) the drift function +bpxq :“ ´γ ∇Wpxq +and +pn1, σpxqq “ pn, ρ Iq +for some +γ, ρ ą 0. +In this context, we have +p35q ðñ ∇2W ě pλ{γq I +for some +λ ą 0. +Also observe that +p37q ðñ α W ` β ` ρ2 +2 Trp∇2Wq ď +` +γ ´ ǫ ρ2˘ +}∇W}2. +The above condition is clearly met when W behaves as }x}m with m ě 1 at infinity; +that is, there exists some sufficiently large radius r such that for any }x} ě r we have +ˇˇTrp∇2Wpxqq +ˇˇ ď c1 }x}pm´2q` +and +}∇Wpxq}2 ě c2 }x}2pm´1q. +17 + +2.4.3 +Hypo-elliptic diffusions +Consider the Rn-valued diffusion (33) with pbpxq, σpxqq “ pAx, Σq, for some matrices +pA, Σq with appropriate dimensions. We assume that A is stable (a.k.a. Hurwitz); +that is its spectral abscissa ςpAq defined below is negative +ςpAq :“ sup tRe pλpAqq : λpAq P SpecpAqu ă 0. +(41) +In the above display SpecpAq denotes the spectrum of the matrix A, and Re pλpAqq +the real part of λpAq. We also assume that R :“ ΣΣ1 is positive semi-definite and the +pair of matrices pA, R1{2q are controllable, in the sense that the pn ˆ n2q-matrices +“ +R1{2, AR1{2 . . . , Ar´1R1{2‰ +has rank n. +(42) +Whenever ςpAq ă 0 we have +Ptpx, dyq “ +1 +a +detp2πCtq +exp +ˆ +´1 +2 py ´ mtpxqq1 C´1 +t +py ´ mtpxqq +˙ +dy +(43) +with the mean value function +x ÞÑ mtpxq :“ etAx ÝÑtÑ8 0 +and the covariance matrices Ct defined for any t ą 0 by +0 ă Ct :“ +ż t +0 +esAResA1 ds ÝÑtÑ8 C8 :“ +ż 8 +0 +esAResA1 ds. +Since A is stable, there exists some norm |.| on Rn such that the corresponding +operator norm satisfies |etA| ď elpAqt for some log-norm parameter lpAq ă 0. This +implies that +|mtpxq| “ |etAx| ď elpAqt |x|. +(44) +This clearly shows that the semigroup Pt of the hypo-elliptic Ornstein-Ulhenbeck +diffusion satisfies (39) and (40). +Let Pt be the Markov semigroup of the Rn-valued linear diffusion +dXtpxq “ pAXtpxq ` apXtpxqqq dt ` Σ dBt +(45) +with some bounded drift function a on Rn, an pn ˆ nq-matrix A satisfying (41), some +n1-valued Brownian motion Bt starting at the origin and some pn ˆ n1q-matrix Σ +satisfying the rank condition (42). +Using the stochastic interpolation formula (cf. Theorem 1.2 in [23]) given by +Xtpxq ´ Xtpxq “ +ż t +0 +ept´sqA1 a pXspxqq ds +we check the almost sure estimate +|Xtpxq ´ Xtpxq| ď c +for some finite constant c ă 8. +This yields the following proposition. +Proposition 2.10. For any v ą 0 and t ą 0 there exists some δt ą 0 such that +V pxq :“ exp pv|x|q ùñ PtpV q{V ď ct{V δt. +18 + +2.4.4 +Langevin diffusion +Consider the Langevin diffusion diffusion flow Xtpzq “ pXtpzq, Ytpzqq P pRr ˆ Rrq +starting at z “ px, yq P pRr ˆ Rrq and given by +dXtpzq +“ +Ytpzq{m dt +dYtpzq +“ +pbpXtpzqq ´ βYtpzq{mq dt ` σ dBt. +In the above display, Bt stands for an r-dimensional Brownian motion Bt starting at +the origin, σ, β, m ą 0 some parameters and b a function of the form +bpxq :“ ´γ x ` apxq +with +γ ą 0 +and +}a} ă 8. +In statistical physics, the above diffusion represents the evolution of N particles +Xtpzq “ pXi +tpzqq1ďiďN P R3N with mass m ą 0, position Xtpzq P R3N and momenta +Ytpzq. In this context, γ ą 0 stands for some friction parameter, and the diffusion +parameter σ ą 0 is related to the Boltzmann constant and the temperature of the sys- +tem. In this context, the function bpxq “ ´∇Wpxq is often described by the gradient +of some potential function W. For instance, for a quadratic confinement we have +Wpxq :“ γ}x}2{2 ` wpxq +with +}∇w} ă 8 +ùñ bpxq “ ´∇Wpxq :“ ´γ x ` apxq +and +apxq “ ∇wpxq. +Notice that Xtpzq can be rewritten in vector form as in (45) with n “ 2r, apx, yq “ +ˆ +0 +apxq +˙ +and the matrices +A “ +ˆ +0 +m´1 Inˆn +´γ Inˆn +´βm´1 Inˆn +˙ +and +Σ :“ +ˆ +0 +0 +0 +σInˆn +˙ +. +(46) +It is a simple exercise to check that A satisfies (41) and (42). +Consider the R2-valued stochastic process Xt “ pqt, ptq defined by +$ +’ +& +’ +% +dqt +“ +β pt +m dt +dpt +“ +´β +ˆBW +Bq pqtq ` σ2 +2 +pt +m +˙ +dt ` σ dBt +(47) +with some positive constants β, m, σ, a Brownian motion Bt, and a smooth positive +function W on R such that for sufficiently large r we have +@ |q| ě r +qBW +Bq pqq ě δ +` +Wpqq ` q2˘ +for some positive constant δ. This condition is clearly met when W behaves as q2l for +certain l ě 1 at infinity. We let V pq, pq be the function on R2 defined by +V pq, pq “ 1 ` 1 +2m p2 ` Wpqq ` ǫ +2 +ˆσ2 +2 q2 ` 2pq +˙ +with +ǫ ă σ2 +2m. +19 + +In this situation, there exists some a ą 0 and c ă 8 such that +LpV q ď ´aV ` c. +(48) +The proof of the above estimate is rather technical, thus it is provided in the appendix +on page 67. +2.5 +Sub-Markov semigroups +Sub-Markov semigroups are prototype-based models of positive integral operators. In +time homogeneous settings, these stochastic models are defined in terms of a stochastic +flow Xtpxq evolving on some metric Polish space pE, dq, some non negative absorption +potential function U on some non necessarily bounded Borel subset E Ă E. For a +given x P E we denote by Tpxq the exit time of the flow Xtpxq from E. +We associate with these objects, the sub-Markov semigroup QrUs +t +defined for any +f P BbpEq and x P E by +QrUs +t +pfqpxq “ E +ˆ +fpXtpxqq 1Tpxqąt exp +ˆ +´ +ż t +0 +UpXspxqqds +˙˙ +. +(49) +The above model can be interpreted as the distribution of a stochastic flow evolving +in an absorbing medium with hard and soft obstacles. Before killing, the flow starts +at x P E and evolves as Xtpxq. Then, it is killed at rate U or as soon as it exits the +set E. In the case E “ E, the flow cannot exit the set E and it is only killed at rate +U. This situation is sometimes referred a sub-Markov semigroup with soft obstacles +represented by the absorbing potential function U on E. When the flow may exit the +set E Ă E, the complementary subset C :“ E ´ E is interpreted as an hard obstacle, +a.k.a. an infinite energy barrier. +We illustrate the V -positive semigroup analysis developed in this article through +three typical examples of solvable sub-Markov semigroups arising in physics and ap- +plied probability. +2.5.1 +The harmonic oscillator +Consider the case E “ E “ R, and let Xtpxq “ Btpxq be a Brownian motion starting +at x P R and let Upxq “ x2{2. In this situation, the semigroup QrUs +t +“ Qt defined +in (49) coincides with the one dimensional harmonic oscillator. For any t ą 0, the +integral operator Qt has a continuous density w.r.t. the uniform measure ν on E given +by +qtpx, yq “ +ÿ +ně1 +eρnt ϕnpxqϕnpyq +(50) +with the L2pνq orthonormal basis eigenstates +ϕnpxq “ p2n´1pn ´ 1q!?πq´1{2 e´x2{2 Hn´1pxq +20 + +associated with the eigenvalues +ρn “ ´pn ´ 1{2q +and the Hermite polynomials +Hnpxq “ p´1qn ex2 Bne´x2. +In this context, the eigenstate associated with the top eigenvalue ρ “ ρ1 “ ´1{2 is +given by the harmonic function +hpxq “ ϕ1pxq “ π´1{4 e´x2{2. +(51) +The spectral resolution of integral operator P h +t px, dyq and its density ph +t px, yq with +respect to the invariant measure +ηh +8pdyq “ +1 +?π e´y2 dy +are given as in (29) and (30) with L2pηh +8q orthonormal basis defined for any n ě 2 by +hn “ p2n´1pn ´ 1q!q´1{2 Hn´1 +and +ρh +n “ ρn ´ ρ1 “ ´pn ´ 1q. +In this context, the h-process is given by the Ornstein-Uhlenbeck diffusion +dXh +t pxq “ B log hpXh +t pxqq dt ` dBt “ ´Xh +t pxq dt ` dBt. +(52) +In the above display, Bt “ Btp0q stands for the one dimensional Brownian motion +starting at the origin. The conjugate formula +Qtphfq{Qtphq “ P h +t pfq ðñ Qtpfq “ eρth P h +t pf{hq +(53) +yields the following proposition. +Proposition 2.11. For any time horizon t ě 0 we have +Qtpx, dyq “ +1 +a +coshptq +exp +ˆ +´x2 +2 pt +˙ +1 +?2πpt +exp +ˆ +´py ´ mtpxqq2 +2pt +˙ +dy +with the mean and variance parameters pmtpxq, ptq defined by +mtpxq “ x{coshptq +and +pt “ tanhptq. +The proof of the above proposition is a direct consequence of the conjugate formula, +thus it is provided in the appendix, on page 65. +Choosing V pxq “ 1 ` |x|n, for some n ě 1, we readily check that +V P C8pEq +and +QtpV q{V ď vt Qtp1q P C0pEq +(54) +where vt is a constant depending only on t. +21 + +2.5.2 +The half-harmonic oscillator +Consider the case E “s0, 8rĂ E “ R, and let Xtpxq “ Btpxq be a Brownian motion +starting at x P R and let Upxq “ x2{2. In this situation, the semigroup QrUs +t +“ Qt +defined in (49) coincides with the harmonic oscillator with an infinite barrier at the +origin BE “ t0u (a.k.a. the half-harmonic oscillator). Using the fact that +ex2{2 1 +2 B2 e´x2{2 “ Upxq ´ 1{2 +we have the conjugate formula +Qtpfqpxq +“ +e´t{2 e´x2{2 E +´ +fpYtpxqq eYtpxq2{2 1T Y pxqąt +¯ +with the Ornstein-Uhlenbeck diffusion +dYtpxq “ ´Ytpxq dt ` dBt +and +T Y pxq :“ inf tt ě 0 : Ytpxq P BEu . +(55) +Note that the stochastic flow Ytpxq coincides with the h-process of the harmonic +oscillator discussed in (52). Thus, by reflection arguments we have +QY +t pfqpxq +:“ +E +` +fpYtpxqq 1T Y pxqąt +˘ +“ E +` +fpBσtpǫtxqq 1Tpǫtxqąt +˘ +“ +ż 8 +0 +fpyq qY +t px, yq dy +with +qY +t px, yq :“ prtpx, yq ´ rtpx, ´yqq. +In the above display, pǫt, σtq stands for the parameters +pǫt, σtq :“ +˜ +e´t, +c +1 ´ ǫ2 +t +2 +¸ +and +rtpx, yq “ +1 +a +2πσ2 +t +exp +ˆ +´ 1 +2σ2 +t +py ´ ǫtxq2 +˙ +. +This yields the following proposition. +Proposition 2.12. For any t ą 0 and x P E “s0, 8r we have +Qtpx, dyq “ +sinh py mtpxqq +P +` +0 ď Z ď mtpxq{?pt +˘ ˆ +1 +?2πpt +exp +ˆ +´y2 ` mtpxq2 +2pt +˙ +νpdyq. +In the above display, νpdyq :“ 1r0,8rpyq dy stands for the trace of the Lebesgue measure +on the half-line, Z is a centered Gaussian variable with unit variance and pmtpxq, ptq +are the mean and variance parameters defined in Proposition 2.11. In addition, the +total mass function Qtp1qpxq is given by the formula +Qtp1qpxq “ 2 +e´ x2 +2 pt +a +coshptq +ˆ P p0 ď Z ď mtpxq{?ptq P C0pEq. +22 + +The proof of the above proposition follows the same lines of arguments as the +proof of Proposition 2.11; it is provided in the appendix, on page 66. +Choosing V pxq “ xn ` 1{x, for some n ě 1, we readily check that +V P C8pEq +and +QtpV q{V ď ct{V +P C0pEq. +(56) +The proof of the above estimate follows elementary but lengthly calculations, thus it +is provided in the appendix on page 68. +For any t ą 0, the integral operator Qt has a continuous density w.r.t. the uniform +measure ν on E given by +qtpx, yq +“ +ÿ +ně1 +eρnt ϕnpxq ϕnpyq +with the L2pνq orthonormal basis eigenstates +ϕnpxq “ +? +2 p22n´1p2n ´ 1q!?πq´1{2 e´x2{2 H2n´1pxq +associated with the eigenvalues +ρn “ ´pp2n ´ 1q ` 1{2q. +In this context, the eigenstate associated with the top eigenvalue ρ “ ρ1 “ ´3{2 is +given for any x Ps0, 8r by the harmonic function +hpxq “ ϕ1pxq “ 2π´1{4 x e´x2{2 “ h0pxq H1pxq +with the ground state h0 of the harmonic oscillator discussed in (51). Note that h +coincides with the restriction on s0, 8r of the first excited state of the harmonic- +oscillator (negative on s ´ 8, 0s and crossing the origin at x “ 0). +The spectral resolution of integral operator P h +t px, dyq and its density ph +t px, yq with +respect to the invariant measure +ηh +8pdyq “ +4 +?π y2 e´y2 1s0,8rpyq dy +are given for any x, y Ps0, 8r as in (29) and (30) with L2pηh +8q orthonormal basis defined +for any n ě 2 and x Ps0, 8r by the odd Hermite functions +hnpxq “ p22np2n ´ 1q!q´1{2 H2n´1pxq{x +and +ρh +n “ ´2pn ´ 1q. +In this context, the h-process is given by the diffusion +dXh +t pxq “ B log hpXh +t pxqq dt ` dBt “ +ˆ +1 +Xh +t pxq ´ Xh +t pxq +˙ +dt ` dBt. +(57) +23 + +2.5.3 +The Dirichlet heat kernel +Let Xtpxq “ Btpxq be a Brownian motion starting at x P E :“s0, 1rĂ E :“ R and +Tpxq be the first time t ě 0 the process Btpxq P BE :“ t0, 1u. Choosing U “ 0 in +(49), the semigroup QrUs +t +“ Qt takes the following form +Qtpfqpxq :“ EpfpBtpxqq 1Tpxqątq. +For any t ą 0, the integral operator Qt has a continuous density w.r.t. the uniform +measure ν on E given by the Dirichlet heat kernel +qtpx, yq “ +ÿ +ně1 +eρnt ϕnpxqϕnpyq +(58) +with the L2pνq orthonormal basis eigenstates +ϕnpxq “ +? +2 sin pnπxq +associated with the eigenvalues +ρn “ ´pnπq2{2. +In this context, the eigenstate hpxq “ ϕ1pxq “ +? +2 sin pπxq associated with the top +eigenvalue ρ “ ρ1 “ ´π2{2 is strictly positive except at the boundary t0, 1u. By +removing the boundary, the semigroup P h +t of the process evolving in the ground state +hpxq on the open interval E :“s0, 1r is a self-adjoint operators on L2pηh +8q with +ηh +8pdxq “ h2pxq νpdxq “ 2 sin2 pπxq 1Epxq dx. +In addition, we have the spectral decomposition (29) with the L2pηh +8q orthonormal +basis eigenstates +hnpxq :“ sin pnπxq{ sin pπxq +associated with the eigenvalues +λn “ ´π2pn2 ´ 1q{2 ă 0. +Our next objective is to estimate the density ph +t px, yq of the integral operator P h +t px, dyq +w.r.t. ηh +8 defined in (30). Recalling that | sin pnyq| ď n| sin pyq|, for any n ě 1 and +y P R, for any x P E we have the diagonal estimate +ph +t px, xq ´ 1 “ +ÿ +ně2 +eρh +nt hnpxq2 +with +hnpxq2 “ +ˆsin pnπxq +sin pπxq +˙2 +ď n2 +so that condition (31) is satisfied. +Observe that the function +V +: x P E ÞÑ V pxq :“ +? +2{hpxq P r1, 8r +24 + +is locally bounded with compact level sets given for any 0 ă ǫ ď 1 by the formulae +Kǫ :“ tx Ps0, 1r : V pxq ď 1{ǫu “ tx : sin pπxq ě ǫu Ă E. +In any dimension we can use the intrinsic ultracontractivity to produce a Lyapunov +function V . Let E be a bounded domain of Rn for some n ě 1 and assume that it is +a C1,α domain for some α ą 0. Denote by qtpx, yq the Dirichlet heat kernel on E. By +[57] one has +qtpx, yq ď ct dpx, BEqdpy, BEq +for some constant ct independent of x and y. Here dpx, BEq denotes the distance from +x to the boundary of E. Set V pxq “ +1 +dpx,BEq. The above intrinsic ultracontractivity +implies +QtpV qpxq “ +ż +E +qtpx, yqV pyqdy ď ct|E| dpx, BEq +which in turn gives QtpV q{V ď ct|E|{V 2 P B0pEq, where |E| stands for the volume of +the bounded set E. +3 +Lyapunov design principles +The aim of this section is to present some general principles to construct Lyapunov +functions for positive semigroups. +Section 3.1 provides equivalent formulations of +the Lyapunov condition in (1) encountered in the literature in terms of exhausting +sequences of compact level sets. +This section also presents simple ways to design +Lyapunov functions for sub-Markov semigroups on normed spaces in terms of their +generators. +Section 3.2 presents some principles to construct Lyapunov functions +for positive semigroups dominated by semigroups with known Lyapunov functions. +Section 3.3 is dedicated to the design of Lyapunov functions for conjugate semigroups. +All the principles discussed in this section are illustrated in Section 5 as well as in +Section 6 in the context of conditional diffusions. +3.1 +Foster-Lyapunov conditions +For time homogeneous models Qs,s`t :“ Qt, the l.h.s. condition in (1) takes the form +QτpV q{V ď Θτ P B0pEq. In terms of the compact sets Kǫ :“ tΘτ ě ǫu, the l.h.s +Lyapunov condition in (1) yields for any τ ą 0 the estimate +QτpV qpxq ď ǫ V pxq ` 1Kǫpxq cǫ +(59) +for any ǫ ą 0 with the parameter cǫ :“ supKǫpV Θτq ă 8. This implies that for any +n ě 1 we have +QτpV qpxq ď ǫn V pxq ` 1Kǫnpxq cǫn +(60) +25 + +where Kǫn Ă E stands for some increasing sequence of compacts sets and cǫn some +finite constants, indexed by a decreasing sequence of parameters ǫn P r0, 1s such that +ǫn ÝÑ 0 as n Ñ 8. +In the reverse angle, assume that QτpV q{V is locally lower +bounded and lower semicontinuous. +In this situation, condition (60) ensures that +QτpV q{V P B0pEq for any τ ą 0. Indeed, for any δ ą 0, there exists some n ě 1 such +that ǫn ă δ and we have +tQτpV q{V ě δu Ă tQτpV q{V ą ǫnu Ă Kǫn. +Since tQτpV q{V ě δu is a closed subset of a compact set it is also compact. +More generally, whenever (60) is met for some exhausting sequence of compact +sets Kǫn, in the sense that for any compact subset K Ă E there exists some n ě 1 +such that K Ă Kǫn we have +inf +K QτpV q{V ě inf +Kǫn QτpV q{V ě ǫn. +This ensures that the function QτpV q{V is necessarily locally lower bounded. In this +situation, we have QτpV q{V P B0pEq as soon as QτpV q{V is lower semicontinuous. +Notice that the sub-level set Vprq :“ tV ď ru of the Lyapunov function V P B8pEq +and the ǫ-super-level sets Kǫ :“ tΘτ ě ǫu of Θτ P B0pEq are equivalent compact +exhausting sequences, in the sense that for any r ě 1 we have +Vprq Ă Kǫr Ă Vprǫq +with +ǫr :“ inf +Vprq Θτ +and +rǫ :“ sup +Kǫr +V. +Whenever E is a locally compact Polish space, the abstract sequence Cn :“ Kǫn +in (60) is automatically exhausting; that is, we have that E “ Yně0Cn with Cn is +included in the interior C˝ +n`1 of the compact set Cn`1. To check this claim, observe +that for any n ě 1 there exists some mn ě n such that +Cn Ă tΘτ ě inf +Cn Θu Ă Cmn Ă tΘτ ě inf +Cmn Θu. +Thus, the exhausting sequence Cn is equivalent to the one defined by the super-level +sets of Θτ. +The rather abstract condition (60) is often presented in the literature as an initial +condition to check on a case-by-case basis to analyze the stability property of time +homogenous sub-Markov semigroups (see for instance [31, 36], as well as Section 17.5 +in [22] in the context of Markov semigroups and the references therein). +We end this section with a brief discussion on condition (60) in the context of the +sub-Markov semigroup discussed in (49). Note that this semigroup can be turned into +a Markov semigroup by sending the killed process into a cemetery state, say ∆, at +the killing time. In this interpretation, functions on E are extended to E∆ “ E Yt∆u +by setting fp∆q “ 0. More interestingly, whenever E is locally compact its topology +coincides with the weak topology induced by C0pEq :“ B0pEqXCbpEq, and inversely (cf. +Proposition 2.1 in [1]). In this context a continuous function f vanishes at infinity +26 + +if and only if its extension to the one point compactification (a.k.a. +Alexandroff +compactification) E∆ :“ E Y t∆u (obtained by setting fp∆q “ 0) is continuous. For +locally compact spaces, we also recall that the one point extension E∆ is compact. +Whenever it exists, the generator LU of these sub-Markov semigroups QrUs +t +are de- +fined on domain of functions DpLUq Ă B0pEq. As expected, the analysis of this class +of models in terms of generators often requires to develop a sophisticated analysis tak- +ing into account the topological structure of the set E. To the best of our knowledge, +there is no simple sufficient condition to check (60) in terms of these generators. +The situation is greatly simplified for sub-Markov semigroups with soft obstacles. +When E “ E is a locally compact normed space pE, }.}q we let L be the generator of +the flow Xtpxq. In this situation, the generator of the sub-Markov semigroup QrUs +t +is +given by LU “ L´U. We further assume that L and LU are defined on some common +domain DpLq Ă BpEq. +Lemma 3.1 ([31]). Let V, V0 P DpLq be a couple of functions such that V, V0 ě 1 and +V pxq ÝÑ}x}Ñ8 8 +and +V pxq{V0pxq ÝÑ}x}Ñ8 8. +(61) +In this situation, condition (60) is satisfied as soon as there exists some finite constant +c0 ă 8 such that +LUpV0q{V0 ď c0 +and +LUpV qpxq{V pxq ÝÑ}x}Ñ8 ´8. +(62) +Note that in this context, the compact sets in (60) are given for some sufficiently +large radii rǫ ą 0 by the closed balls: +Kǫ “ Bp0, rǫq :“ tx P E : }x} ď rǫu. +(63) +3.2 +Semigroup domination +For a given p ě 1 we clearly have +V P B8pEq ðñ V p P B8pEq +and +BV 1{ppEq Ă BV pEq Ă BV ppEq. +We say that a V -positive semigroup Qs,t is p-dominated by a collection of integral +operators Qs,t on BV ppEq and we write Q !p Q as soon as for any non negative +function f P BV pEq and any s ď t we have +Qs,tpfq ď ct´sppq Qs,tpf pq1{p. +To simplify notation, when p “ 1 we write Q ! Q instead of Q !1 Q. Observe that +Q !p Q +ùñ +@s ď t +pQs,tpV q{V qp ď ct´sppqp Qs,tpV pq{V p. +This yields for any τ ą 0 and θτ P B0pEq the Lyapunov estimate +Qs,s`τpV pq{V p ď θp +τ ùñ Qs,s`τpV q{V ď cτ θτ. +(64) +27 + +We illustrate the above domination property with the Langevin diffusion flow X paq +t +pzq “ +pXtpzq, Ytpzqq P pRn ˆ Rnq starting at z “ px, yq P pRn ˆ Rnq and defined by the hypo- +elliptic diffusion +dXtpzq +“ +Ytpzq{m dt +dYtpzq +“ +papXtpzqq ´ γXtpzq ´ βYtpzq{mq dt ` σ dBt. +(65) +In the above display, σ, γ, β, m ą 0 stands for some parameters and a some Lipschitz +function on Rn, with n ě 1. Notice that when a “ 0, the flow X p0q +t +pzq resumes to +an hypo-elliptic Ornstein-Ulhenbeck on R2n. +Consider a bounded open connected +domain D Ă Rn and set +@z P E :“ D ˆ Rn +T paqpzq :“ inf +! +t ě 0 : X paq +t +pzq P BE +) +. +We associate with these objects, the sub-Markov semigroup defined for any f P BbpEq +and z “ px, yq P E by +Qpaq +t pfqpzq :“ E +´ +fpX paq +t +pzqq 1T paqpzqąt +¯ +. +In this situation, we have +sup +D +a ă 8 +ùñ +@p ą 1 +Qpaq !p Qp0q. +(66) +The proof of the above assertion is a direct consequence of Girsanov’s theorem and +H¨older’s inequality. For the convenience of the reader, a detailed proof is provided in +the appendix on page 70. +To emphasize the role of the absorption in sub-Markov semigroups we return to the +class of models discussed in (49). We let Pt be the free evolution Markov semigroup +associated with the stochastic flow Xtpxq. Assume that QrUs +t +p1q P B0pEq and +}QrUs +t +pV q{V } ă 8 +for some t ą 0 and V P B8pEq. +(67) +Applying H¨older’s inequality and choosing Vp :“ V 1{p P B8pEq with p ą 1 we readily +check the estimate +QrUs +t +pVpq{Vp ď ctppq QrUs +t +p1q1´1{p P B0pEq. +(68) +The next lemma provides several practical conditions to check the uniform estimate +(67) for sub-Markov semigroups associated with soft obstacles. +Lemma 3.2. Consider the sub-Markov semigroup discussed in (49) when E “ E is +a locally compact normed space pE, }.}q. Assume that the generators L and LU of +the flows Pt and QrUs +t +are defined on some common domain DpLq Ă BpEq. In this +situation, for any V P B8pEq X DpLq and parameter a ą 0 we have +LUpV q ď ´aV ` c +ùñ +@t ě 0 +}QrUs +t +pV q{V } ă 8. +(69) +Whenever U P B8pEq X DpLq, for any a0 ě 0 and a1 P R we have +LpUq ď a0 ` a1U +ùñ +@t ě 0 +}QrUs +t +pUq} ă 8. +(70) +28 + +The proof of the above lemma follows essentially the same lines of arguments as +the proof of Lemma 3.1; thus it is provided in the appendix, on page 69. +Whenever E “ E and the absorption potential function U is bounded, we have +P ! QrUs ! P. In this context, there is no hope to have that QrUs +t +p1q P B0pEq for +some t ą 0. Nevertheless, for any V P B8pEq and any time horizon t ą 0 we have +QrUs +t +pV q{V P B0pEq ðñ PtpV q{V P B0pEq. +In this situation, the design of Lyapunov functions V satisfying (1) or equivalently +Foster-Lyapunov conditions of the form (60) is equivalent to the problem of finding a +Lyapunov function for the Markov semigroup Pt. +Whenever Pt is stable, in the sense that it has a Lyapunov V P B8pEq such that +PtpV q{V P B0pEq for some t ą 0, then the domination property QrUs ! P yields +automatically a Lyapunov function for QrUs +t +. +Whenever Pt is not necessarily stable but we have }PtpV q{V } ă 8 for some t ą 0 +and V P B8pEq, applying (68) the domination property QrUs ! P ensures that for +any p ą 1 we have Vp :“ V 1{p P B8pEq and +QrUs +t +p1q P B0pEq +ùñ +QrUs +t +pVpq{Vp P B0pEq. +Last, but not least, note that the above discussion extends without difficulties to +time varying models. +3.3 +Some conjugacy principles +For any given V P B8pEq, observe that for any positive function H, +H P B0,V pEq ðñ V H :“ V {H P B8pEq. +Thus, Qt is a V -positive semigroup on BV pEq if and only if the H-conjugate semigroup +QH +t pfq :“ QtpfHq{H is a V H-positive semigroup on BV HpEq. In this situation, any +semigroup Q ! QH dominated by QH yields for any s ě 0 and t ą 0 the Lyapunov +estimate +Qs,s`tpV Hq{V H ď ct QtpV q{V P B0pEq. +To get one step further, observe that +QtpV q{V “ Qtp1q QtpV q{V. +In this notation, for any H P B0,V pEq and any V -positive semigroup Qt on BV pEq +such Qtp1q P B0pEq and +ˇˇˇˇˇˇQt +ˇˇˇˇˇˇ +V ă 8 we have +Q ! QH ùñ Qs,s`tpV Hq{V H ď ct Qtp1q P B0pEq. +(71) +29 + +We illustrate the above comparison principles with an elementary example. Let +E :“ R and W P B8pRq be some non negative function. Consider the stochastic flow +XW +t pxq of a one-dimensional Langevin diffusion on E with generator +Lpfq “ 1 +2 e2WB +` +e´2WBf +˘ +. +(72) +We associate with a given open connected interval E Ă E, the sub-Markov semigroup +Qt on BbpEq defined by +Qtpfqpxq :“ EpfpXW +t pxqq 1T W +BEpxqątq +with +T W +BEpxq :“ inf +␣ +t ě 0 : XW +t pxq P BE +( +. +(73) +Observe that +H :“ e´W ùñ U :“ H´11 +2 B2H “ 1 +2 +` +pBWq2 ´ B2W +˘ +. +(74) +When W “ 0 the flow X0 +t pxq “ Btpxq coincides with the Brownian flow Btpxq starting +at x. Thus, by a change of probability we check that +Qt “ QH +t +with +Qtpfqpxq :“ E +´ +fpBtpxqq 1T 0 +BEpxqąt e´ şt +0 UpBspxqq ds¯ +. +(75) +Whenever E “s0, 1r the semigroup Qt is dominated by the Dirichlet heat kernel on +s0, 1r. When E “ R, respectively E “s0, 8r, and Upxq ě c ` ς x2{2, for some c ă 8 +and ς ą 0, the semigroup Qt is dominated by the harmonic oscillator, respectively the +half-harmonic oscillator. All of these dominating semigroups are completely solvable +with Qtp1q P B0pEq and known Lyapunov functions. +4 +Boundary problems +Let pE, dq be a locally compact Polish space with a distinguish complete metric d : +px, yq P E2 ÞÑ dpx, yq P R`. +We recall that these metric spaces are complete σ- +compact and locally compact metric spaces, thus they have the Heine-Borel property, +that is each closed and bounded subsets in E are compact. +We also recall that a subspace E Ă E is Polish if and only if it is the intersection +of a countable collection of open subsets. The distance from x P E to a measurable +subset A Ă E is denoted by +dpx, Aq :“ inf tdpx, yq : y P Au. +We also denote by BE :“ E ´E˝ the boundary of some domain (open and connected) +E Ă E, where E and E˝ stand for the closure and the interior of a subset E. +In the further development of the article, χ stands for some decreasing positive +function χ on s0, 8r such that for any 0 ă α ă 1 we have +lim +αÑ0 χpαq “ `8 +χpαq ă 1{α +and +χpαq :“ +ż α +0 +χpuqdu ă 8. +30 + +Definition 4.1. We associate with χ the function VB P CpEq defined by +VB : x P E ÞÑ VBpxq :“ χpdpx, BEqq Ps0, 8r. +(76) +For instance, we can choose χpuq “ 1{u1´ǫ, for some ǫ Ps0, 1r. For any r ą 0 the +r-sub-level sets of VB are given by the closed subsets +VBprq :“ tx P E : VBpxq ď ru “ tx P E : dpx, BEq ě χ´1prqu. +Note that VB P C8pEq as soon as E is compact. +4.1 +Bounded domains +Let E Ă E :“ Rn be some bounded domain with locally Lipschitz boundary BE, for +some n ě 1. Consider a semigroup of integral operators +Qtpx, dyq “ qtpx, yq dy +(77) +having for any t ą 0 a bounded density px, yq P E2 ÞÑ qtpx, yq P r0, 8r w.r.t. the trace +of the Lebesgue measure νpdyq “ dy on E. In this situation, we have the following +lemma. +Lemma 4.2. For any t ą 0 we have +VB P C8pEq +and +}QtpVBq} ď ct +ż +E +χpdpx, BEqq dx ă 8. +(78) +The proof of the above lemma follows from an elementary change of variable +formulae, thus it is provided in the appendix, on page 70. +The estimate (78) clearly applies to the class of sub-Markov semigroups QrUs +t +de- +fined in (49) for any choice of the absorption potential function, as soon as the semi- +group QrUs ! Q is dominated by a collection of integral operators Qtpx, dyq having +a bounded density qtpx, yq on E2 w.r.t. the Lebesgue measure on E. For instance, +when the transition semigroup of the free evolution flow Xtpxq in (49) has a density +ptpx, yq for any non negative function f on E and any x P E we have +QrUs +t +pfqpxq ď +ż +qtpx, yq fpyq dy +with +qtpx, yq :“ ptpx, yq 1Epyq. +We summarize the above discussion with the following proposition. +Proposition 4.3. Assume that QrUs ! Q is dominated by a collection of integral +operators Qt satisfying (77). Then, +QrUs +t +pVBq{VB ď ct{VB P B0pEq. +31 + +The choice of the Lyapunov function V is clearly not unique. For instance, when +E “s0, 1r instead of VB we can choose V pxq :“ 1{?x ` 1{?1 ´ x. For the Dirichlet +heat kernel discussed in Section 2.5.3 we can also choose V pxq “ 1{ sin pπxq. +We emphasize that sub-Markov integral operators on the compact interval E “ +r0, 1s with a positive continuous density w.r.t. the Lebesgue measure on E arise when +the free evolution process is reflected at both sides of the interval. In this context +the process is not conditioned by any type of non absorption at the boundaries. In +this context, the unit function V “ 1 belongs to B8pEq. +In the same vein, sub- +Markov integral operators with mixed boundary conditions on the left-closed interval +E “ r0, 1r, or respectively on the right-closed interval E “s0, 1s arise when the free +evolution process is reflected at the Neumann boundary BNE :“ t0u and non absorbed +at the Dirichlet boundary BDE “ t1u, or respectively reflected at BNE :“ t1u and non +absorbed at BDE “ t0u. More generally, consider a bounded domain Ω Ă Rn with +Lipschitz boundary BΩ “ BDΩYBNΩ consisting of two disjoint connected components +BDΩ and BNΩ closed in Rn, and set E :“ Ω Y BNΩ. In this notation, the function +VBpxq :“ χ pdpx, BDEqq belongs to C8pEq. In addition, for any bounded density qtpx, yq +on E2 we have the uniform estimate +ż +E +qtpx, yq VBpyq dy ď ct +ż +E +VBpyq dy ă 8. +The above estimate also holds for the function VBpxq “ χ pdpx, BEqq. +4.2 +Unbounded domains +When the domain E is not bounded the function VB R B8pEq. In this context, one +natural way to design a Lyapunov function V P B8pEq is to consider an auxiliary +function VE P C8pEq with VEpxq ě 1 for any x P E. In this situation, we have +V :“ VB ` VE P C8pEq. +To check this claim, observe that the sub-level sets of VB are given by the closed +subsets +VBprq :“ tVB ď ru “ tx P E : dpx, BEq ě χ´1prqu Ă E +and we have the compact inclusion +Vprq :“ tV ď ru Ă VEprq X VBprq +with +VEprq :“ tx P E : VEpxq ď ru. +This yields the following easily checked proposition. +Proposition 4.4. For any t ą 0 we have +}QtpVBq} _ }QtpVEq} ă 8 +ùñ +QtpV q{V ď ct{V P B0pEq. +When Qtp1q P B0pEq we also have +}QtpVBq} _ }QtpVEq{VE} ă 8 ùñ QtpV q{V ď ct Qtp1q P B0pEq. +32 + +The design of a function VE is rather flexible. For instance, assume that Q ! P is +dominated by some Markov integral operators Pt on BbpEq such that }PtpVEq{VE} ă 8 +for some VE P B8pEq. In this situation, we have }QtpVEq{VE} ă 8 as well as +}VE Qtp1q} ă 8 ùñ }QtpVEq} ă 8. +For instance, when Pt satisfies the sub-Gaussian estimates (39) on E “ Rn we can +choose VEpxq :“ 1`}x}k, for some k ě 1, as soon as the function Qtp1qpxq ÝÑ}x}Ñ8 0 +faster than }x}´k. +When the domain E and its boundary BE are both non necessarily bounded, it +may happens that Qtp1q P B0pEq but QtpVBq R BbpEq. In this situation, we can use +the following proposition. +Proposition 4.5. Assume there exists some VE P C8pEq with VEpxq ě 1 for any x P E +and such that +}QtpVBq{VE} _ }QtpVEq{VE} _ }Qtp1qVE} ă 8. +Then we have +QtpV q{V ď ct{V P B0pEq. +Proof. Using the following decompositions +QtpVBq “ Qtp1qVE QtpVBq{VE +and +QtpVEq “ Qtp1qVE QtpVEq{VE +and applying Proposition 4.4 we have +}QtpVBq} _ }QtpVEq} ă 8 +and therefore +QtpV q{V ď ct{V P B0pEq. +This ends the proof of the proposition. +The case Qtp1q R B0pEq can also be handle whenever the pair pVB, VEq can be +chosen so that +@δ ą 0 +VB V δ +E P C8pEq. +(79) +For instance we can choose for some v ą 0 and ǫ Ps0, 1r the functions +VEpxq :“ exp pv}x}q +and +χpuq :“ 1{u1´ǫ. +Observe that +dpx, BEq ď }x} ` dp0, BEq +and +VBpxq ě χp}x} ` p1 _ dp0, BEqqq +and for any m ě 0 and δ ą 0 we have +VEpxq ě cvpm, δq p1 ` }x}qpm`1q{δ. +33 + +This implies that +VEpxqδVBpxq ě c2 +p1 ` }x}qm`1 +p}x} ` p1 _ dp0, BEqqq1´ǫ ě c p1 ` }x}qm`ǫ. +Using the fact that VEpxq ě 1 for any x P E, this implies that +tx P E : VEpxqδVBpxq ď ru Ă tx P E : c p1 ` }x}qm`ǫ ď ru X tx P E : VBpxq ď ru. +We conclude that V δ +E VB has compact level sets and (79) is satisfied. +In this context, we have the following proposition. +Proposition 4.6. Consider a couple of functions pVB, VEq satisfying (79). Assume +there exists some parameters t ą 0, δt ą 0 and ǫ ě 0 such that +QtpVEq{VE ď ct{V δt +E +and +QtpVBq ď ct V ǫ δt +E +. +(80) +In this situation, for any p ą 1 ` ǫ we have +V :“ V 1´1{p +E +V 1{p +B +P C8pEq +as well as +Qt pV q {V ď ct{pV δtǫp +E +V 1{p +B +q P C0pEq +with +ǫp :“ p1 ´ p1 ` ǫq{pq ą 0. +Proof. Observe that for any p ą 1 ` ǫ we have +VBV p´1 +E +P C8pEq +and therefore +V :“ V 1{p +B +V 1´1{p +E +P C8pEq. +In the same vein, for any ǫ ě 0 we have +p79q ùñ VBV pδtǫp +E +P C8pEq +and therefore +V 1{p +B +V δtǫp +E +P C8pEq. +On the other hand, using H¨older’s inequality, we have +Qt pV q {V +ď +pQtpVEq{VEq1´1{p pQtpVBq{VBq1{p +ď +ctp1qpQtpVBq{pV δtpp´1q +E +VBqq1{p ď ctp2qp1{pV pδtp1´p1`ǫq{pq +E +VBqq1{p. +This ends the proof of the proposition. +The design of a function VE satisfying (80) is rather flexible. For instance, (80) is +automatically satisfied when Q ! P is dominated by some Markov integral operators +Pt on BbpEq such that +PtpVEqpxq{VEpxq ď ctp1q{VEpxqδt. +Section 2.4 discusses a variety of Lyapunov functions VE satisfying the above condi- +tion for Markov diffusion semigroups. These Lyapunov functions can also be designed +34 + +using the domination principles presented in Section 3.2. For instance, consider the +semigroup Qt :“ Qpaq +t +associated with the Langevin diffusion flow on a cylinder dis- +cussed in (66). In this situation, combining (64) with Proposition 2.9, for any v ě 0 +and t ą 0 there exists some finite constant δt ą 0 such that +VEpxq :“ exp pv|x|q ùñ QtpVEq{VE ď ct{V δt +E . +Next, we illustrate the r.h.s. condition in (80) when qt are sub-Gaussian densities; +in the sense that for any x, y P E we have +qtpx, yq ď ct gtpx, yq +with +gtpx, yq :“ +1 +p2πσ2 +t qn{2 exp +ˆ +´ 1 +2σ2 +t +}y ´ mtpxq}2 +˙ +(81) +for some parameter σt ą 0 and some non necessarily bounded function mt on E. +Proposition 4.7. Let ϕ be a Lipschitz function on Rn´1 with uniformly bounded +gradient and set +E :“ tx “ pxiq1ďiďn P Rn : xn ą ϕpx´nqu +with +x´n :“ pxiq1ďiăn P Rn´1. +Then the r.h.s. condition in (80) is met with ǫ “ 0 for any positive semigroup sat- +isfying (81). The same property holds when the boundary BE can be decomposed as +a finite union of graphs of differentiable functions on Rn´1 with uniformly bounded +gradients. +Proof. We choose α ą 0 sufficiently small so that for any +x P DαpEq :“ tx P E : dpx, BEq ď αu +there exists a projection x P BE with dpx, BEq “ }x ´ x}. Let C̟pxq be an interior +cone with a given base vertex x “ px´n, ϕpx´nqq P BE and a given half-opening angle +̟ around the axis Apxq :“ tpx´n, xnq : xn ě ϕpx´nqu. For any x P Apxq there exists +a projection px P BC̟pxq on the boundary BC̟pxq with +dpx, BC̟pxqq “ dpx, pxq “ cos +´π +2 ´ ̟ +¯ +pxn ´ ϕpx´nqq ď dpx, xq +On the other hand, for any y P BE we have +z :“ py´n, ϕpx´nqq +ùñ +0 ď π +2 ´ ̟ ď y +yxz +and +tanpy +yxzq “ |ϕpx´nq ´ ϕpy´nq| +}x´n ´ y´n} +. +This yields the estimate +cos +´π +2 ´ ̟ +¯ +ě cos +´ +y +yxz +¯ +“ +1 +b +1 ` tan2py +yxzq +35 + +from which we conclude that +0 ď xn ´ ϕpx´nq ď κ dpx, xq +with +κ :“ +a +1 ` }∇ϕ}2 +and +}∇ϕ} :“ sup +yPRn´1 }∇ϕpyq} ă 8. +This implies that +ż +DαpEq +χ pdpy, BEqqqtpx, yq dy +ď ctp1q +ż +DαpEq +χ ppyn ´ ϕpy´nqq {κqq exp +ˆ +´ 1 +2σ2 +t +ppyn ´ ϕpy´nqq ` pϕpy´nq ´ pmtpxqqnqq2 +˙ +ˆ exp +ˆ +´ 1 +2σ2 +t +}y´n ´ pmtpxqq´n}2 +˙ +dyndy´n. +Using the change of variables +z :“ pyn ´ ϕpy´nqq {κ ùñ dyn “ κ dz +we find that +ż +DαpEq +χ pdpy, BEqqqtpx, yq dy +ď κ ctp1q χpαq +ż +Rn´1 exp +ˆ +´ 1 +2σ2 +t +}y´n ´ pmtpxqq´n}2 +˙ +dy´n ď ctp2q. +On the other hand, for any α ą 0 we have +ż +E´DαpEq +χ pdpy, BEqq qtpx, yq dy ď χ pαq }Qtp1q} ď ctp3q. +This ends the proof of the proposition. +4.3 +Smooth boundaries +Next, we illustrate the Lyapunov conditions on VB in the context of absolutely con- +tinuous sub-Markov semigroup of the form (77) with a bounded density qtpx, yq on +a non necessarily bounded domain E Ă Rn with smooth non necessarily bounded +C2-boundary with uniformly bounded interior curvature. +36 + +We assume that there exists α ą 0 sufficiently small so that every point of the +α-offset of BE (a.k.a. α-tubular neighborhood) defined by +TubαpBEq :“ tx P Rn : dpx, BEq ď αu +lies on some normal ray passing through a point on BE and no two normal rays passing +through different points of BE intersect in TubαpBEq. We let Npzq be the unit normal +vector at z P TubαpBEq pointing inward E, and let DrpEq the closed subset defined +for any r ď α by +DrpEq :“ tx P E : dpx, BEq ď ru +and +D´rpEq :“ tx P Rn ´ E : dpx, BEq ď ru. +In this notation, the inverse of the normal coordinate map +F : pz, rq P BE ˆ r´α, αs ÞÑ Fpz, rq “ z ` r Npzq P TubαpBEq +(82) +is given for any x P TubαpBEq by +F ´1pxq “ pprojBEpxq, dαpx, BEqq +where projBEpxq stands for the projection of x P TubαpBEq onto BE and dαpx, BEq +stands for the signed distance function +dαpx, BEq :“ dpx, BEq 1DαpEqpxq ´ dpx, BEq 1D´αpEqpxq P r´α, αs. +In addition, the inward normal Npxq at any x on the C2 boundary BE is given by +∇dαpx, BEq “ Npxq. +The Hessian of the signed distance function on the boundary BE gives the Weingarten +map Wpxq. With this notation at hand, we have +ż +DαpEq +fpdpy, BEqq qtpx, yq dy “ +ż α +0 +fprq qB +t px, rq dr +with the level-set density function +qB +t px, rq +:“ +ż +BEr +qtpx, yq σB,rpdyq +(83) +“ +ż +BE +qt px, z ` rNpzqq |det pI ´ r Wpzqq| σBpdzq. +In the above display, σB,rpdzq stands for the Riemannian volume measure on the r- +extended boundary +BEr :“ tx P E : dpx, BEq “ ru. +37 + +Moreover, since E has uniformly bounded interior curvature, for any r ď α we +have +κBpαq :“ sup |det pI ´ r Wpzqq| ă 8 +and +κ´ +B pαq :“ sup |det pI ` r Wpyqq| ă 8. +In the above display, the supremum is taken over all z P BE, y P BEr, and r ď +α. Several examples of hypersurface boundaries satisfying the above conditions are +discussed in Section 7 (cf. for instance Proposition 7.4). +We denote by qB +t ě qB +t the function defined as qB +t by replacing qt by qt. Using the +fact that +QtpVBqpxq ď χpαq ` +ż α +0 +χprq qB +tpx, rq dr +we readily check the following proposition. +Proposition 4.8. For any t ą 0 we have +sup +0ďrďα +sup +xPE +qB +t px, rq ă 8 +ùñ +QtpVBq ď χpαq Qtp1q ` ctpαq χpαq +sup +0ďrďα +sup +xPE +qB +t px, rq ă 8 +ùñ +QtpVBq ď χpαq ` ctpαq χpαq. +(84) +When the boundary BE is bounded, for any t ą 0 we have the estimate +}QtpVBq} ď ctpαq +ˆ +χpαq ` χpαq +sup +0ďrďα σB,r pBErq +˙ +. +(85) +We end this section with some practical tools to estimate the level-set density +functions discussed in Section 4.3. Most of our estimates are based on the following +technical lemma. +Lemma 4.9. Consider a couple of non negative functions f, g on Rn and some pa- +rameter α ą 0 such that +sup +}u}ďα +fpz ` uq ď ιpαq gpzq +for some ιpαq ă 8. +In this situation, we have the uniform estimate +sup +0ďrďα +ż +BEr +fpzq σB,rpdzq ď ιpαq κBpαq +ż +BE +g pzq σBpdzq +as well as the co-area estimate +ż +BE +fpzq σBpdzq ď 1 +α ιpαq κ´ +B pαq +ż +DαpEq +gpzq dz. +The proof of the above lemma is provided in the appendix, on page 72. +Note that the level-set density function defined in (83) can be estimated for any +0 ď r ď α by the formula +qB +t px, rq +ď +κBpαq +ż +BE +qt px, z ` rNpzqq +σBpdzq. +38 + +Proposition 4.10. Assume that qtpx, yq ď ̟t gtpx, yq is dominated by some proba- +bility density y ÞÑ gtpx, yq on Rn for some t ą 0 and some parameter ̟t ă 8. In +addition, we have +sup +}u}ďα +gtpx, y ` uq ď ιtpαq gα,tpx, yq +(86) +for some probability density y ÞÑ gα,tpx, yq and some ιtpαq ă 8. In this situation, we +have the uniform density estimates +sup +0ďrďα sup +xPE +qB +t px, rq ď ̟t ιtpαqκ´ +B pαqκBpαq{α. +(87) +Proof. By (86) for any 0 ď r ď α we have +qB +t px, rq ď ̟t κBpαq +ż +BE +gt px, z ` rNpzqq +σBpdzq. +On the other hand, we have +ż +DαpEq +gt,αpx, yq dy ď 1. +The estimate (87) is now a direct consequence of the co-area estimate stated in +Lemma 4.9. This ends the proof of the proposition. +We illustrate the above condition when qt are the sub-Gaussian densities discussed in +(81). In this situation, using the fact that 2a1b ď 1 +ǫ}a}2 ` ǫ}b}2 for any 0 ă ǫ ă 1 and +}u} ď α we check that +´ 1 +2σ2 +t +}py ` uq ´ mtpxq}2 ď ´p1 ´ ǫq +2σ2 +t +}y ´ mtpxq}2 ` +1 +2σ2 +t +ˆ1 +ǫ ´ 1 +˙ +α2. +In this context, condition (86) is met with the gaussian density +gα,tpx, yq :“ e +´ +1 +2σtpǫq2 }y´mtpxq}2 +p2πσtpǫq2qn{2 +with +ιtpαq :“ ct e +α2{ǫ +2σtpǫq2 +and +σtpǫq2 :“ σ2 +t {p1 ´ ǫq. +5 +Riccati type processes +5.1 +Positive diffusions +Consider the Riccati type diffusion on E “s0, `8r defined for any x P E by +dXtpxq “ +` +a0 ` a1 Xtpxq ´ b Xtpxq2˘ +dt`σ1pXtpxqq dB1 +t `σ2pXtpxqq dB2 +t , X0pxq “ x +39 + +for some Brownian motion pB1 +t , B2 +t q on R2, the diffusion functions +σ1pxq :“ ς1 +?x +σ2pxq :“ ς2 x +and the parameters +a1 P R +a0 ą ς2 +1 +b ą 0 +and +ς1, ς2 ě 0. +Applying Itˆo’s formula, we readily check that +BtEpXtpxqq ď Ricc pEpXtpxqqq +and +BtEp1{Xtpxqq ď Ricc´ pEp1{Xtpxqqq +with the Riccati drift functions defined by +Riccpzq :“ a0 ` a1z ´ bz2 +and +Ricc´pzq :“ a´ +0 ` a´ +1 z ´ b´z2 +(88) +with the parameters +a´ +0 :“ b +a´ +1 :“ pς2 +2 ´ a1q +and +b´ :“ a0 ´ ς2 +1. +Consider the Lyapunov function V P B8pEq defined by V pxq :“ x ` 1{x. By well +known properties of Riccati flows, for any t ą 0 we have }PtpV q} ă 8. For a more +thorough discussion on this class of one-dimensional Riccati diffusions, we refer to the +article [4]. +5.2 +Matrix valued diffusions +Let E and E be the space of pn ˆ nq-positive semi-definite and definite matrices +respectively. Also let λ1pxq ě . . . ě λnpxq denote the ordered eigenvalues of x P E. +Let Wt denotes an pn ˆ nq-matrix with independent Brownian entries. Also let A be +an pn ˆ nq-matrix with real entries and let R, S P E. We associate with these objects +the E-valued diffusion +dXt “ pAXt ` XtA1 ` R ´ XtSXtq dt ` ǫ +2 +” +X1{2 +t +dWt R1{2 ` R1{2 dW1 +t X1{2 +t +ı +. +Whenever ǫ ď 2{ +? +n ` 1, the diffusion Xt has a unique strong solution that never +hits the boundary BE “ E ´ E. In addition, the transition semigroup Pt of Xt is +strongly Feller and admits a smooth density w.r.t. the Lebesgue measure on E, thus +it is irreducible. Furthermore, when ǫ2p1 ` nq{2 ď λnpRq{λ1pRq then the function +V pxq “ Trpxq ` Trpx´1q is a Lyapunov function with compact level subsets. For a +detailed proof of the above assertion for more general classes of Riccati matrix valued +diffusions we refer to [5] (see for instance the stability Theorem 2.4 and Section 5.4 +in [5]). +40 + +5.3 +Logistic birth and death process +Let Xtpxq be the stochastic flow on E :“ N ´ t0u with generator L defined for any +f P BbpEq and x ě 2 by +Lpfqpxq “ Jpx, x ´ 1q pfpx ´ 1q ´ fpxqq ` Jpx, x ` 1q pfpx ` 1q ´ fpxqq +and for x “ 1 by +Lpfqp1q “ Jp1, 2qpfp2q ´ fp1qq. +In the above display, the birth and death rates in the above display are given by +Jpx, x ` 1q :“ λb x ` υb +and +Jpx, x ´ 1q :“ λd x ` λl xpx ´ 1q ` υd +(89) +for some non negative parameters λd, λb, υb, υd ě 0 and λl ą 0. Consider the identity +function V : x P E ÞÑ V pxq “ x. For any x ě 2 we have +LpV qpxq “ Jpx, x ` 1q ´ Jpx, x ´ 1q “ pF ˝ V q pxq +with the concave function +z P R` ÞÑ Fpzq :“ pυb ´ υdq ` pλb ` λl ´ λdq z ´ λl z2 P R. +(90) +Observe that +LpV qp1q ´ FpV p1qq “ Jp1, 2q ´ Fp1q “ Jp1, 0q “ υd ` λd. +This yields the estimate +PtpLpV qqpxq +“ +Ptp1r2,8r LpV qqpxq ` Ptp1t1u LpV qqpxq +“ +PtppF ˝ V qqpxq ` Ptp1t1uqpxq pLpV qp1q ´ FpV p1qqq +ď +FpPtpV qqpxq ` Jp1, 0q +from which we check that +BtPtpV qpxq +ď +Ricc pPtpV qpxqq +with the Riccati drift function defined in (88) with the parameters +a0 :“ υb ` λd +a1 :“ λb ` λl ´ λd +and +b :“ λl ą 0. +By well known properties of Riccati flows, for any t ą 0 we conclude that }PtpV q} ă 8. +41 + +5.4 +Multivariate birth and death processes +We denote by e :“ tei, 1 ď i ď nu the collection of column vector ei on t0, 1un with +entries eipjq “ 1i“j and with a slight abuse of notation we denote by 0 the null state +in Nn. Let Xtpxq be a stochastic flow on E “ Nn ´ t0u with generator L defined by +Lpfqpxq :“ +ÿ +yPE +Jpx, yq pfpyq ´ fpxqq. +(91) +Let λ, µ, υ, ς be some column vectors and let C, D some pd ˆ dq-matrices with real +entries such that for any 1 ď i ď d and any x P E we have +Jpx, x ` eiq :“ υi ` xi pλi ` pCxqiq ě 0 +and +Jpx, x ´ eiq :“ ςi ` xi pµi ` pDxqiq ě 0. +We also set +Jpx, yq “ 0 +as soon as +|x ´ y| ě 2. +We further assume that +|υ| ě |ς| +B :“ pD ´ Cq ě b I ą 0 +for some b ą 0. +and we set +a0 :“ |υ| ´ |ς| ě 0 +a1 :“ _1ďiďnpλi ´ µiq +and for any x P Nn +}x} :“ +˜ ÿ +1ďiďn +x2 +i +¸1{2 +ě |x| :“ +ÿ +1ďiďn +xi. +Consider the Lyapunov function +x P E ÞÑ V pxq “ |x| P N`. +Note that V is locally bounded with finite level sets and for any x P E ´ e we have +LpV qpxq +“ +ÿ +1ďiďn +ppυi ` xi pλi ` pCxqiqq ´ pςi ` xi pµi ` pDxqiqqq . +In this situation, we have the formula +LpV qpxq “ a0 ` pλ ´ µq1x ´ x1Bx ď a0 ` a1|x| ´ b}x}2. +(92) +On the other hand, for any y “ ej we have +LpV qpyq “ +ÿ +1ďiďn +Jpy, y ` eiq. +42 + +This implies that +PtpLpV qq +“ +Ptp1E´e LpV qq ` Ptp1e LpV qq +“ +a0 ` a1 PtpV q ´ b PtpV 2q +` +ÿ +1ďjďn +Ptp1ejq +` +LpV qpejq ´ +` +a0 ` pλ ´ µq1ej ´ e1 +jBej +˘˘ +from which we readily check that +BtEpV pXtpxqqq ď a` +0 ` a1EpV pXtpxqqq ´ b pEpV pXtpxqqqq2 +with +a` +0 +:“ +a0 ` +ÿ +1ďjďd +˜ ÿ +1ďiďd +Jpej, ej ` eiq ´ +` +|υ| ´ |ς| ` pλ ´ µq1ej ´ e1 +jpD ´ Cqej +˘ +¸ +“ +a0 ` +ÿ +1ďjďd +` +|ς| ` µ1ej ` e1 +jDej +˘ +. +We conclude that }PtpV q} ă 8. +The semigroup analysis discussed above can be +extended without difficulties to more general process on countable spaces models +satisfying condition (92). The extension to time varying models can also be handle +using a more refined analysis on time varying Riccati equations. +We also mention, that the case |υ| “ 0 “ |ς| coincides with the competitive and +multivariate Lotka-Volterra birth and death process discussed in Theorem 1.1 in [10]. +6 +Some conditional diffusions +6.1 +Coupled harmonic oscillators +Consider the Rn-valued diffusion (33) with pbpxq, σpxqq “ pAx, Σq, for some non neces- +sarily stable drift matrix A and some diffusion matrix Σ with appropriate dimensions. +We associate with a given semi-definite positive pn ˆ nq matrix S ě 0 the potential +function +Upxq :“ 1 +2 x1Sx +and we set +R “ ΣΣ1. +(93) +We assume that the pairs pA, R1{2q and pA1, S1{2q are both controllable. Let Qt “ QrUs +t +be the sub-Markov semigroup defined in (49) on the Euclidean space E “ E “ Rn. +As shown in [13], the leading-triple pρ, h, η8q discussed in (24) is given by +ρ +“ +´TrpRq8q{2 “ ´Trpp8Sq{2 +hpxq +“ +exp p´x1q8x{2q +and +η8pdxq “ exp p´x1p´1 +8 x{2q +a +detp2πp8q +dx, +(94) +43 + +with the positive fixed points p8 and q8 of the dual algebraic Riccati matrix equation +Ap8 ` p8A1 ` R ´ p8Sp8 “ 0 +and +A1q8 ` q8A ` S ´ q8Rq8 “ 0. +In this context, the h-process, denoted pXh +t pxqqtě0 and defined by the stochastic dif- +ferential equation +dXh +t pxq “ AhXh +t pxq dt ` Σ dBt +with +Ah :“ A ´ R q8. +(95) +Our controllability conditions ensures that Ah is a stable matrix. Note that Xh +t pxq +is an Rn-valued Gaussian random variable with mean mh +t pxq and covariance matrix +ph +t P Rnˆn given for any t ą 0 by +mh +t pxq “ exp +` +Aht +˘ +x +and +ph +t “ +ż t +0 +exp +` +Ahs +˘ +R exp +` +pAhq1s +˘ +ds ą 0. +This yields the explicit formula +P h +t px, dyq “ +1 +a +detp2πph +t q +exp +ˆ +´1 +2py ´ mh +t pxqq1pph +t q´1py ´ mh +t pxqq +˙ +dy. +Moreover the invariant measure ηh +8 “ ηh +8Ph +t is unique and given by +ηh +8pdxq “ +1 +a +detp2πph +8q +exp +ˆ +´1 +2y1pph +8q´1y +˙ +dy +with the limiting covariance matrix +ph +8 :“ +ż 8 +0 +exp +` +Ahs +˘ +Σ2 exp +` +pAhq1s +˘ +ds “ pp´1 +8 ` q8q´1 ą 0. +For any time horizon t ě 0 and any measurable function F on the set Cpr0, ts, Rnq +of continuous paths from r0, ts into Rn we have the path space exponential change of +measure Feynman-Kac formula +E +ˆ +FpXtpxqq exp +ˆż t +0 +UspXspxqq ds +˙˙ +“ eρt hpxq E +` +FpXh +t pxqq{hpXh +t pxqq +˘ +with the historical processes +Xtpxq :“ pXspxqq0ďsďt, +Xh +t pxq :“ pXh +s pxqq0ďsďt +and +UspXspxqq :“ UspXspxqq. +This yields the conjugate formulae +Qtpfq “ eρt h P h +t pf{hq. +We denote by pmtpxq, ptq P pRnˆRnˆnq the mean and covariance parameters satisfying +the linear evolution and the Riccati matrix differential equations +$ +& +% +Btmtpxq +“ +pA ´ ptSq mtpxq +Btpt +“ +Apt ` ptA1 ` Σ2 ´ ptSpt +with +pm0pxq, p0q “ px, 0q. +(96) +The next proposition provides an explicit description of these semigroups. +44 + +Proposition 6.1 ([13]). For any time horizon t ą 0 we have pt ą 0 and +Qtpx, dyq “ +1 +a +detp2πptq +exp +ˆ +´1 +2py ´ mtpxqq1p´1 +t py ´ mtpxqq +˙ +dy +(97) +as well as +´2 log Qtp1qpxq “ x1 +ˆż t +0 +F 1 +sSFs ds +˙ +x ` +ż t +0 +TrpSpsq ds +with the fundamental matrix semigroup Ft starting at F0 “ I given by +BtFt “ pA ´ ptSq Ft. +Observe that the normalized Markov operator Qt satisfies (39) and (40) with the +parameters +ct “ +1 +a +detp2πptq +, σ2 +t “ λmaxpptq +and +ǫτ “ |eτpA´p8Sq| ÝÑ 0 as τ Ñ 8 +(98) +for some matrix norm |.|. The r.h.s. assertion is a direct consequence of the Floquet +representation theorem presented in [3] (cf. (1.3) and Theorem 1.1) and the fact that +pA´ p8Sq is a stable matrix. Applying Lemma 2.9 for any v ě 0 and t ą 0 there also +exists some finite constant δt ą 0 such that +V pxq :“ exp pv|x|q ùñ QtpV q{V ď ct{V δt. +Using Proposition 6.1, for any k ě 0 and t ě 0 it is also readily checked that +V pxq :“ p1 ` }x}qk ùñ }QtpV q{V } ă 8 +and +}QtpV q} ă 8. +6.2 +Half-harmonic linear diffusions +For one dimensional models, the coupled harmonic oscillator discussed in Section 6.1 +resumes to one dimensional linear diffusion +dXtpxq “ a Xtpxq dt ` dBt +and the potential +Upxq “ ςx2{2 +(99) +for some parameters ς ą 0 and a P R. We set β :“ a ` +? +a2 ` ς. In this notation, the +leading pair pρ, hq “ pρ1, ϕ1q is given by +ρ “ ´β{2 +and +hpxq “ ppβ ´ aq{πq1{4 exp +` +´βx2{2 +˘ +. +(100) +The quasi-invariant measure is therefore given by +η8pdxq “ +c ς +2πβ exp +` +´ςx2{p2βq +˘ +dx. +45 + +Therefore, the h-process resumes to the Ornstein-Uhlenbeck diffusion +dXh +t pxq “ ´b Xh +t pxq dt ` dBt +(101) +with the invariant measure +ηh +8pdxq :“ +c +b +π exp +` +´b x2˘ +dx +with +b :“ pβ ´ aq “ +a +a2 ` ς ą 0. +Note that any Ornstein-Uhlenbeck process can be seen as the h-process associated +with a non absorbed (possibly transient) linear diffusion evolving in some quadratic +potential well. +In this context, Proposition 6.1 is also satisfied with the mean and variance pa- +rameters +$ +& +% +Btmtpxq +“ +pa ´ ptςq mtpxq +Btpt +“ +2apt ` 1 ´ ςp2 +t +with +pm0pxq, p0q “ px, 0q. +(102) +We also have +´2 +ς log Qtp1qpxq “ pt ` χt x2 +(103) +with +χt :“ +ż t +0 +exp +ˆ +´2 +ż s +0 +pa ´ puςqdu +˙ +ds +and +pt :“ +ż t +0 +psds. +The half-harmonic semigroup associated with the flow Xtpxq is defined for any +x P E :“s0, 8r and f P BbpEq by the formulae +Qtpfqpxq +:“ +E +ˆ +fpXtpxqq 1Tpxqąt exp +" +´ +ż t +0 +UpXspxqq ds +*˙ +. +(104) +In the above display, Tpxq stands for the hitting time of the origin. +In terms of +the h-process of the flow in the harmonic potential (101) we also have the conjugate +formula +Qtpfqpxq +“ +etρ e´βx2{2 E +´ +fpYtpxqq eβYtpxq2{2 1T Y pxqąt +¯ +(105) +with the parameters pρ, βq defined in (100) and the Ornstein-Uhlenbeck diffusion flow +defined by +dYtpxq “ ´b Ytpxq dt ` dBt +with +b :“ pβ ´ aq ą 0. +In the above display, T Y pxq stands for the hitting time of the origin by the flow Ytpxq +starting at x ą 0. Arguing as in Section 2.5.2 we check that +Qtpx, dyq +“ sinh py mtpxqq exp +` +´ ς +2 pχt x2 ` ptq +˘ +ˆ +c +2 +πpt +exp +ˆ +´y2 ` mtpxq2 +2pt +˙ +1s0,8rpyq dy +46 + +with the parameters pmtpxq, ptq and pχt, ptq defined in (102) and (103). +Arguing as in (56), choosing the Lyapunov function V pxq “ xn ` 1{x, for some +n ě 1, we readily check that +V P C8pEq +and +QtpV q{V ď ct{V +P C0pEq. +(106) +6.3 +Linear diffusions in some domains +Consider the one-dimensional stochastic flow Ytpxq of an Ornstein-Uhlenbeck +dYtpxq “ ´b Ytpxq dt ` dBt +for some +b ą 0. +In the above display, Bt is a one-dimensional Brownian motion starting at the origin. +For a given x P E :“s0, 8r, we let T Y pxq be the hitting time of the origin by the flow +Ytpxq starting at x ą 0. Consider the semigroup +QY +t pfqpxq :“ EpfpYtpxqq 1T Y pxqątq. +Choosing pa, ς, β, ρq “ p0, b2, b, ´b{2q in (100), formula (105) takes the form +QY +t pfqpxq “ e´ρt Hpxq´1QtpfHqpxq +with +Hpxq “ exp +` +´bx2{2 +˘ +with the semigroup Qt defined in (104) with Upxq “ b2x{2. +For any given n ě 1 we have +V pxq :“ xn ` 1{x ùñ V P C8pEq +and +V H :“ V {H P C8pEq. +Using (106) we conclude that +V H P C8pEq +and +QY +t pV Hq{V H “ e´ρt QtpV q{V ď ct{V P C0pEq. +The long time behavior of the positive semigroup QY +t is also studied in [46], and +more recently in [55] in terms of the tangent of the h-process. +More generally, consider the Rn-valued diffusion flow Xtpxq defined in (33) with +pbpxq, σpxq “ pAx, Σq, for some matrices pA, Σq with appropriate dimensions. Assume +that R :“ ΣΣ1 is positive semi-definite and the pair of matrices pA, R1{2q are control- +lable. In this situation, the Markov semigroup Pt of the stochastic flow Xtpxq satisfies +the sub-Gaussian estimate (81) for some parameters pσt, mtpxqq. +Consider a domain E Ă Rn with C2-boundary with uniformly bounded interior +curvature. For any given x P E, let Qt be the sub-Markov semigroup +Qtpfqpxq :“ EpfpXtpxqq 1Tpxqątq +with +Tpxq :“ inf tt ě 0 : Xtpxq P BEu . +(107) +We clearly have QtpVBq ď PtpVBq, with the function VB defined in (76). When E is +non necessarily bounded but its boundary BE is bounded we known from (85) that +}QtpVBq} ă 8. For non necessarily bounded boundaries the sub-gaussian property +(81) ensures that }QtpVBq} ă 8. +47 + +When E is bounded, applying Lemma 4.2 (see also Proposition 4.3) we have +VB P C8pEq +and +QtpVBq{VB ď ct{VB P C0pEq. +For unbounded domains we need to ensure that A is stable so that (44) is satisfied +for some norm |.| on Rn. In this situation, applying Proposition 2.10 for any t ą 0 +there exists some δt ą 0 such that +VEpxq :“ exp pv|x|q ùñ QtpVEq{VE ď ct{V δt +E . +Applying Proposition 4.6 with ǫ “ 0, for any p ą 1 we conclude that +Vp :“ V 1´1{p +E +V 1{p +B +P C8pEq +and +Qt pVpq {Vp ď ct Θp,t +(108) +with the function +Θp,t :“ 1{pV δtp1´1{pq +E +V 1{p +B +q P C0pEq. +6.4 +Langevin diffusions in some domains +Consider the semigroup Qt of the one-dimensional Langevin diffusion defined in (73) +with E “s0, 8r and a quadratic confinement potential +Wpxq “ x2{2 ùñ Hpxq :“ e´W pxq “ e´x2{2 +and +U :“ 1 +2 +` +x2 ` 1 +˘ +. +In this case, the semigroup Qt defined in (75) coincides with the semigroup of the +half-harmonic oscillator discussed in Section 2.5.2. By (56) for any n ě 1 we have +V pxq :“ xn ` 1{x ùñ QtpV q{V ď ct{V +P C0pEq. +Notice that +V Hpxq :“ V pxq{Hpxq “ xn ex2{2 ` ex2{2 +x . +(109) +Using (75) we conclude that +V H P C8pEq +and +QtpV Hq{V H “ QtpV q{V ď ct{V P C0pEq. +More generally, consider the case E “s0, 8r with at least a quadratic confinement +potential U, in the sense that +Upxq “ 1 +2 +` +pBWq2 ´ B2W +˘ +pxq ě U2pxq :“ c ` ς x2{2 +for some ς ą 0. +In this situation, Qt ! QrU2s is dominated by the semigroup QrU2s of the half-harmonic +oscillator discussed in Section 2.5.2. Arguing as in (109) we have +H :“ e´W +V H :“ V {H P C8pEq +and +QtpV Hq{V H ď ct{V P C0pEq. +48 + +For instance, whenever the confinement potential W is chosen so that +Wpxq ě ǫ0 log x ` W1pxq +for some 0 ď ǫ0 ă 1 +and some function W1 ě 1 such that W1pxq ÝÑxÑ8 8 we have +H “ e´W ùñ V Hpxq :“ V pxq{Hpxq “ xn eW pxq ` eW pxq +x +ě xn eW pxq ` eW1pxq +x1´ǫ0 . +Using (75) we conclude that +V H P C8pEq +and +QtpV Hq{V H “ QtpV q{V ď ct{V P C0pEq. +We illustrate the above result, with the logistic diffusion discussed in [9]. Consider +the generalized Feller diffusion +dYtpxq :“ +` +2a Ytpxq ´ p8b{σ2q Ytpxq2˘ +dt ` σ +a +Ytpxq dBt +starting at x P E :“s0, 8r. In the above display, Bt is a one dimensional Brownian +motion starting at the origin and a, b, σ ą 0 some parameters. Observe that +Xtpxq :“ p2{σq +a +Ytpxq ùñ dXtpxq “ ´BW pXtpxqq dt ` dBt +with the potential function +BWpxq “ 1 +2x ´ a x ` b x3 +with +a, b ą 0. +Thus, choosing +Wpxq “ 1 +2 log x ` bx4 +4 ´ ax2 +2 +we readily check that +V Hpxq :“ eW pxqpxn ` 1{xq “ pxn´1{2 ` 1{?xq eb x4 +4 ´a x2 +2 ùñ V H P C8pEq. +More generally, consider the Langevin diffusion flow +Xtpzq “ pXtpzq, Ytpzqq P pRn ˆ Rnq +starting at z “ px, yq P pRn ˆ Rnq and defined by the hypo-elliptic diffusion (65). We +further assume that supD a ă 8 for some bounded open connected domain D Ă Rn +with C2-boundary, and for any z P E :“ D ˆ Rn and f P BbpEq we set +Qtpfqpzq :“ E +` +fpXtpzqq 1T paqpzqąt +˘ +with +Tpzq :“ inf tt ě 0 : Xtpzq P BDu . +We know from (66) that for any q ą 1 we have Q !q Q is q-dominated by the sub- +Markov semigroup Qt associated with the Ornstein-Uhlenbeck diffusion on E defined +in (107), with the matrices pA, Σq defined in (46). In terms of the functions pVp, Θp,tq +defined in (108), combining (64) with (108) for any p, q ą 1 we conclude that +QtpVp,qq{Vp,q ď ctpp, qq Θp,q,t +with the collection of Lyapunov functions +Vp,q :“ V 1{q +p +P C8pEq +and the function +Θp,q,t :“ Θ1{q +p,t P C0pEq. +49 + +6.5 +Coupled oscillators in some domains +Consider the Rn-valued diffusion Xtpxq and the quadratic potential function U dis- +cussed in Section 6.1, for some n ě 2 and set E :“s0, 8rˆRn´1. +Let Qt be the +semigroup defined for any f P BbpEq and x P E by the formulae +Qtpfqpxq +:“ +E +ˆ +fpXtpxqq 1Tpxqąt exp +ˆ +´ +ż t +0 +UpXspxqqds +˙˙ +(110) +with the quadratic function U in (93) and the exit time Tpxq given by +Tpxq :“ inf tt ě 0 : Xtpxq P BEu +with +BE “ t0u ˆ Rn´1. +In terms of the h-process Ytpxq :“ Xh +t pxq associated with the leading pair pρ, hq defined +in (95) we also have the conjugate formula +Qtpfq “ eρt h QY +t pf{hq +with +QY +t pfqpxq :“ E +` +fpYtpxqq 1T Y pxqąt +˘ +. +In the above display, T Y pxq stands for the boundary hitting time +T Y pxq :“ inf tt ě 0 : Ytpxq P BEu . +When n “ 2, the linear diffusion Xtpxq associated to the matrices A1,2 “ A2,1 “ +A2,2 “ 0 and A1,2 “ 1 and Σ1,1 “ Σ1,2 “ Σ2,1 “ 0 and Σ2,2 “ 1 coincides with the +integrated Wiener process model discussed in [45, 35, 48]. In a seminal article [48], +McKean obtained the joint distribution of the pair pTpxq, X2 +Tpxqq in the absence of soft +absorption, that is when U “ 0. To the best of our knowledge, an explicit descrip- +tion of the distribution of this pair and the corresponding sub-Markov semigroup is +unknown in more general situations. +Observe that for any x P E and any non negative function f P BbpRnq we have +Qtpfqpxq ď Qtpfqpxq :“ eρt hpxq E pfpYtpxqq{hpYtpxqqq . +The semigroup Qt defined above coincides with the semigroup of the coupled harmonic +oscillator discussed in Section 6.1. +We know from (98) that Qt satisfies the sub- +Gaussian estimates (39) with +ct “ +1 +a +detp2πptq +and +σ2 +t “ λmaxpptq +with the solution pt of the Riccati-matrix equation (96). Using Proposition 6.1 for +any k ě 1 we have +VEpxq :“ 1 ` }x}k ùñ }QtpVEq{VE} ă 8 ùñ QtpVEq ď ct Qtp1qVE. +Recalling that Qtp1qpxq tends to 0 exponentially fast as }x} Ñ 8, this implies that +@t ą 0 +}QtpVEq} ă 8. +50 + +On the other hand for any y “ py1, y´1q P E “ ps0, 8rˆRn´1q, the distance to the +boundary is given by dpy, BEq “ y1. In terms of the function VB defined in (76) his +implies that +QtpVBqpxq ď +ż +Qtpx, dyq 1s0,1rpy1q χpy1q ` χp1q Qtp1qpxq +from which we check that }QtpVBq} ă 8. Applying Proposition 4.4 we conclude that +V :“ VB ` VE P C8pEq +and +QtpV q{V ď ct{V P C0pEq. +The same analysis applies by replacing the half line E1 by the unit interval E1 :“s0, 1r. +In this context, the boundary is given by the two infinite potential walls +BE “ pt0u ˆ Rn´1q Y pt1u ˆ Rn´1q +and +dpx, BEq “ x1 ^ p1 ´ x1q. +More generally, consider a domain E Ă Rn with C2-boundary with uniformly bounded +interior curvature. +In this situation, the sub-Gaussian property (81) ensures that +}QtpVBq} ă 8 and therefore +}QtpVBq} ď }QtpVBq} “ }Qtp1qQtpVBq} ă 8. +Applying Proposition 4.4, we conclude that +V :“ VB ` VE P C8pEq +and +and +QtpV q{V ď ct{V P C0pEq. +7 +Some hypersurface boundaries +7.1 +Defining functions and charts +Consider a smooth function y P Rn´1 ÞÑ ϕpyq P R with non empty and connected +level set, for some n ě 2. +Consider a domain E in Rn with a smooth boundary +BE “ ϕ´1pt0uq defined as the null level set of the function +x “ pxiq1ďiďn P Rn ÞÑ ϕpxq :“ ϕpx´nq ´ xn +with +x´n :“ pxiq1ďiăn P Rn´1. +Consider the column vectors ∇ϕpx´nq :“ pBxiϕpx´nqq1ďiăn. In this notation, the unit +normal vector Npxq at x P BE is given by the column vectors +Npxq “ ∇ϕpxq +}∇ϕpxq} “ +1 +a +1 ` }∇ϕpx´nq}2 +ˆ +∇ϕpx´nq +´1 +˙ +. +Observe that the vector Npxq is the outward-pointing normal direction to E as soon +as E “ ϕ´1 ps ´ 8, 0rq and the inward-pointing normal direction to E when E “ +ϕ´1 ps0, `8rq. +51 + +Consider the column vectors ei :“ p1ipjqq1ďiăn, with 1 ď i ă n. In this notation, +the pn ´ 1q tangential column vectors Tipxq at x P BE are given for any 1 ď i ă n by +the column vectors +Tipxq :“ +ˆ +ei +Bxiϕpx´nq +˙ +. +The inner product gpxq on the tangent space TxpBEq (a.k.a. the first fundamental +form on BE) is given by the Gramian matrix +gpxq “ pTipxq1Tjpxqq1ďi,jăn “ TpxqTpxq1 +with +Tpxq1 :“ pT1pxq, . . . , Tn´1pxqq . +This yields the matrix formula +gpxq “ +` +I, ∇ϕpx´nq +˘ ˆ +I +∇ϕpx´nq1 +˙ +“ I ` ∇ϕpx´nq∇ϕpx´nq1. +In this notation, the projection projTxpBEq on the tangent space TxpBEq is defined for +any column vector V “ pV iq1ďiďn P Rn by +projTxpBEqpV q :“ pT1pxq, . . . , Tn´1pxqq gpxq´1 +¨ +˚ +˝ +T1pxq1 +... +Tn´1pxq1 +˛ +‹‚ +¨ +˚ +˝ +V 1 +... +V n +˛ +‹‚. +In matrix notation, the projection of m column vectors Vi P Rn, with i P t1, . . . , mu +and any m ě 1 takes the synthetic form +projTxpBEqpV1, . . . , Vmq +“ +` +Tpxq1gpxq´1Tpxq +˘ +pV1, . . . , Vmq +“ +` +projTxpBEqpV1q, . . . , projTxpBEqpVmq +˘ +. +Equivalently, if gpxqi,j denotes the pi, jq-entry of the inverse matrix gpxq´1, the pro- +jection of a column vector V P Rn onto TxpBEq is defined by +projTxpBEqpV q “ +ÿ +1ďi,jăn +gpxqi,j pTjpxq1V q Tipxq. +7.2 +The shape matrix +Consider the Monge parametrization +ψ : θ “ pθiq1ďiăn P S :“ Rn´1 ÞÑ ψpθq “ +ˆ +θ +ϕpθq +˙ +P BE Ă Rn. +(111) +In this chart, the tangent vectors and the normal unit vector at x “ ψpθq are given +for any 1 ď i ă n by +T ψ +i pθq :“ Bθiψpθq “ Ti pψpθqq P TxpBEq +and +Nψpθq :“ Npψpθqq P TK +x pBEq. +52 + +For any 1 ď i, j ă n we have +pBθiψpθqq1 Nψpθq “ 0 +ùñ Ωpψpθqqi,j :“ +` +Bθi,θjψpθq +˘1 Nψpθq “ ´ pBθiψpθqq1 BθjNψpθq. +Observe that for x “ ψpθq, +BθiNψpθq “ +ÿ +1ďkďn +pBxkNq pxq Bθiψkpθq “ p∇Npxqq1 Bθiψpθq +from which we check that for any 1 ď i, j ă n the coefficients of the second funda- +mental form can be computed as follows: +Ωpxqi,j “ ´ pBθiψpθqq1 p∇Npxqq1 Bθiψpθq. +We set +` +BNψpθq +˘1 :“ +` +Bθ1Nψpθq, . . . , Bθn´1Nψpθq +˘ +P +` +TψpθqpBEq +˘n´1 . +In this notation, for any x “ ψpθq we have the matrix formulation +Ωpxq +:“ +´Bψpθq +` +BNψpθq +˘1 “ +´` +Bθi,θjψpθq +˘1 Npxq +¯ +1ďi,jăn +“ +´ +∇2ϕpθq +a +1 ` }∇ϕpθq}2 +with +∇2ϕpθq :“ +` +Bθi,θjϕpθq +˘ +1ďi,jăn . +We also readily check the matrix formulation of the Weingarten’s equations +` +BNψpθq +˘1 +“ +` +pBψpθqq1 gpψpθqq´1˘ +pBψpθqq +` +BNψpθq +˘1 “ ´ pBψpθqq1 W pxq . +In the above display, Wpxq stands for the shape matrix (a.k.a. the Weingarten map +or the mixed second fundamental form) defined by +Wpxq +:“ +gpxq´1Ωpxq +“ +´ +1 +a +1 ` }∇ϕpx´nq}2 pI ` ∇ϕpx´nq∇ϕpx´nq1q´1 ∇2ϕpx´nq. +We summarize the above discussion in the following proposition. +Proposition 7.1. For any 1 ď i ă n we have the Weingarten’s equations +BθiNψpθq “ ´ +ÿ +1ďkăn +W pψpθqqk,i Bθkψpθq P TψpθqpBEq. +Example 7.2. For n “ 2 we have x P R ÞÑ ϕpxq “ ϕpxq ´ x, so that the boundary +BE “ ϕ´1pt0uq coincides with the graph of the function ϕ. In this context, the metric +and Weingarten map at x P BE “ tx “ px1, x2q P R2 : x2 “ ϕpx1qu take the form +gpxq “ 1 ` }Bϕpx1q}2 +and +Wpxq “ ´ +1 +p1 ` }Bϕpx1q}2q3{2 B2ϕpx1q. +53 + +Example 7.3. For n “ 3, the boundary BE is given by the surface in R3 defined +BE :“ tx “ pxiq1ďiď3 P R3 : x3 “ ϕpx1, x2qu. +The Monge parametrization is given by +ψ : θ “ pθ1, θ2q P R2 ÞÑ ψpθq “ +¨ +˝ +θ1 +θ2 +ϕpθ1, θ2q +˛ +‚P BE Ă R3. +In this situation, the tangent vectors at x P BE are given by +T1pxq “ +¨ +˝ +1 +0 +Bx1ϕpxq +˛ +‚ +and +T2pxq “ +¨ +˝ +0 +1 +Bx2ϕpxq +˛ +‚. +In the same vein, whenever E “ tx P R3 : ϕpx1, x2q ď x3u the outward pointing unit +normal at x P BE is given by +Npxq “ +1 +a +1 ` pBx1ϕpxqq2 ` pBx2ϕpxqq2 +¨ +˝ +Bx1ϕpxq +Bx2ϕpxq +´1 +˛ +‚. +The inner product gpxq is easily computed and given by +gpxq “ +ˆ +1 ` pBx1ϕpxqq2 +pBx1ϕpxqqpBx2ϕpxqq +pBx1ϕpxqqpBx2ϕpxqq +1 ` pBx2ϕpxqq2 +˙ +. +The inverse metric is given by +gpxq´1 “ +1 +detpgpxqq +ˆ +1 ` pBx2ϕpxqq2 +´pBx1ϕpxqqpBx2ϕpxqq +´pBx1ϕpxqqpBx2ϕpxqq +1 ` pBx1ϕpxqq2 +˙ +with +detpgpxqq “ 1 ` pBx1ϕpxqq2 ` pBx2ϕpxqq2 “ 1 ` }∇ϕpxq}2. +The second fundamental form is also given by +Ωpxq “ ´ +1 +a +1 ` }∇ϕpxq}2 +ˆ +B2 +x1ϕpxq +Bx1,x2ϕpxq +Bx1,x2ϕpxq +B2 +x2ϕpxq +˙ +and the Weingarten map is defined by +Wpxq “ ´ +1 +p1 ` }∇ϕpxq}2q3{2 +ˆ +¨ +˝ +p1 ` pBx2ϕpxqq2qB2 +x1ϕpxq ´ pBx1ϕpxqqpBx2ϕpxqqBx1,x2ϕpxq +p1 ` pBx2ϕpxqq2qBx1,x2ϕpxq ´ ´pBx1ϕpxqqpBx2ϕpxqqB2 +x2ϕpxq +´pBx1ϕpxqqpBx2ϕpxqqB2 +x1ϕpxq ` p1 ` pBx1ϕpxqq2qBx1,x2ϕpxq +´pBx1ϕpxqqpBx2ϕpxqqBx1,x2ϕpxq ` p1 ` pBx1ϕpxqq2qB2 +x2ϕpxq +˛ +‚. +54 + +7.3 +Surface and volume forms +The surface form σB on the boundary BE expressed in the chart ψ introduced in (111) +is given by +` +σB ˝ ψ´1˘ +pdθq “ +a +det pgpψpθqqq dθ +with the Gramian of the coordinate chart +gpψpθqq +:“ +Gram +` +Bθ1ψpθq, . . . , Bθn´1ψpθq +˘ +:“ +pBψpθqq pBψpθqq1 “ I ` ∇ϕpθqp∇ϕpθqq1 +with the coordinates tangent vectors Bψpθq “ T ψpθq :“ Tpψpθqq. To check this claim +recall that the surface area spaced by the column vectors +Bψpθq1 :“ +` +Bθ1ψpθq, . . . , Bθn´1ψpθq +˘ +is equal to the volume of the parallelepided generated by the column vectors +pBψpθq1, Npψpθqqq :“ +` +Bθ1ψpθq, . . . , Bθn´1ψpθq, Npψpθqq +˘ +which is given by the determinant of the column vectors, so that +` +σB ˝ ψ´1˘ +pdθq “ |det pBψpθq1, Npψpθqqq | dθ. +On the other hand, we have +ˆ +Bψpθq +Npψpθqq1 +˙ ` +Bψpθq1, Npψpθqq +˘ +“ +¨ +˚ +˚ +˚ +˝ +pBθ1ψpθqq1 +... +pBθ1ψpθqq1 +Npψpθqq1 +˛ +‹‹‹‚ +` +Bθ1ψpθq, . . . , Bθ1ψpθq, Npψpθqq +˘ +“ +ˆ +pBψpθqq pBψpθqq1 +0n´1,1 +01,n´1 +1 +˙ +. +This implies that +|det pBψpθq1, Npψpθqqq | +“ +b +|det +` +pBψpθq1, Npψpθqqq1 pBψpθq1, Npψpθqqq +˘ +| +“ +b +det +` +pBψpθqq pBψpθqq1˘ +. +Using the determinant perturbation formula w.r.t. rank-one matrices det pI ` uv1q “ +1 ` v1u which is valid for any column vectors u, v P Rn we check that +det pI ` ∇ϕpθq∇ϕpθq1q “ 1 ` }∇ϕpθq}2. +This yields the formula +` +σB ˝ ψ´1˘ +pdθq “ +a +1 ` }∇ϕpθq}2 dθ. +55 + +The mapping F defined in (82) can also be rewritten as a chart ψ on DrpEq defined +for any pθ, uq P pS ˆ r0, rsq defined by +ψpθ, uq :“ Fpψpθq, uq “ ψpθq ` u Npψpθqq P DrpEq. +The Jacobian matrix of ψ is given by +Jacpψqpθ, uq “ +` +Bθ1ψpθ, uq, . . . , Bθn´1ψpθ, uq, Npψpθqq +˘ +. +By Proposition 7.1 we have +Bθiψpθ, uq +“ +Bθiψpθq ` u BθiNψpθq +“ +Bθiψpθq ´ u +ÿ +1ďkăn +Bθkψpθq W pψpθqqk,i . +This yields the formula +` +Bθ1ψpθ, uq, . . . , Bθn´1ψpθ, uq +˘ +“ +` +Bθ1ψpθ, uq, . . . , Bθn´1ψpθq +˘ +pI ´ u Wpψpθqqq +from which we check that +|det +` +Jacpψqpθ, uq +˘ +| “ +a +detpgpψpθqq |det pI ´ u Wpψpθqqq| . +Note that ψpθ, 0q “ ψpθq, and for any given u ă r, the mapping θ ÞÑ ψpθ, uq is a chart +on BEu. This yields the following proposition. For the convenience of the reader, a +more detailed proof of the next proposition is provided in the appendix on page 72. +Proposition 7.4. For any u ď r, the surface form σB,u on the boundary BEu expressed +in the chart θ P S ÞÑ ψpθ, uq :“ Fpψpθq, uq is given by the formula +` +σB,u ˝ ψp., uq´1˘ +pdθq “ |det pI ´ u Wpψpθqqq| +` +σB ˝ ψ´1˘ +pdθq +with +|det pI ´ u Wpψpθqqq| +“ |det +˜ +I ` +u +a +1 ` }∇ϕpθq}2 pI ` ∇ϕpθq∇ϕpθq1q´1 ∇2ϕpθq +¸ +|. +In addition, the volume form σDrpEq on DrpEq expressed in the chart ψ is given by +´ +σDrpEq ˝ ψ +´1¯ +pdpθ, uqq “ |det pI ´ u Wpψpθqqq| +` +σB ˝ ψ´1˘ +pdθq du. +Using Jacobi’s formula for the derivative of determinants, we also have +Bu log det pI ´ u Wpxqq “ ´Tr +` +pI ´ u Wpxqq´1 Wpxq +˘ +. +The level-set density function defined in (83) expressed in the chart ψ is given by +the formula +qB +t px, rq +“ +ż +S +qt px, ψpθq ` r Npψpθqqq |det pI ´ r Wpψpθqqq| +a +detpgpψpθqqq dθ. +56 + +7.4 +Boundary decompositions +For some given coordinate index k P t1, . . . , nu and x “ pxiq1ďiďn P Rn we set +x´k :“ pxiqiPI +with +I :“ t1, . . . , nu ´ tku +We further assume that +BE “ tx P Rn : x´k P S +and +ϕpx´kq “ xku “ ϕ´1pt0uq +is defined as the null level set of some global defining function of the form +ϕ : x P tpxiq1ďiďn P Rn : x´k P Su ÞÑ ϕpxq :“ ϕpx´kq ´ xk P R +for some open domain S Ă Rn´1. +Example 7.5 (Cylindrical boundaries). Let 1 ď k ď n1 and n “ n1 ` n2 for some +n1 ą 1 and n2 ě 1. Consider a domain S of the form S “ p pS ˆ Rn2q with pS Ă Rn1´1 +and assume that +@y P Rn1 s.t. y´k P pS +and +@z P Rn2 +we have +ϕpy´k, zq :“ pϕpy´kq. +In this situation, the set BE is a cylindrical boundary given by the formula +BE “ B pE ˆ Rn2 +with +B pE :“ +! +y P Rn1 : y´k P pS +and +pϕpy´kq “ yk +) +. +In this context, the coordinates of the outward normal by +Njpxq “ +ǫ +a +1 ` }∇ϕpx´kq}2 +` +1Ipjq Bxjϕpx´kq ` 1kpjq p´1q +˘ +with the orientation parameter ǫ “ 1 when E “ ϕ´1ps ´ 8, 0rq; and ǫ “ ´1 when +E “ ϕ´1ps0, `8rq. In the same vein, the entries T j +i pxq of the tangent vectors Tipxq +indexed by i P I are given for any 1 ď j ď n by +T j +i pxq “ 1ipjq ` 1kpjq Bxiϕpx´kq. +Consider the pn ˆ pn ´ 1qq-matrix +Tpxq1 :“ pT1pxq, . . . , Tk´1pxq, Tk`1pxq, . . . , Tnpxqq . +In this notation, the inner product gpxq on the tangent space TxpBEq is given by the +ppn ´ 1q ˆ pn ´ 1qq-square Gramian matrix +gpxq +“ +TpxqTpxq1 “ I ` ∇ϕpx´kq∇ϕpx´kq1 +57 + +with the gradient column vector +∇ϕpx´kq :“ pBxiϕ px´kqqiPI “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +Bx1ϕ px´kq +... +Bxk´1ϕ px´kq +Bxk`1ϕ px´kq +... +Bxnϕ px´kq +˛ +‹‹‹‹‹‹‹‚ +P Rn´1. +We check this claim using the fact that for any i1, i2 P I we have +Ti1pxq1Ti2pxq +“ +ÿ +1ďjďn +` +1i1pjq ` 1kpjq Bxi1ϕpx´kq +˘ ` +1i2pjq ` 1kpjq Bxi2ϕpx´kq +˘ +“ +1i1“i2 ` Bxi1ϕpx´kq Bxi2ϕpx´kq. +The parametrization of the hyper surface BE is now given by the chart function +ψ : θ “ pθiqiPI P S ÞÑ ψpθq P BE +with +@1 ď j ď n +ψjpθq :“ 1Ipjq θj ` 1kpjq ϕpθq. +For any 1 ď j ď n and i1, i2 P I observe that +Bθi1ψpθq “ T ψ +i1pθq :“ Ti1pψpθqq +and +Bθi1,θi2 ψjpθq “ 1kιpjq Bθi1,θi2ϕpθq. +This implies that +´ +Bθi1,θi2 ψpθq +¯1 +Npψpθqq “ ´ǫ +p∇2ϕpθqqi1,i2 +a +1 ` }∇ϕpx´kq}2 +with +∇2ϕpθq :“ +` +Bθi1,θi2ϕpθq +˘ +pi1,i2qPI2 . +We set pBψpθqq1 :“ Tpψpθqq1 and Nψpθq :“ Npψpθqq. In this notation, we also have +` +BNψpθq +˘1 +:“ +` +Bθ1Nψpθq, . . . , Bθkι´1Nψpθq, Bθkι`1Nψpθq, . . . , BθnNψpθq +˘ +“ ´Wpψpθqq :“ ´gpψpθqq´1Ωpψpθqq +with +Ωpψpθqq “ ´ǫ +∇2ϕpθq +a +1 ` }∇ϕpθq}2. +Example 7.6. For the cylindrical boundary discussed in Example 7.5, the inner prod- +uct and the Weingarten map on the boundary B pE are given for any y P B pE by the +matrices +pgpyq “ Ipn1´1,n1´1q ` ∇pϕpy´kq∇pϕpy´kq1 +and +x +Wpyq :“ ǫ pgpyq´1 +∇2 pϕpy´kq +a +1 ` }∇pϕpy´kq}2 +58 + +with the gradient column vector and the Hessian matrix given by +∇pϕpy´kq +:“ +pByi pϕpy´kqqiPpI +∇2 pϕpy´kq +:“ +` +Byi1,yi´2 pϕpy´kq +˘ +i1,i2PpI +with +pI :“ t1, . . . , n1u ´ tku. +Observe that +detppgpyqq “ 1 ` }∇pϕpy´kq}2 +and +∇2ϕpy´k, zq “ +ˆ +∇2 pϕpy´kq +0pn1´1,n2q +0pn2,n1´1q +0pn2,n2q +˙ +. +In this case, the inner product and the Weingarten map on the boundary BE are given +for any point x “ py, zq P pB pE ˆ Rn2q by the matrices +gpxq “ +ˆ +pgpyq +0pn1´1,n2q +0pn2,n1´1q +Ipn2,n2q +˙ +and +Wpxq “ +ˆ x +Wpyq +0pn1´1,n2q +0pn2,n1´1q +0pn2,n2q +˙ +. +Observe that the above matrices are bounded (w.r.t. any matrix norm) as soon as B pE +is bounded. +More generally, assume that the boundary BE Ă YιPJ Opιq Ă Rn admits a finite +covering by open connected subsets Opιq Ă Rn indexed by some finite set J . In +addition, there exists some local defining smooth functions ϕι with non vanishing +gradients on Opιq such that +BEpιq :“ BE X Opιq “ ϕ´1 +ι +pt0uq +and +Epιq :“ E X Opιq “ ϕ´1 +ι +ps0, 8rq . +Up to shrinking the set Opιq, by the implicit function theorem there is no loss of +generality to assume that the defining functions are given by +ϕι : x “ pxiq1ďiďn P Opιq ÞÑ ϕιpxq “ ϕιpx´kιq ´ xkι +for some parameter 1 ď kι ď n and some smooth function ϕι on some ball Spιq Ă Rn´1. +We set Iι :“ t1, . . . , nu ´ tkιu. In this notation, the parametrization of the hyper +surface BEpιq is now given by the smooth homeomorphism +ψι : θ “ pθiqiPIι P Spιq ÞÑ ψιpθq P BEpιq +with +ψj +ι pθq :“ 1Iιpjq θj ` 1kιpjq ϕιpθq. +(112) +The first and second fundamental forms on Tx pBEpιqq as well as the Weingarten map +at x P BEpιq are given by +gιpxq +“ +I ` ∇ϕιpx´kιq∇ϕpx´kιq1 +Ωιpxq +“ +´ +∇2ϕιpx´kιq +a +1 ` }∇ϕιpx´kιq}2 +and +Wιpxq :“ gιpxq´1Ωιpxq. +59 + +The atlas A “ pψι, SιqιPJ represents a collection of local coordinate systems of the +boundary BE “ YιPJ BEpιq. In this situation, the surface form on BE and the volume +form σDrpEq on DrpEq expressed in the atlas A are defined by the formulae +σA +B pdθq +:“ +ÿ +ιPJ +πι pψιpθqq 1Spιqpθq +a +1 ` }∇ϕιpθq}2 dθ +σA +DrpEqpdpθ, uqq +:“ +ÿ +ιPJ +πι pψιpθqq 1Spιqpθq |det pI ´ u Wιpψιpθqqq| +a +1 ` }∇ϕιpθq}2 du. +In the above display, πι : BE ÞÑ r0, 1s stands for some partition of unity subordinate +to the open cover of the boundary induced by the atlas. +Example 7.7. Observe that the metric in the graph model discussed in Example 7.2 +is not necessarily bounded. In this context, we can also use for any a ă a` ă b´ ă b +a covering of the form +Op0q “sa, brˆR +Op´1q “sb´, `8rˆR +and +Op1q “s ´ 8, a`rˆR. +For instance when ϕpzq “ z2 and pa, a`, b´, bq “ p´2, ´1, 1, 2q we have +BEp0q +“ +tpx1, x2q Ps ´ 2, 2rˆs4, 8r : x2 “ ϕ0px1qu +BEp1q +“ +tpx1, x2q Ps ´ 8, ´1rˆs1, `8r : x1 “ ϕ1px2qu +BEp´1q +“ +tpx1, x2q Ps1, 8rˆs1, `8r : x1 “ ϕ´1px2qu +with the functions +ϕ0pzq “ z2 +and +@ǫ P t´1, 1u +ϕǫpzq “ ´ǫ?z. +Whenever E is the sub-graph of ϕ, the parameter ǫ P t´1, 1u plays the role of the +orientation and the outward pointing unit normal vector at x P BEp0q and y P BEpǫq +are given by +N0pxq “ +1 +a +1 ` 4x2 +1 +ˆ +2x1 +´1 +˙ +and +Nǫpyq “ +ǫ +a +1 ` 1{p4y2q +ˆ +´1 +´ǫ{?4y2 +˙ +. +The tangent vectors at x P BEp0q and at y P BEpǫq are defined by +T0pxq “ +ˆ +1 +2x1 +˙ +and +Tǫpyq “ +ˆ +´ǫ{p?4y2q +1 +˙ +. +The above sub-graphs can be described with 3 charts tψ0, ψ`1, ψ´1u defined for any +ǫ P t´1, 1u by +ψ0 : θ Ps ´ 2, 2rÞÑ ψ0pθq “ +ˆ +θ +θ2 +˙ +and +ψǫ : θ Ps1, 8rÞÑ ψǫpθq :“ +ˆ +´ǫ +? +θ +θ +˙ +. +60 + +In this situation, the tangent vectors are given by +Bθψ0pθq “ T0pψ0pθqq “ +ˆ +1 +2θ +˙ +and +Bθψǫpθq “ Tǫ pψǫpθqq “ +ˆ +´ǫ{p +? +4θq +1 +˙ +. +In this context, for any θ Ps ´ 2, 2r we have +gpψ0pθqq “ 1 ` 4θ2 +and +Wpψ0pθqq “ ´2 +` +1 ` 4θ2˘´3{2 . +In addition, for any θ Ps1, 8r we have +gpψǫpθqq “ 1 ` 1{p4θq +and +Wpψǫpθqq “ ´2ǫ p1 ` 4θq´3{2 . +Observe that the metric expressed in the chart tψ0, ψ`1, ψ´1u is defined in terms of +bounded functions. +Example 7.8. Consider the hyperbolic paraboloid boundary +BE +“ +tpy1, y2, y3q P R3 : y3 “ y2 +1 ` y2 +2u +“ +BEp0q Y BEp1, 1q Y BEp1, ´1q Y BEp2, 1q Y BEp2, ´1q. +In the above display, BEp0q and BEpi, ǫq with i P t1, 2u and ǫ P t´1, 1u stands for the +partition defined for any ǫ P t´1, 1u by +BEp0q +:“ +ty P R3 : py1, y2q P S0 +y3 “ ϕ0py1, y2q :“ y2 +1 ` y2 +2u +BEp1, ǫq +:“ +ty P R3 : py1, y3q P S +y2 “ ϕ1,ǫpy1, y3q :“ ǫ +b +y3 ´ y2 +1u +BEp2, ǫq +:“ +ty P R3 : py2, y3q P S +y1 “ ϕ2,ǫpy1, y2q :“ ǫ +b +y3 ´ y2 +2u +with the sets +S0 +:“ +tpy1, y2q P R2 : y2 +1 ` y2 +2 ă 2u +S +:“ +tpy2, y3q P R2 : y3 ą 1 +& +|y2| ă +a +3y3{4u. +On the truncated boundary BEp0q we use a single chart defined by +ψ0 : θ “ pθ1, θ2q P S0 ÞÑ ψ0pθq “ +¨ +˝ +θ1 +θ2 +θ2 +1 ` θ2 +2 +˛ +‚P BEp0q. +On BEp1, ǫq we use the chart defined by +ψ1,ǫ : θ “ pθ1, θ3q P S ÞÑ ψ1,ǫpθq “ +¨ +˝ +θ1 +ǫ +a +θ3 ´ θ2 +1 +θ3 +˛ +‚P BEp1, ǫq. +61 + +Finally, on BEp2q we use the chart defined by +ψ2,ǫ : θ “ pθ2, θ3q P S ÞÑ ψ2,ǫpθq “ +¨ +˝ +ǫ +a +θ3 ´ θ2 +2 +θ2 +θ3 +˛ +‚P BEp2, ǫq. +For any θ “ pθ1, θ2q P S0 we have +Bθ1ψ0pθq “ +¨ +˝ +1 +0 +2θ1 +˛ +‚ +and +Bθ2ψ0pθq “ +¨ +˝ +0 +1 +2θ2 +˛ +‚. +In this chart, the metric is given by +gpψ0pθqq “ +ˆ +1 ` 4θ2 +1 +4θ1θ2 +4θ1θ2 +1 ` 4θ2 +2 +˙ +and +gpψ0pθqq´1 “ +1 +1 ` 4pθ2 +1 ` θ2 +2q +ˆ +1 ` 4θ2 +2 +´4θ1θ2 +´4θ1θ2 +1 ` 4θ2 +1 +˙ +. +In addition, the outward pointing unit normal at ψ0pθq P BEp0q is given by +N0 pψ0pθqq “ +1 +a +1 ` 4pθ2 +1 ` θ2 +2q +¨ +˝ +2θ1 +2θ2 +´1 +˛ +‚ +and +Ω0 pψ0pθqq “ +1 +a +1 ` 4pθ2 +1 ` θ2 +2q +ˆ +´2 +0 +0 +´2 +˙ +. +For any θ “ pθ1, θ3q P S we have +Bθ1ψ1,ǫpθq “ +¨ +˚ +˝ +1 +´ǫθ1 +? +θ3´θ2 +1 +0 +˛ +‹‚ +and +Bθ3ψ1,ǫpθq “ +¨ +˚ +˝ +0 +ǫ +2? +θ3´θ2 +1 +1 +˛ +‹‚. +In this chart, the metric is given by +gpψ1,ǫpθqq “ +˜ +1 ` +θ2 +1 +θ3´θ2 +1 +´ +θ1 +2pθ3´θ2 +1q +´ +θ1 +2pθ3´θ2 +1q +1 ` +1 +4pθ3´θ2 +1q +¸ +and +gpψ1,ǫpθqq´1 “ +1 +1 ` +θ2 +1 +θ3´θ2 +1 ` +1 +4pθ3´θ2 +1q +˜ 1 ` +1 +4pθ3´θ2 +1q +θ1 +2pθ3´θ2 +1q +θ1 +2pθ3´θ2 +1q +1 ` +θ2 +1 +θ3´θ2 +1 +¸ +. +In addition, the outward pointing unit normal at ψ1,ǫpθq P BEp1, ǫq is given by +N1,ǫ pψ1,ǫpθqq “ +´ǫ +b +1 ` +θ2 +1 +θ3´θ2 +1 ` +1 +4pθ3´θ2 +1q +¨ +˚ +˝ +´ǫθ1 +? +θ3´θ2 +1 +´1 +ǫ +2? +θ3´θ2 +1 +˛ +‹‚ +and +Ω1,ǫ pψ1,ǫpθqq “ +´ǫ +b +1 ` +θ2 +1 +θ3´θ2 +1 ` +1 +4pθ3´θ2 +1q +˜ +ǫθ3 +pθ3´θ2 +1q3{2 +´ǫθ1 +pθ3´θ2 +1q3{2 +´ǫθ1 +pθ3´θ2 +1q3{2 +´ +ǫ +4pθ3´θ2 +1q3{2 +¸ +. +Finally, for any θ “ pθ2, θ3q P S we have +Bθ2ψ2,ǫpθq “ +¨ +˚ +˝ +´ǫθ2 +? +θ3´θ2 +2 +1 +0 +˛ +‹‚ +and +Bθ3ψ2,ǫpθq “ +¨ +˚ +˝ +ǫ +2? +θ3´θ2 +2 +0 +1 +˛ +‹‚. +In this chart, the metric and the outward pointing unit normal at ψ2,ǫpθq P BEp2, ǫq +are given by +gpψ2,ǫpθqq “ +˜ +1 ` +θ2 +2 +θ3´θ2 +2 +´ +θ2 +2pθ3´θ2 +2q +´ +θ2 +2pθ3´θ2 +2q +1 ` +1 +4pθ3´θ2 +2q +¸ +and +N2,ǫ pψ2,ǫpθqq “ ´ +ǫ +b +1 ` +θ2 +2 +θ3´θ2 +2 ` +1 +4pθ3´θ2 +2q +¨ +˚ +˝ +´1 +´ǫθ2 +? +θ3´θ2 +2 +ǫ +2? +θ3´θ2 +2 +˛ +‹‚. +62 + +Appendix +Proof of (9) +We have +Ps,s`tpV q ď V ` +ż s`t +s +p´aPs,upV q ` cq du “ V ` ct ´ a +ż t +0 +Ps,s`upV q du +and +ż t +0 +Ps,s`upV q du +“ +tPs,s`tpV q ´ +ż t +0 +uPs,s`upLs`upV qq du ě tPs,s`tpV q ´ ct2{2. +Combining the above estimates, we readily check that +Ps,s`tpV q ď p1 ` atq´1V ` ct 1 ` at{2 +1 ` at +ď p1 ` atq´1V ` ct. +This ends the proof of (9). +Proof of Proposition 2.7 +We have the following almost sure estimate }∇Xτpxq}2 ď e´λτ, where }A}2 stands +for the spectral norm of a matrix A. This yields for any x, y P Rn the almost sure +estimate +}Xτpxq ´ Xτpyq} ď e´λτ }x ´ y}. +(113) +Applying the above to y “ 0 we find that +PτpV qpxq ď PτpV qp0q V pxq1´δ +with +δ “ 1 ´ e´λτ. +Next, we check that P X +τ pV qp0q ă 8. We have +Xup0q “ +ż u +0 +pbp0q ds ` σ dBsq ` +ż u +0 +ż 1 +0 +∇bpǫXsp0qq1 Xsp0q dǫ ds. +This implies that +}Xup0q} ď β pu ` }Bu}q ` β +ż u +0 +}Xsp0q} ds +with β :“ σ _ }bp0q} _ }∇b}. Applying Gr¨onwall lemma we check that +}Xup0q} +law +ď β pu ` }Bu}q ` β2 +ż u +0 +ps ` }Bs}q eβpu´sq ds. +63 + +On the other hand, we have +ż u +0 +}Bs} ds “ u +ż 1 +0 +}Bus} ds +law +“ u3{2 +ż 1 +0 +}Bs} ds. +This yields the rather crude estimate +}Xup0q} +law +ď β +` +u ` u1{2 }B1} +˘ +` β2 u2{2 ` β2eβu u3{2 +ż 1 +0 +}Bs} ds. +For any a ě 0 by Jensen’s inequality +E +´ +ea +ş1 +0 }Bs} ds¯ +ď +ż 1 +0 +E +` +ea}Bs}˘ +ds ď ea2r{2. +It is now an elementary exercise to check that Epev}Xτ p0q}q ă 8. This ends the proof +of the proposition. +Proof of Proposition 2.8 +Consider the function +ftpxq :“ exp +ˆ +2ǫ +ˆ +e´αt Wpxq ´ β 1 ´ e´αt +α +˙˙ +ùñ ´Bt log ftpxq “ 2ǫ e´αt pαW ` βq. +In the same vein, we check that +Bxiftpxq{ftpxq +“ +2ǫ e´αt BxiW +Bxi,xjftpxq{ftpxq +“ +2ǫ e´αt ` +2ǫ e´αt BxiWBxjW ` Bxi,xjW +˘ +. +This implies that +pLpftq ´ Btftq {ft +“ 2ǫ e´αt ` +pαW ` βq ` LpWq ` ǫ e´αt ΓLpW, Wq +˘ +. +Combining the above with (37) we find that +pLpftqpxq ´ Btftpxqq ď ´2 ǫ2 e´αt ` +1 ´ e´αt˘ +ΓLpW, Wq ftpxq ď 0. +This yields the interpolation formula +E pf0pXtpxqqq ´ ftpxq “ +ż t +0 +E pBsft´spXspxqqq ds ď 0. +We check (38) after some elementary manipulations, thus there are skipped. This +ends the proof of the proposition. +64 + +Proof of Proposition 2.11 +Notice that +Xh +t pxq +law +“ ǫt x ` σt Z “ Bσt pǫt xq +with +ǫt :“ e´t +and +σt :“ +c +1 ´ ǫ2 +t +2 +and some centered Gaussian random variable Z with unit variance. The conjugate +formula (53) yields the integral operator equation +Qtpx, dyq “ e´t{2 e´x2{2 +1 +a +2πσ2 +t +exp +ˆ +´py ´ ǫtxq2 +2σ2 +t +` y2 +2 +˙ +dy. +Observe that +´py ´ ǫtxq2 +σ2 +t +` y2 “ ´ 1 +pt +ˆ +y ´ +ǫt +1 ´ σ2 +t +x +˙2 +` x2 +ǫ2 +t +1 ´ σ2 +t +with +pt :“ 1 ´ ǫ2 +t +1 ` ǫ2 +t +“ tanhptq +ðñ +Btpt “ 1 ´ p2 +t +with +p0 “ 0. +(114) +We check this claim using the fact that +1 +σ2 +t +“ +2 +1 ´ ǫ2 +t +“ 1 ` 1 ` ǫ2 +t +1 ´ ǫ2 +t +“ 1 ` 1 +pt +. +On the other hand, we have +1 ´ σ2 +t +ǫt +“ coshptq +and +Bt log coshptq “ pt “ tanhptq. +This implies that +ż t +0 +ps ds “ log coshptq +and +ǫt +1 ´ σ2 +t +“ +1 +coshptq “ exp +ˆ +´ +ż t +0 +psds +˙ +. +We also have +1 ´ σ2 +t “ 1 ´ 1 ´ ǫ2 +t +2 +“ 1 ` ǫ2 +t +2 +ùñ 1 ´ +ǫ2 +t +1 ´ σ2 +t +“ 1 ´ ǫ2 +t +1 ` ǫ2 +t +“ pt. +This implies that +Qtp1qpxq “ e´t{2 +?pt +σt +exp +ˆ +´x2 +2 pt +˙ +“ e´t{2 hpxq P h +t p1{hqpxq. +Notice that +e´t{2 +?pt +σt +“ +d +ǫt +1 ´ ǫ2 +t +1 ` ǫ2 +t +2 +1 ´ ǫ2 +t +“ +d +2 +1{ǫt ` ǫt +“ +1 +a +coshptq +and +Btmtpxq “ ´pt mtpxq +and +Btpt “ 1 ´ p2 +t +with +pm0pxq, p0q “ px, 0q. +This ends the proof of the proposition. +65 + +Proof of Proposition 2.12 +Notice that +et{2 ex2{2 Qtpx, dyq +“ +1 +a +2πσ2 +t +ˆ +exp +ˆ +´py ´ ǫtxq2 +2σ2 +t +` y2 +2 +˙ +´ exp +ˆ +´py ` ǫtxq2 +2σ2 +t +` y2 +2 +˙˙ +1r0,8rpyq dy. +This implies that +Qtp1qpxq “ e´ x2 +2 +tanhptq +a +coshptq +ˆ +ż 8 +0 +1 +?2πpt +ˆ +exp +ˆ +´py ´ mtpxqq2 +2pt +˙ +´ exp +ˆ +´py ` mtpxqq2 +2pt +˙˙ +dy. +We conclude that +Qtp1qpxq “ e´ x2 +2 +tanhptq +a +coshptq +ˆ P p´mtpxq{?pt ď Z ď mtpxq{?ptq +“ 2 e´ x2 +2 +tanhptq +a +coshptq +ˆ P +˜ +0 ď Z ď +x +a +sinhptq coshptq +¸ +ÝÑ 0 +as x Ñ 8 or x Ñ 0 or as t Ñ 8. +In the above display, Z stands for some centered Gaussian random variable with unit +variance. Note that we have used the fact that +mtpxq{?pt “ +x +coshptq +a +tanhptq +“ +x +a +sinhptq coshptq +. +In addition, we have +Qtpx, dyq “ +1 +P +´ +0 ď Z ď x{ +a +sinhptq coshptq +¯ +ˆ +1 +2?2πpt +ˆ +exp +ˆ +´py ´ mtpxqq2 +2pt +˙ +´ exp +ˆ +´py ` mtpxqq2 +2pt +˙˙ +1r0,8rpyq dy. +This ends the proof of the proposition. +66 + +Proof of (48) +The generator of the process (47) is defined by +Lpfqpq, pq “ β p +m +Bf +Bq ´ β +ˆBW +Bq ` σ2 +2 +p +m +˙ Bf +Bp ` σ2 +2 +B2f +Bp2 . +Recalling that 2pq ď p2 ` q2, we prove that +V pq, pq +ď +1 +2 +ˆ 1 +m ` ǫ +˙ +p2 ` ǫ +2 +ˆσ2 +2 ` 1 +˙ +q2 ` Wpqq +ď +C‹pǫq +` +1 ` p2 ` q2 ` Wpqq +˘ +with +C‹pǫq :“ max +"1 +2 +ˆ 1 +m ` ǫ +˙ +, ǫ +2 +ˆσ2 +2 ` 1 +˙ +, 1 +* +. +On the other hand, we have +LpV q +“ +β p +m +ˆBW +Bq ` ǫ σ2 +2 q ` ǫ p +˙ +´β +ˆBW +Bq ` σ2 +2 +p +m +˙ ´ p +m ` ǫ q +¯ +` σ2 +2m +“ +´β +„ 1 +m +ˆ σ2 +2m ´ ǫ +˙ +p2 ` ǫ q BW +Bq + +` σ2 +2m. +Under our assumptions, this implies that for any |q| ě r we have +LpV q +ď +´β +„ 1 +m +ˆ σ2 +2m ´ ǫ +˙ +p2 ` ǫ δ +` +Wpqq ` q2˘ +` σ2 +2m +ď +´C‹pǫ, δq +` +1 ` p2 ` q2 ` Wpqq +˘ +` cmpǫ, δq +with +C‹pǫ, δq :“ β min +"ˆ 1 +m +ˆ σ2 +2m ´ ǫ +˙ +, ǫ δ +˙* +and +cmpǫ, δq :“ C‹pǫ, δq ` σ2 +2m. +We conclude that for any |q| ą r, +pV ´1LpV qqpq, pq +ď +´C‹pǫ, δq p1 ` p2 ` q2 ` Wpqqq ´ cmpǫ, δq +V pq, pq +ď +´C‹pǫ, δq p1 ` p2 ` q2 ` Wpqqq ´ cmpǫ, δq +C‹pǫq p1 ` p2 ` q2 ` Wpqqq +“ +´C‹pǫ, δq +C‹pǫq ` cmpǫ, δq +C‹pǫq +1 +1 ` p2 ` q2 ` Wpqq +ď +´ +„C‹pǫ, δq +C‹pǫq ´ cmpǫ, δq +C‹pǫq +1 +1 ` p2 ` q2 + +. +67 + +We choose r sufficiently large to satisfy +|p| ą r +or +|q| ą r +ñ C‹pǫ, δq +C‹pǫq ´ cmpǫ, δq +C‹pǫq +1 +p2 ` q2 ě C‹pǫ, δq +C‹pǫq ´ cmpǫ, δq +C‹pǫq +1 +r2 ě a :“ C‹pǫ, δq +2C‹pǫq ą 0, +and we set +Kr :“ tpq, pq P R2 : |p| _ |q| ď ru. +In this notation, we have +LpV q ď ´aV 1E´Kr ` sup +Kr +LpV q ď ´aV ` c +with +c :“ sup +Kr +LpV q ` a sup +Kr +V. +Proof of (56) +Observe that for any 0 ă y ď 1 and z P E “s0, 8r we have +sinh pyzq ď y sinh pzq +and +sinh pzq ď 1 +2 ez. +This implies that +ż 8 +0 +Qtpx, dyq 1 +y ď sinh pmtpxqq e´ x2 +2 ppt`e´2t{ptq +a +coshptq +ˆ +ż 8 +0 +c +2 +πpt +exp +ˆ +´ y2 +2pt +˙ +dy. +from which we check that +ż 8 +0 +Qtpx, dyq 1 +y 1s0,1spyq ď +exp +´ +´ +´ +x2 +2 ppt ` e´2t +pt q ´ e´tx +¯¯ +2 +a +coshptq +. +On the other hand, for any n ě 1 we have +ż 8 +0 +Qtpx, dyq yn +ď 1 +2 +1 +a +coshptq +c +2 +πpt +exp +ˆ +´ǫ2 +tx2 +2pt +´ x2 +2 pt +˙ ż 8 +0 +yn exp +ˆ +yǫtx ´ y2 +2pt +˙ +dy. +Notice that +yǫtx ´ y2 +2pt +“ ´ 1 +2pt +py ´ ǫtxptq2 ` x2 +2 ǫ2 +t pt +so that +ż 8 +0 +Qtpx, dyq yn ď 1 +2 +1 +a +coshptq +c +2 +πpt +ˆ exp +ˆ +´x2 +2 +ˆ +p1 ´ ǫ2 +tq pt ` ǫ2 +t +pt +˙˙ ż 8 +0 +yn exp +ˆ +´ 1 +2pt +py ´ ǫtxptq2 +˙ +dy. +68 + +For any n ě 1, we conclude that +V pxq :“ xn ` 1{x ùñ V P C8pEq +and +}QtpV q} ă 8. +This ends the proof of (56). +Proof of Lemma 3.2 +To simplify notation, we write Qt instead of QrUs +t +. For any V P B8pEq X DpLq we +have +QtpV q +“ +V ` +ż t +0 +QspLpV q ´ UV q ds +ď +V ` +ż t +0 +r´a QspV q ` c Qsp1qs ds “ V ` c +ż t +0 +Qsp1q ds ´ a +ż t +0 +QspV qds. +On the other hand, through integration by parts we have +ż t +0 +QspV qds +“ +rs QspV qst +0 ´ +ż t +0 +s d +dsQspWq ds +“ +t QtpV q ´ +ż t +0 +s QspLpV q ´ UV +looooomooooon +ďc +q ds ě t QtpV q ´ c +ż t +0 +s Qsp1qds. +This implies that +QtpV q +ď +V ` c +ż t +0 +Qsp1q ds ´ a +ˆ +t QtpV q ´ c +ż t +0 +s Qsp1qds +˙ +from which we conclude that +QtpV q +ď +V +1 ` at ` c +ż t +0 +Qsp1qds ùñ QtpV q ď +V +1 ` at ` ct. +This ends the proof of (69). Now, we come to the proof of (70). We have the forward +evolution equation given for any f P DpLq by +BtQtpfq “ QtpLUpfqq. +Applying the above to f “ U we readily check that +BtQtpUq ď a0 ` a1 QtpUq ´ QtpU2q ď a0 ` a1 QtpUq ´ pQtpUqq2{Qtp1q +from which we find the Riccati estimates +BtQtpUq ď a0 ` a1 QtpUq ´ pQtpUqq2 ùñ @t ą 0 +}QtpUq} ă 8. +This ends the proof of the lemma. +69 + +Proof of (66) +By Girsanov theorem we have +Qpaq +t pfqpzq “ E +` +fpX 0 +t pzqq Ztpzq 1T 0pzqąt +˘ +with the exponential martingale +Ztpzq “ exp +ˆ 1 +σ +ż t +0 +apXspzqq1 dBu ´ +1 +2σ2 +ż t +s +}apXspzqq}2 du +˙ +. +By H¨older’s inequality, for any non negative function f on E, any z P E and any +conjugate parameters p, q ą 1 with 1{p ` 1{q “ 1 we have +Qpaq +t pfqpzq ď E +` +Ztpzqq 1T 0pzqąt +˘1{q Qpaq +t pf pqpzq1{p. +On the other hand, we have +E +` +Ztpzqq 1T 0pzqąt +˘ +“ +E +ˆ +Ztpzq exp +ˆqpq ´ 1q +2σ2 +ż t +s +}apXspzqq}2 du +˙ +1T 0pzqąt +˙ +ď +ctppq :“ exp +ˆ +pt +2ppp ´ 1qσq2 sup +D +a +˙ +with the exponential martingale +Ztpzq “ exp +ˆ q +σ +ż t +0 +apXspzqq1 dBu ´ q2 +2σ2 +ż t +s +}apXspzqq}2 du +˙ +. +This ends the proof of roof of (66). +Proof of Lemma 4.2 +For any z P BE there exists some open ball Bpz, rq Ă Rn with r ą 0 and some +C1-mapping g from Rn´1 into R such that +E X Bpz, rq +“ +tx P Bpz, rq : xn ă gpx´nqu +BE X Bpz, rq +“ +tx P Bpz, rq : xn “ gpx´nqu +with +x´n :“ px1, . . . , xn´1q. +We make the change of variables +Epz, rq :“ E X Bpz, rq +ÞÑ ςpxq :“ px´n, xn ´ gpx´nqq P Opz, rq :“ ςpEpz, rqq Ă pRn´1 ˆ R`q +with Jacobian +∇ςpxq “ +ˆ +Ipn´1qˆpn´1q +´∇gpx´nq +0 +1 +˙ +. +70 + +Observe that +ς : x P E0pz, rq :“ pBE X Bpz, rqq +ùñ ςpxq “ px´n, 0q P O0pz, rq :“ ςpE0pz, rqq Ă pRn´1 ˆ t0uq. +The inverse is given by +y P Opz, rq ÞÑ ς´1pyq “ py´n, yn ` gpy´nqq P Epz, rq +ùñ ∇ς´1pyq “ +ˆ +Ipn´1qˆpn´1q +∇gpy´nq +0 +1 +˙ +. +On the other hand we have +}ςpxq ´ ςpxq} +“ +` +}x´n ´ x´n}2 ` p|xn ´ xn| ` |gpx´nq ´ gpx´nq|q2˘1{2 +ď +` +}x´n ´ x´n}2 ` 2|xn ´ xn|2 ` 2}∇g}2}x´n ´ x´n}2˘1{2 +ď +cpgq }x ´ x} +with +cpgq :“ +a +2 _ p1 ` 2}∇g}2q ě 1. +In the same vein, we have +}ς´1pyq ´ ς´1pyq} ď cpgq }y ´ y} +so that +1 +cpgq }y ´ y} ď }ς´1pyq ´ ς´1pyq}. +For any x P Epz, rq and x P E0pz, rq we have ςpxq P O0pz, rq and +}x ´ x} “ }ς´1pςpxqq ´ ς´1pςpxqq} ě +1 +cpgq }ςpxq ´ ςpxq} ě +1 +cpgq |ςpxqn|. +Taking the infimum of all x P E0pz, rq this implies that +dpx, E0pz, rqq ě +1 +cpgq |ςpxqn| +and +dpς´1pyq, E0pz, rqq ě +1 +cpgq |yn| +for any x P Epz, rq and y P Opz, rq. We conclude that +ż +Epz,rq +χ pdpx, E0pz, rqqq dx +“ +ż +Opz,rq +χ ` +dpς´1pyq, E0pz, rqq +˘ +|det +` +ς´1pyq +˘ +| dy +ď +1 +cpgq +sup +yPOpz,rq +|det +` +ς´1pyq +˘ +| +ż +Opz,rq +χpynq dy ă 8. +We end the proof of the lemma by covering BE by finitely many boundary coordinates +patches pEpzi, riq, giq1ďiďn, for some zi P BE, ri ą 0 and some local defining functions +gi. +71 + +Proof of Lemma 4.9 +Using the change of variable formulae +ż +BEr +fpzq σB,rpdzq “ +ż +BE +f pz ` rNpzqq |det pI ´ r Wpzqq | σBpdzq +and +ż +BE +fpzq σBpdzq “ +ż +BEr +f pz ´ rNpzqq |det pI ` r Wpzqq | σB,rpdzq +we check that +ż +BEr +fpzq σB,rpdzq ď κBpαq +ż +BE +f pz ` rNpzqqq σBpdzq +and +ż +BE +fpzq σBpdzq ď κ´ +B pαq +ż +BEr +f pz ´ rNpzqq +σB,rpdzq. +This yields the estimate +ż +BEr +fpzq σB,rpdzq ď ιpαq κBpαq +ż +BE +gpzq σBpdzq. +In the same vein, we have +ż +BE +fpzq σBpdzq ď ιpαqκ´ +B pαq +ż +BEr +g pzq +σB,rpdzq. +Integrating w.r.t. the parameter r P r0, αs we check the co-area estimate +α +ż +BE +fpzq σBpdzq +ď +ιpαqκ´ +B pαq +ż α +0 +dr +ż +BEr +gpzq σB,rpdzq +“ +ιpαq κ´ +B pαq +ż +DαpEq +gpzq dz. +This ends the proof of the lemma. +Proof of Proposition 7.4 +For any given θ :“ pθ1, . . . , θnq P pRn´1 ˆ r0, rsq we set θ´n :“ pθ1, . . . , θn´1q. In this +notation, we have +ψ : θ P pRn´1 ˆ r0, rsq ÞÑ ψpθq +:“ +Fpψpθ´nq, θnq +“ +ψpθ´nq ` θn Npψpθ´nqq P DrpEq. +72 + +The volume form σDrpEq on DrpEq expressed in the chart ψ is given by +´ +σDrpEq ˝ ψ +´1¯ +pdθq “ |det +` +Jacpψqpθq +˘ +| “ +c +det +´` +Bψpθq +˘ ` +Bψpθq +˘1¯ +dθ. +Arguing as above, we have +` +Bψpθq +˘1 “ +ˆ´ +Bθ´nψpθq +¯1 +, Bθnψpθq +˙ +P +´ +TψpθqpDrpEqq +¯n +with the tangent vectors +´ +Bθ´nψpθq +¯1 +:“ +´ +Bθ1ψpθq, . . . , Bθn´1ψpθq +¯ +and +Bθnψpθq “ Npψpθ´nqq. +In addition, we have +´ +Bθ´nψpθq +¯1 +“ +` +Bψpθ´nq +˘1 ` θn +` +BpNpψpθ´nqqq +˘1 “ +` +Bψpθ´nq +˘1 ` +I ´ θn Wpψpθ´nqq +˘ +. +This yields the formula +´ +Bθ´nψpθq +¯ ´ +Bθ´nψpθq +¯1 +“ gpψpθ´nqq +` +I ´ θn Wpψpθ´nqq +˘2 +from which we check that +` +Bψpθq +˘ ` +Bψpθq +˘1 “ +ˆ +gpψpθ´nq +` +I ´ θn Wpψpθ´nqq +˘2 +0n´1 +0 +1 +˙ +. +We conclude that +c +det +´` +Bψpθq +˘ ` +Bψpθq +˘1¯ +“ +b +detpgpψpθ´nqq +ˇˇdet +` +I ´ θn Wpψpθ´nqq +˘ˇˇ +and therefore +´ +σDrpEq ˝ ψ +´1¯ +pdθq “ +ˇˇdet +` +I ´ θn Wpψpθ´nqq +˘ˇˇ dθn +` +σB,0 ˝ ψ´1˘ +pdθ´nq. +For any given θn “ u P r0, rs, the volume form σB,u on the boundary BEu expressed +in the boundary chart +ψp., uq : θ P Rn´1 ÞÑ ψpθ, uq “ Fpψpθq, uq P BEu +is given by +` +σB,u ˝ ψp., uq´1˘ +pdθq “ |det pI ´ u Wpψpθqqq| pσB,0 ˝ ψ´1q pdθq. +This ends the proof of the proposition. +73 + +References +[1] A.R. 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Journal of Theoretical Probability, vol. 31, no. 3, 1322– +1355 (2018). +79 + diff --git a/rtE1T4oBgHgl3EQf3AU7/content/tmp_files/load_file.txt b/rtE1T4oBgHgl3EQf3AU7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29c3ffede084f1b695aba2edc961643e74e07c3d --- /dev/null +++ b/rtE1T4oBgHgl3EQf3AU7/content/tmp_files/load_file.txt @@ -0,0 +1,2755 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf,len=2754 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='03484v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='PR] 9 Jan 2023 A Lyapunov approach to stability of positive semigroups: An overview with illustrations Marc Arnaudon1, Pierre Del Moral2 & El Maati Ouhabaz1 1University of Bordeaux, Institut de Math´ematiques de Bordeaux, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' E-Mail: marc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='arnaudon@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='fr, Elmaati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='Ouhabaz@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='fr 2Centre de Recherche Inria Bordeaux Sud-Ouest, Talence, 33405, FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' E-Mail: pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='del-moral@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='fr January 10, 2023 Abstract The stability analysis of possibly time varying positive semigroups on non necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, physics and molecular chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ar- ticle presents an overview of some recent Lyapunov-based approaches, focusing principally on practical and powerful tools for designing Lyapunov functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' These techniques include semigroup comparisons as well as conjugacy princi- ples on non necessarily bounded manifolds with locally Lipschitz boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' All the Lyapunov methodologies discussed in the article are illustrated in a variety of situations, ranging from conventional Markov semigroups on general state spaces to more sophisticated conditional stochastic processes possibly re- stricted to some non necessarily bounded domains, including locally Lipschitz and smooth hypersurface boundaries, Langevin diffusions as well as coupled harmonic oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Keywords: Integral operators, semigroups, Markov and Sub-Markov semi- groups, harmonic oscillators, Langevin diffusions, Lyapunov function, hyper- surfaces, shape matrices, boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Mathematics Subject Classification: Primary 47D08, 60J25, 47D06, 47D07, 47H07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' secondary 47B65, 37A30, 37M25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 1 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Description of the models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 55 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 Boundary decompositions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 57 2 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Description of the models Let BpEq be the algebra of locally bounded measurable functions on a locally compact Polish space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We denote by BbpEq Ă BpEq the sub-algebra of bounded measurable functions endowed with the supremum norm }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Qs,t, be a semigroup of positive integral operators on BbpEq indexed by a continuous time indices s, t P T “ R` :“ r0, 8r or by a discrete time index set T “ N, with s ď t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a given uniformly positive function V P BpEq, we let BV pEq Ă BpEq be the sub-space of functions f P BpEq equipped with the norm }f}V :“ }f{V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also let B8pEq Ă BpEq be the subalgebra of locally bounded and uniformly positive functions V that grow at infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is, supK V ă 8 for any compact set K Ă E, and for any r ě V‹ :“ infE V ą 0 the r-sub-level set Vprq :“ tV ď ru Ă E is a non empty compact subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We denote by B0pEq :“ t1{V : V P B8u Ă BbpEq the sub-algebra of positive functions, locally lower bounded and that vanish at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a given V P B8pEq, consider the subspace B0,V pEq :“ tf P BpEq : |f|{V P B0pEqu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We say that Qs,t is a V -positive semigroup on BV pEq for some Lyapunov function V P B8pEq as soon as there exists some τ ą 0 and some function Θτ P B0pEq such that for any 0 ă f P BV pEq and s ă t we have 0 ă Qs,tpfq P B0,V pEq as well as Qs,s`τpV q{V ď Θτ and sup |t´s|ďτ ` |||Qs,t||| _ |||Qs,t|||V ˘ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (1) The growth conditions stated above are discussed in some details in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' As shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1, the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' criterion in (1) can be seen as a uniform Foster- Lyapunov condition (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' drift condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Foster-Lyapunov criterion dates back to the 1950s with the seminal articles [32, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' These criteria are nowadays an essential tool to analyze the stability properties of Markov semigroups on general state spaces [7, 26, 37, 39, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Their use in the context of positive semigroup arising in discrete time nonlinear filtering goes back to the pioneering articles [27, 64], based on coupling techniques developed in [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The extension of Foster-Lyapunov criterion to discrete or continuous time varying positive semigroups and their normalized versions on general state spaces were further developed in [21], extending Dobrushin’s ergodic coefficient techniques introduced in [11, 12] and further developed in [15, 16, 17, 14, 22] to unbounded state space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Recall that the Dobrushin’s ergodic coefficient of a Markov semigroup is the oper- ator norm of the Markov integral operator acting on probability measures equipped with the total variation norm (see for instance [16] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, the V -Dobrushin’s ergodic coefficient of a Markov transition is defined as the operator norm of the Markov integral operator acting on probability measures 3 equipped with the V -norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this operator theoretical framework, the contraction w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' V -norms is deduced by coupling the Foster-Lyapunov criterion with a local contraction on a sufficiently large compact sub-level set of the Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' A brief overview on this subject is provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The local contraction on the compact sub-level sets of the Lyapunov function is generally an easily verifiable condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This property is often deduced from a Doeblin type local minorization property of integral operators on the compact sub- level sets of the Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, this local minorization condition is satisfied as soon as the semigroup is lower bounded by an absolutely continuous integral operator (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' transition kernel operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This class of models includes hypo-elliptic diffusion semigroups as well as some regular jump processes on non necessarily bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Even for diffusion semigroups with smooth densities on bounded manifolds with entrance boundaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' boundary states that cannot be reached from the inside), the existence of a sufficiently strong Lyapunov function is essential to ensure the stability of the semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the transition densities are null on entrance bound- ary states so that the local minorization condition alone applied to some exhausting sequence of compact subsets is not sufficient to ensure the stability of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The exhausting sequence of compact subsets needs to be equivalent to the sub-level sets of some sufficiently strong Lyapunov function near entrance boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a more thorough discussion on this subject we refer to Section 2 and the article [21], see also the series of Riccati-type diffusions discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The general problem of constructing Lyapunov functions for positive semigroups, including for Markov semigroups often requires to have some good intuition about a candidate for a Lyapunov function on some particular class of model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' As for determin- istic dynamical systems, the design of Lyapunov functions for sub-Markov semigroups associated with a non-absorbed stochastic process requires to use some physical in- sight on the stability and the behavior of the free evolution stochastic process near possible absorbing boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Constructing Lyapunov functions for general classes of positive semigroups is well known as a very hard problem in system theory as well as in applied probability literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The main subject of this article is to find practical ways to design these Lyapunov functions for various classes of positive semigroups that have been discussed in the literature, including conditional diffusions on manifolds with Neumann and Dirichlet boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We did our best to cover the subject as broadly as possible, we also refer to the article [21] for additional historical and reference pointers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Due to the vast literature on this subject we apologize for possible omissions of some important contributions due to the lack of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The remainder of this article is structured as follows: In Section 2, we begin with a brief review of V -norm contraction theorems and semigroup stability properties stemming from an assumed Lyapunov structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 is dedicated to time varying Markov semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The extension of these re- sults to time varying positive semigroups are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3, 4 we present some consequences of these results in the context of time homogenous models, including existence of ground states and quasi-invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5 present different tools to design Lyapunov functions for continuous time Markov semigroups and sub-Markov semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also illustrate these results through different examples of semigroups arising in physics and applied probability, including overdamped Langevin diffusions, Langevin and hypo-elliptic diffusions, as well as typical examples of solvable one-dimensional sub-Markov semigroups such as the harmonic oscillator, the half-harmonic oscillator and the Dirichlet heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' General comparison and conjugacy principles to construct Lyapunov functions for positive semigroups are provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Boundary problems are discussed in some details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We then turn in Section 5 to the design of Lyapunov func- tions for Riccati type processes, including positive definite matrix valued diffusions, logistic and multivariate birth and death processes arising respectively in Ensemble Kalman-Bucy filter theory and population dynamic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In Section 6 we illustrate the power of the Lyapunov approach in the context of multivariate conditional diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 7 is dedicated to illustrations with explicit computations of geometrical objects for the Lyapunov functions discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 in the context of hypersurface Dirichlet boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Some basic notation We denote by CpEq Ă BpEq the sub-algebra of continuous functions and by CbpEq Ă CpEq the sub-algebra of bounded continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also set CV pEq :“ BV pEq X CpEq, C0pEq :“ B0pEq X CpEq and C8pEq :“ B8pEq X CpEq and C0,V pEq :“ B0,V pEq X CpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that none of the sub-algebras B0pEq and B8pEq have an unit unless E is compact, the null function 0 R B0pEq but the unit function 1 P C0,V pEq as soon as V P B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let MbpEq be the set of bounded signed measures µ on E equipped with the total variation norm }µ}tv :“ |µ|pEq{2, where |µ| :“ µ` ` µ´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' It stands for the total variation measure associated with a Hahn-Jordan decomposition µ “ µ` ´ µ´ of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Also let PpEq Ă MbpEq be the subset of probability measures on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' With a slight abuse of notation, we denote by 0 and 1 the null and unit scalars as well as the null and unit function on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The action of Qs,t on BbpEq is given for any f P BbpEq by the formulae Qs,tpfqpxq :“ ż Qs,tpx, dyq fpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (2) The left action of Qs,t on MbpEq is given for any η P MbpEq by the formulae pη Qs,tqpdyq :“ ż ηpdxq Qs,tpx, dyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (3) In this notation, the semigroup property takes the following form Qs,uQu,t “ Qs,t with Qs,s “ I, the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (4) 5 In the above display, Qs,uQu,t is a shorthand notation for the composition Qs,u˝Qu,t of the left or right-action operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Unless otherwise stated, all the semigroups discussed in this article are indexed by conformal indices s ď t in the set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To avoid repetition, we often write Qs,t without specifying the order s ď t of the indices s, t P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We denote by MV pEq be the space of measures µ P MbpEq equipped with the operator V -norm |||µ|||V :“ |µ|pV q, and by PV pEq Ă MV pEq be the convex set of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with a function h P B0,V pEq the Boltzmann-Gibbs transformation Ψh : µ P PV pEq ÞÑ Ψhpµq P PV hpEq (5) with the probability measure Ψhpµqpdxq :“ hpxq µphq µpdxq and V h :“ V {h P B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also denote by |||Q|||V the operator norm of a bounded linear operator Q : f P BV pEq ÞÑ Qpfq P BV pEq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is |||Q|||V :“ supt}Qpfq}V : f P BV pEq such that }f}V ď 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (6) In terms of the V -conjugate semigroup f P BbpEq ÞÑ QV pfq :“ QpV fq{V P BbpEq we have |||Q|||V “ }QV p1q} “ ˇˇˇˇˇˇQV ˇˇˇˇˇˇ :“ supt}QV pfq} : f P BbpEq such that }f} ď 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a given measurable function f and a given measurable subset, we use the short- hand notation ´8 ď inf A f :“ inf xPA fpxq ď sup A f :“ sup xPA fpxq ď `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a given s P T and τ P T with τ ą 0, we consider the time mesh rs, 8rτ:“ ts ` nτ P rs, 8r : n P Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Throughout, unless otherwise is stated we write c for some positive constants whose values may vary from line to line, and we write cα, as well as cpβq and cαpβq when their values may depend on some parameters α, β defined on some parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also set a ^ b “ minpa, bq, a _ b “ maxpa, bq, and a` “ a _ 0 for a, b P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Regularity conditions The irreducibility condition f ą 0 ùñ Qs,tpfq ą 0 is satisfied if and only if we have Qs,tp1q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We check this claim by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that Qs,tp1q ą 0 and consider a function f ą 0 and some x P E such that Qs,tpfqpxq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this case, for any ǫ ą 0 we would have 0 “ ǫ Qs,t p1fěǫq pxq ď Qs,tpfqpxq by Fatou’s lemma we would find the contraction lim inf ǫÑ0 Qs,t p1fěǫqp xq “ 0 ě Qs,tp1qpxq ùñ Qs,tp1qpxq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Without further mention, all semigroups Qs,t considered in this article are assumed to be semigroups of positive integral operators Qs,t on BbpEq satisfying the irreducibility condition Qs,tp1q ą 0 for any s ď t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that the condition 0 ă f P BV pEq ùñ @s ă t 0 ă Qs,tpfq P B0,V pEq is met as soon as Qs,t is a strong V -Feller semigroup, in the sense that for any s ă t we have Qs,tpBV pEqq Ă CV pEq and when we have Qs,tpV q{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To check this claim, observe that for any positive function f P BV pEq and s ă t the function Qs,tpfq is positive and continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' and thus locally lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, whenever }f}V ď 1, for any s ă t we have the comparison property Qs,tpfq{V ď Qs,tpV q{V P B0pEq ùñ Qs,tpfq{V P B0pEq ðñ Qs,tpfq P C0,V pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In summary, a strong V -Feller semigroup Qs,t is V -positive on BV pEq as soon as there exists some τ ą 0 and some function Θτ P B0pEq such that the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (1) is met and for any s ă t we have Qs,tpV q{V P B0pEq and Qs,s`τpV q{V ď Θτ P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When V P C8pEq, we say that Qs,t is a V -positive semigroup on CV pEq as soon as Qs,tpCV pEqq Ă C0,V pEq for any s ă t and condition (1) is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' A V -Feller semigroup Qs,t for some V P C8pEq, in the sense that for any s ă t we have Qs,tpCV pEqq Ă CV pEq, is also said to be V -positive on CV pEq as soon as there exists some τ ą 0 and some function Θτ P B0pEq such that the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (1) is met and for any s ă t we have Qs,tpV q{V P C0pEq and Qs,s`τpV q{V ď Θτ P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Last but not least, observe that positive semigroups Qs,t with continuous time indices s ď t P R` can be turned into discrete time models by setting Qp,n “ Qpτ,nτ for any p ď n P N and some parameter τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Up to a time rescaling, the parameter τ ą 0 arising in the definition of a discrete time V -positive semigroups Qp,n can be chosen as the unit time parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (1) is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 7 2 A brief review on semigroups 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 A V -norm contraction theorem The aim of this section is to present some stability theorems for uniform V -positive semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We first examine the situation where Qs,t “ Ps,t is a semigroup of Markov integral operators Ps,t on BbpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that the Lyapunov condition stated in the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' of p1q ensures the following geometric drift condition Ps,s`τpV q ď ǫτ V ` cτ (7) some parameter ǫτ P r0, 1r and some finite constant cτ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The geometric drift condition (7) ensures that the sequence |||Ps,s`nτ|||V indexed by s ě 0 and n ě 1 is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (1) applied to Qs,t “ Ps,t ensures that the operator norms of Ps,t are uniformly bounded w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' any time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More precisely, whenever (7) is met we have the equivalence sup sě0 sup těs |||Ps,t|||V ă 8 ðñ sup |t´s|ďτ |||Ps,t|||V ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (8) Note that (8) is automatically satisfied whenever (7) is met for any τ ą 0 with supτPr0,1s cτ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, consider the Markov transition semigroup Ps,t of a continuous time stochastic flow Xs,tpxq on some locally compact normed vector space pE, }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' }q with generator Lt defined on some common domain DpLq Ă BpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, for any non negative function V P DpLq and any parameters a ą 0, c ă 8 and τ ą 0 we have LτpV q ď ´aV ` c ùñ p7q and p8q with ǫτ “ p1 ` aτq´1 ă 1 and cτ “ cτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (9) The above estimate is rather well known, a detailed proof is provided in the appendix on page 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Further examples of Markov diffusion semigroups on Rn satisfying (7) are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We further assume there exists some r0 ě 1 and some function ατ : r P rr0, 8r ÞÑ ατprq P s0, 1s, such that for any r ě r0 we have sup px,yqPVprq2 }δxPs,s`τ ´ δyPs,s`τ}tv ď 1 ´ ατprq with Vprq :“ tV ď ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (10) Consider the V -norm operator βV pPs,tq (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the V -Dobrushin coefficient) of Ps,t defined by βV pPs,tq :“ sup µ,ηPPV pEq |||pµ ´ ηqPs,t|||V {|||µ ´ η|||V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (11) In this notation, conditions (7), (8) and (10) ensure the existence of some parameter τ ą 0 such that sup |t´s|ďτ βV pPs,tq ă 8 and sup sě0 βV pPs,s`τq ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (12) 8 The proof of the above assertion can be found in [21] (see also [22] in the context of time homogeneous models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The next exponential contraction theorem is a direct consequence of the operator norm estimates (12) and it is valid on abstract measurable spaces as well as for any function V ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Ps,t be a semigroup of Markov integral operators Ps,t on some measurable state space E satisfying condition (12) for some function V ě 1 and some parameter τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, there exists a parameter b ą 0 and some finite constant c ă 8 such that for any s ď t and µ, η P PV pEq we have the exponential estimate |||pµ ´ ηqPs,t|||V ď c e´bpt´sq |||µ ´ η|||V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (13) In particular, the above exponential Lipschitz estimates are met as soon as conditions (7), (8) and (10) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The estimates (13) also hold for any s ě 0 and t P rs, 8rτ as soon as (7) and (10) are satisfied for some τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 is based on discrete time type V -norm operator contrac- tion techniques combining the geometric drift condition (7) with the total variation estimates (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (8) is a technical condition only made for con- tinuous time semigroups to ensure that (13) also holds for continuous time indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For time homogeneous semigroups Pt :“ Ps,s`t the contraction estimate (13) en- sures the existence of a single invariant probability measure µ8 “ µ8Pt P PV pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, similar approaches are presented in the article [37], simplifying the Foster-Lyapunov methodologies and the small-sets return times estimation techniques developed in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 can be seen as an extension of Harris’ theorem to time varying Markov semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The operator-theoretic framework discussed above pro- vides a very direct proof based on the V -Dobrushin coefficient (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a more thorough discussion on this subject we refer to [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that the strength of conditions (7) and (10) depends on the strength of the function V : when the function V is bounded, the geometric drift condition (7) and the uniform norm condition (8) are trivially met but in this case condition (10) is a uniform contraction condition on the state E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the reverse angle, when V P B8pEq is a function with compact sub-level sets, the geometric drift condition (7) combined with (8) ensures that µPs,t is a tight collection of probability measures indexed by s ď t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the local contraction condition (10) is met if and only if for any s ě 0 and any px, yq P Vprq2 there exists some probability measure µ on E (that may depends on the parameters pτ, r, s, x, yq) such that @z P tx, yu δzPs,s`τpdyq ě ατprq µpdyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, the above condition is met as soon as Ps,s`τpx, dyq ě ps,s`τpx, yq ντpdyq (14) 9 for some Radon positive measure ντ on E and some density function ps,s`τ, satisfying for any r ě r0 the local minorization condition 0 ă inf sPT inf Vprq2 ps,s`τ and 0 ă ντpVprqq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (15) For locally compact Polish spaces condition 0 ă ντpVprqq ă 8 is met as soon as V has compact sub-levels sets Vprq with non empty interior and ντ is a Radon measure of full support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is ντ is finite on compact sets and strictly positive on non-empty open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For time homogeneous models, also note that the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' minorization condition (15) is satisfied as soon as px, yq P pE˝q2 ÞÑ pτpx, yq is a continuous positive function on the interior E˝ of the set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Several illustrations of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 in the context of diffusion processes on Euclidean spaces as well as in Section 5 in the context of Riccati- type diffusion on positive definite matrix spaces and multivariate birth and death jump type processes on countable state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The stability of Markov semigroups on man- ifolds with entrance boundaries can also be analyzed using the Lyapunov techniques developed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, as shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1, any absolutely con- tinuous Markov semigroup Ps,t on a bounded connected subset E Ă Rn with locally Lipschitz boundary BE satisfies the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 with the (non unique) Lyapunov function V pxq “ 1{ a dpx, BEq and the distance to the boundary defined for any x P E by dpx, BEq :“ inf t}x ´ y} : y P BEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We illustrate the above discussion with some elementary one dimensional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a one dimensional Brownian on the compact interval E “ r0, 1s with reflected boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, Pt :“ P0,t coincides with the Neu- mann heat semigroup on r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, recalling that the Neumann heat kernel is smooth and strictly positive on the compact interval r0, 1s, the conditions of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 are satisfied with the unit Lyapunov function V pxq “ 1, as well as for any of the Lyapunov functions V pxq “ 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='x, V pxq “ 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 1 ´ x or V pxq “ 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='x`1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 1 ´ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The same reasoning applies to the one dimensional positive Riccati-type diffusions with an entrance boundary at the origin discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Reflecting this class of positive diffusions at x “ 1, the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 are satisfied on E “s0, 1s with the Lyapunov functions V pxq “ 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='x as well as for V pxq “ 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='x ` 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 ´ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Normalized semigroups For non necessarily Markov V -positive semigroups Qs,t one natural idea is to normalise the semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any probability measure η P PV pEq we let Φs,tpηq P PV pEq be the normalised distribution defined for any f P BV pEq by the formula Φs,tpηqpfq :“ ηQs,tpfq ηQs,tp1q and we set Qs,tpfqpxq :“ Qs,tpfqpxq Qs,tp1qpxq “ Φs,tpδxqpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (16) 10 The mapping Φs,t is a well defined semigroup on PV pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The denormalisation formula connecting these semigroups is given for any t P rs, `8rτ by µQs,tpfq “ Φs,tpµqpfq ź uPrs,trτ Φs,upµqpQu,u`τp1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (17) with rs, trτ:“ ts ` nτ P rs, tr : n P Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To check this claim, observe that for any t :“ s ` nτ we have Φs,s`pτpµqpQs`pτ,s`pp`1qτp1qq “ µQs,s`pp`1qτp1q{µQs,s`pτp1q and therefore ś 0ďpăn Φs,s`pτpµqpQs`pτ,s`pp`1qτp1qq “ µQs,s`nτp1q The above formula coincides with the product formula relating the unnormalised operators Qs,t with the normalised semigroup Φs,t discussed in [14, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2], see also [17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1] and [19, Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We strengthen (14) and assume that for any s ě 0 and τ ą 0, the integral operator Qt,t`τ has a density qs,s`τ with respect to some Radon positive measure ντ on E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is we have that Qs,s`τpx, dyq “ qs,s`τpx, yq ντpdyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (18) We also assume there exists some r0 ą 1 such that for any r ě r0 we have 0 ă ιrpτq :“ inf sPT inf Vprq2 qs,s`τ ď sup sPT sup Vprq2 qs,s`τ ă 8 and ντpVprqq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (19) In this situation, for any r ě r0 and r ě r we have the uniform estimate inf Vprq Qs,s`τp1q ě \uf6ber,rpτq :“ inf Vprq Qs,s`τp1Vprqq ě ιrpτq ντpVprqq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with a given µ P PV pEq and some function H P B0,V pEq the finite rank (and hence compact) operator f P BV pEq ÞÑ T µ,H s,t pfq :“ Qs,tpHq µspQs,tp1qq µtpfq P CV pEq with the flow of measures µt “ Φs,tpµsq starting at µ0 “ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' With this notation at hand, one has the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 ([21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a V -positive semigroups Qs,t with a density (18) sat- isfying (19) for some parameter τ ą 0 and some r0 ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, there exists a parameter b ą 0 such that for any µ, η P PV pEq and any s ě 0 and t P rs, 8rτ we have the local Lipschitz estimate |||Φs,tpµq ´ Φs,tpηq|||V ď cpµ, ηq e´bpt´sq |||µ ´ η|||V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (20) 11 For any pµ, Hq P pPV pEqˆB0,V pEqq there exists some finite constant cHpµq ă 8 such that for any s ě 0 and t P rs, 8rτ we have ˇˇˇˇ ˇˇˇˇ ˇˇˇˇ Qs,t µsQs,tp1q ´ T µ,H s,t ˇˇˇˇ ˇˇˇˇ ˇˇˇˇ V ď cHpµq e´bpt´sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (21) For continuous time semigroups, the above estimates also hold for any continuous time indices s ď t as soon as for any r ě r0 there exists some r ě r such that infδPr0,τs \uf6ber,rpδq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 is based on discrete time type V -norm operator contrac- tion techniques combining the geometric drift condition stated in the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' of (1) with the local minorization condition stated in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The condition infδPr0,τs \uf6ber,rpδq ą 0 is a technical condition only made for continuous time semigroups to ensure that (20) and (21) also hold for continuous time indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Theses regularity conditions are rather flexible as we will now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Absolutely continuous integral operators arise in a natural way in discrete time settings [17, 14, 27, 64] and in the analysis of continuous time elliptic diffusion absorp- tion models [2, 29, 30, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In connection to this, two-sided estimates for stable-like processes are provided in [6, 43, 59, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Two sided Gaussian estimates can also be obtained for some classes of degenerate diffusion processes of rank 2, that is when the Poisson brackets of the first order span the whole space [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This class of diffusions includes frictionless Hamiltonian kinetic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Diffusion density estimates can be extended to sub-Markovian semigroups using the multiplicative functional methodology developed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever the trajecto- ries of these diffusion flows, say t ÞÑ Xtpxq, where x P E is the initial position, are absorbed on the smooth boundary BE of a open connected domain E, for any τ ą 0 the densities qτpx, yq of the sub-Markovian semigroup Qτ (with respect to the trace of the Lebesgue measure on E) associated with the non absorption event are null at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Nevertheless, whenever these densities are positive and continuous on the open set E2 for some τ ą 0, they are uniformly positive and bounded on any compact subset of E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' thus condition (19) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, whenever Tpxq stands for first exit time from E and Trpxq the first exit time from the compact level set Vprq Ă E starting from x P Vprq, for any δ P r0, τs and r` ą r we have the estimate Qδp1Vpr`qqpxq :“ E ` 1Vpr`qpXδpxqq 1Tpxqąδ ˘ ě P ` Tr`pxq ą δ ˘ ě P ` Tr`pxq ą τ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, we have inf xPVprq P ` Tr`pxq ą τ ˘ ą 0 ùñ inf δPr0,τs inf Vprq Qδp1q ě inf δPr0,τs \uf6ber,r`pδq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (22) Whenever the interior E` :“ Vpr`q˝ is a connected domain, the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' estimate in (22) is met as soon as the sub-Markovian semigroup Q` τ associated with the non absorption 12 event at the boundary BE` has a continuous density px, yq P E2 ` ÞÑ q` τ px, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To check this claim, observe that for any x P Vprq we have P ` Tr`pxq ą τ ˘ “ Q` τ p1qpxq ě Q` τ p1Vprqqpxq “ ż q` τ px, yq 1Vprqpyq ντpdyq ě ντpVprqq inf Vprq2 q` τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' It is out of the scope of this article to review the different classes of absolutely con- tinuous operators and related two-sided Gaussian estimates arising in the analysis of continuous time elliptic diffusion and particle absorption models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a more thorough discussion on this topic we refer to the series of reference pointers presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Needless to say that the design of Lyapunov functions is a crucial and challenging problem in the stability analysis of positive semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We have chosen to concen- trate our review on presenting practical and general principles for designing Lyapunov functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Time homogenous models For time homogeneous models we use the notation pΦt, Qt, Qtq :“ pΦ0,t, Q0,t, Q0,tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' As expected for time homogeneous semigroups a variety of results follow almost im- mediately from the estimates obtained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Following [21], these results include the existence of an unique leading eigen-triple pρ, η8, hq P pR ˆ PV pEq ˆ B0,V pEqq with η8phq “ 1 (23) in the sense that for any t P T we have Qtphq “ eρt h and η8Qt “ eρt η8 or equivalently Φtpη8q “ η8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (24) The eigenfunction h is sometimes called the ground state and the fixed point measure η8 the quasi-invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any x P E we also have the product series formulation 0 ă hpxq :“ ź ně0 ␣ 1 ` e´ρτ rΦnτpδxqpQτp1qq ´ Φnτpη8qpQτp1qqs ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, choosing pµ, Hq “ pη8, hq in (21), we readily check that T η8,h s,s`tpfq “ Tpfq :“ h η8phq η8pfq and ˇˇˇˇˇˇe´ρt Qt ´ T ˇˇˇˇˇˇ V ď chpη8q e´bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any η P PV pEq we have the conjugate formulae ΨhpΦtpηqq “ ΨhpηqP h t (25) 13 with the Doob h-transform of Qt defined by the Markov semigroup P h t : f P BV hpEq ÞÑ P h t pfq :“ e´ρt 1 h Qtphfq P BV hpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that η8 “ Φtpη8q ðñ ηh 8 :“ Ψhpη8q “ ηh 8P h t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The Markov semigroup P h t is sometimes called the transition semigroup of the h- process, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the process evolving in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We further assume that Qt is a sub-Markov semigroup of self-adjoint operators on L2pνq with respect to some locally finite measure ν on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, there exists an orthonormal basis pϕnqně1 associated with a decreasing sequence of eigenvalues ρn ď 0 such that Qtpx, dyq “ ÿ ně1 eρnt ϕnpxq ϕnpyq νpdyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (26) In this context, the formulae (24) are satisfied with the parameters pρ, hq “ pρ1, ϕ1q and η8pdxq “ Ψhpνqpdxq :“ 1 νphq hpxq νpdxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that in this case h has unit norm νph2q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The spectral resolution (26) yields for any t ě 0 and f P L2pνq the following decomposition e´ρtQtpfqpxq ´ hpxq η8phq η8pfq “ ÿ ně2 eρh nt ϕnpxq νpϕnfq with ρh n “ ρn ´ ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (27) This yields the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any time horizon t ě 0 and any f P L2pνq we have the expo- nential estimates ››››e´ρtQtpfq ´ h η8phq η8pfq ›››› L2pνq ď eρh 2 t ` νpf 2q ´ νphfq2˘1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (28) Whenever Qt is a positive semigroup of self-adjoint operators on L2pνq the Doob h-transform P h t is a semigroup of self-adjoint operators on L2pηh 8q and we have the following spectral decomposition Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ě 0 and f P L2pηh 8q we have P h t px, dyq “ ηh 8pdyq ` ÿ ně2 eλnt hnpxq hnpyq ηh 8pdyq (29) with the L2pηh 8q orthonormal basis phnqně2 defined for any n ě 2 by hn :“ ϕn{h and λn “ ρn ´ ρ1 ă 0 and ηh 8 “ Ψh2pνq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 14 Note that the density of the integral operator P h t px, dyq w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' ηh 8pdyq is given by ph t px, yq “ e´ρ1t qtpx, yq hpxqhpyq “ 1 ` ÿ ně2 eλnt hnpxq hnpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (30) We further assume that h P B0pEq and P h t is ultra contractive, in the sense that for any t ą 0 we have ˇˇˇˇˇˇP h t ˇˇˇˇˇˇ L2pηh8qÞÑL8pηh8q “ e´ρ1t sup px,yqPE2 qtpx, yq hpxqhpyq “ sup px,yqPE2 ph t px, yq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (31) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that νpEq ă 8 and h P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, for any t ą 0 (31) holds and the mapping x ÞÑ ş ph t px, yq νpdyq is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' and locally lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the function V :“ 1{h P B8pEq and for any t ą 0 we have QtpV q{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, for any t ą 0 we have QtpV q{V ď ct{V 2 P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (32) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 Markov diffusion semigroups This section is mainly concerned with the design of Lyapunov functions for continuous time Markov semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To simplify notation, we only consider time homogeneous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' All the semigroups discussed in this section satisfy condition (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Thus, by (14) the contraction theorem, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 applies to all the Markov semigroups discussed in this section as soon as the transition semigroups have a continuous density with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 presents some elementary principles based on spectral conditions on the drift function and a simple way to design Lyapunov functions in terms of the generator of diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' These generator-type techniques are illustrated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 in the context of overdamped Langevin diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The design of Lyapunov functions for hypo-elliptic diffusions and Langevin diffusions are discussed respectively in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Some general principles Consider the Markov semigroup Pt of a diffusion flow Xtpxq on E “ Rn defined by dXtpxq “ bpXtpxqq dt ` σpXtpxqq dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (33) In the above display, Bt is a n1-dimensional Brownian motion starting at the origin for some n ě 1, b is a differentiable drift function from Rn into itself with gradient-matrix ∇b “ pBxibjq1ďi,jďn, and σ stands for some diffusion function from Rn into Rnˆn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We set Σ2 :“ σσ1, where σ1pxq :“ σpxq1 stands for the transposition of the matrix σpxq, so that Σ2pxq :“ σpxqσ1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The absolutely continuity of the transition semigroup Ptpx, dyq “ PpXtpxq P dyq “ ptpx, yqνpdyq for some continuous transition densities 15 ptpx, yq (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the Lebesgue measure νpdyq) is ensured as soon as pb, σq are globally Lipschitz continuous and the diffusion matrix is invertible or more generally satisfying a parabolic H¨ormander condition (see for instance [53, 58, 56] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The generator L of the diffusion flow Xtpxq and its carr´e du champ operator ΓL are given respectively by the formula Lpfq :“ b1∇f ` 1 2 Tr ` Σ2∇2f ˘ and ΓLpf, gq :“ p∇fq1Σ2∇g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (34) The next proposition provides a rather elementary way to design a Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that σpxq “ σ0 for some σ0 P Rnˆn1 and we have ∇b ` p∇bq1 ď ´2λ I for some λ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (35) Then for any v ą 0 and t ą 0 there exists some δt ą 0 such that V pxq :“ exp pv}x}q ùñ PtpV q{V ď ct{V δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (36) The proof of the above proposition is rather technical, thus it is provided in the appendix on page 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The next proposition is a slight extension of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='6 [49] on reversible semi- groups to stochastic flows in Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' It provides a rather simple way to design Lyapunov functions in terms of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume there exists some α ą 0, β P R and 0 ă ǫ ă 1 such that α W ` β ` LpWq ď ´ǫ ΓLpW, Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (37) In this situation, for any t ą 0 we have V :“ exp p2ǫWq ùñ Pt pV q {V ď vt{V δt (38) with the parameters vt “ exp ` ´2βǫ p1 ´ e´αtq{α ˘ and δt :“ p1 ´ e´αtq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The proof of the above proposition follows word-for-word the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='6 in [49], thus it is provided in the appendix on page 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We further assume that Pt satisfies for any t ą 0 the sub-Gaussian estimate Ptpx, dyq ď ct exp ˆ ´ 1 2σ2 t }y ´ mtpxq}2 ˙ dy (39) for some parameters σt ą 0 and some some function mt on Rn such that }mtpxq} ď ct p1 ` }x}q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 16 In this situation, for any n ě 1 and t ě 0 we have V pxq :“ 1 ` }x}n ùñ }PtpV q{V } ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More refined estimates can be found when the function mt is such that |mtpxq| ď ǫt |x| with ǫt Ps0, 1r (40) for some norm |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='| on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, observe that any v ě 0 and any centered Gaussian random variable Y on Rn with identity covariance matrix In we have e´v|x| E ´ ev|mtpxq`σ2 t Y |¯ ď ct e´vp1´ǫtq|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a Markov semigroup Pt satisfying the sub-Gaussian estimate (39) as well as (40) for some norm |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='| on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Then for any v ě 0 and t ą 0 there also exists some finite constant δt ą 0 such that V pxq :“ exp pv|x|q ùñ PtpV q{V ď ct{V δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Overdamped Langevin diffusion Let Wpxq be some twice differentiable potential function from Rn into R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The over- damped Langevin diffusion is defined by choosing in (33) the drift function bpxq :“ ´γ ∇Wpxq and pn1, σpxqq “ pn, ρ Iq for some γ, ρ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, we have p35q ðñ ∇2W ě pλ{γq I for some λ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Also observe that p37q ðñ α W ` β ` ρ2 2 Trp∇2Wq ď ` γ ´ ǫ ρ2˘ }∇W}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The above condition is clearly met when W behaves as }x}m with m ě 1 at infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is, there exists some sufficiently large radius r such that for any }x} ě r we have ˇˇTrp∇2Wpxqq ˇˇ ď c1 }x}pm´2q` and }∇Wpxq}2 ě c2 }x}2pm´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Hypo-elliptic diffusions Consider the Rn-valued diffusion (33) with pbpxq, σpxqq “ pAx, Σq, for some matrices pA, Σq with appropriate dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We assume that A is stable (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Hurwitz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is its spectral abscissa ςpAq defined below is negative ςpAq :“ sup tRe pλpAqq : λpAq P SpecpAqu ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (41) In the above display SpecpAq denotes the spectrum of the matrix A, and Re pλpAqq the real part of λpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also assume that R :“ ΣΣ1 is positive semi-definite and the pair of matrices pA, R1{2q are controllable, in the sense that the pn ˆ n2q-matrices “ R1{2, AR1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Ar´1R1{2‰ has rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (42) Whenever ςpAq ă 0 we have Ptpx, dyq “ 1 a detp2πCtq exp ˆ ´1 2 py ´ mtpxqq1 C´1 t py ´ mtpxqq ˙ dy (43) with the mean value function x ÞÑ mtpxq :“ etAx ÝÑtÑ8 0 and the covariance matrices Ct defined for any t ą 0 by 0 ă Ct :“ ż t 0 esAResA1 ds ÝÑtÑ8 C8 :“ ż 8 0 esAResA1 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Since A is stable, there exists some norm |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='| on Rn such that the corresponding operator norm satisfies |etA| ď elpAqt for some log-norm parameter lpAq ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that |mtpxq| “ |etAx| ď elpAqt |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (44) This clearly shows that the semigroup Pt of the hypo-elliptic Ornstein-Ulhenbeck diffusion satisfies (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Pt be the Markov semigroup of the Rn-valued linear diffusion dXtpxq “ pAXtpxq ` apXtpxqqq dt ` Σ dBt (45) with some bounded drift function a on Rn, an pn ˆ nq-matrix A satisfying (41), some n1-valued Brownian motion Bt starting at the origin and some pn ˆ n1q-matrix Σ satisfying the rank condition (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the stochastic interpolation formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 in [23]) given by Xtpxq ´ Xtpxq “ ż t 0 ept´sqA1 a pXspxqq ds we check the almost sure estimate |Xtpxq ´ Xtpxq| ď c for some finite constant c ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any v ą 0 and t ą 0 there exists some δt ą 0 such that V pxq :“ exp pv|x|q ùñ PtpV q{V ď ct{V δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 Langevin diffusion Consider the Langevin diffusion diffusion flow Xtpzq “ pXtpzq, Ytpzqq P pRr ˆ Rrq starting at z “ px, yq P pRr ˆ Rrq and given by dXtpzq “ Ytpzq{m dt dYtpzq “ pbpXtpzqq ´ βYtpzq{mq dt ` σ dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, Bt stands for an r-dimensional Brownian motion Bt starting at the origin, σ, β, m ą 0 some parameters and b a function of the form bpxq :“ ´γ x ` apxq with γ ą 0 and }a} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In statistical physics, the above diffusion represents the evolution of N particles Xtpzq “ pXi tpzqq1ďiďN P R3N with mass m ą 0, position Xtpzq P R3N and momenta Ytpzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, γ ą 0 stands for some friction parameter, and the diffusion parameter σ ą 0 is related to the Boltzmann constant and the temperature of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the function bpxq “ ´∇Wpxq is often described by the gradient of some potential function W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, for a quadratic confinement we have Wpxq :“ γ}x}2{2 ` wpxq with }∇w} ă 8 ùñ bpxq “ ´∇Wpxq :“ ´γ x ` apxq and apxq “ ∇wpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that Xtpzq can be rewritten in vector form as in (45) with n “ 2r, apx, yq “ ˆ 0 apxq ˙ and the matrices A “ ˆ 0 m´1 Inˆn ´γ Inˆn ´βm´1 Inˆn ˙ and Σ :“ ˆ 0 0 0 σInˆn ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (46) It is a simple exercise to check that A satisfies (41) and (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the R2-valued stochastic process Xt “ pqt, ptq defined by $ ’ & ’ % dqt “ β pt m dt dpt “ ´β ˆBW Bq pqtq ` σ2 2 pt m ˙ dt ` σ dBt (47) with some positive constants β, m, σ, a Brownian motion Bt, and a smooth positive function W on R such that for sufficiently large r we have @ |q| ě r qBW Bq pqq ě δ ` Wpqq ` q2˘ for some positive constant δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This condition is clearly met when W behaves as q2l for certain l ě 1 at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We let V pq, pq be the function on R2 defined by V pq, pq “ 1 ` 1 2m p2 ` Wpqq ` ǫ 2 ˆσ2 2 q2 ` 2pq ˙ with ǫ ă σ2 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 19 In this situation, there exists some a ą 0 and c ă 8 such that LpV q ď ´aV ` c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (48) The proof of the above estimate is rather technical, thus it is provided in the appendix on page 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5 Sub-Markov semigroups Sub-Markov semigroups are prototype-based models of positive integral operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In time homogeneous settings, these stochastic models are defined in terms of a stochastic flow Xtpxq evolving on some metric Polish space pE, dq, some non negative absorption potential function U on some non necessarily bounded Borel subset E Ă E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a given x P E we denote by Tpxq the exit time of the flow Xtpxq from E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with these objects, the sub-Markov semigroup QrUs t defined for any f P BbpEq and x P E by QrUs t pfqpxq “ E ˆ fpXtpxqq 1Tpxqąt exp ˆ ´ ż t 0 UpXspxqqds ˙˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (49) The above model can be interpreted as the distribution of a stochastic flow evolving in an absorbing medium with hard and soft obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Before killing, the flow starts at x P E and evolves as Xtpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Then, it is killed at rate U or as soon as it exits the set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the case E “ E, the flow cannot exit the set E and it is only killed at rate U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This situation is sometimes referred a sub-Markov semigroup with soft obstacles represented by the absorbing potential function U on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When the flow may exit the set E Ă E, the complementary subset C :“ E ´ E is interpreted as an hard obstacle, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' an infinite energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We illustrate the V -positive semigroup analysis developed in this article through three typical examples of solvable sub-Markov semigroups arising in physics and ap- plied probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 The harmonic oscillator Consider the case E “ E “ R, and let Xtpxq “ Btpxq be a Brownian motion starting at x P R and let Upxq “ x2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the semigroup QrUs t “ Qt defined in (49) coincides with the one dimensional harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0, the integral operator Qt has a continuous density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the uniform measure ν on E given by qtpx, yq “ ÿ ně1 eρnt ϕnpxqϕnpyq (50) with the L2pνq orthonormal basis eigenstates ϕnpxq “ p2n´1pn ´ 1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='πq´1{2 e´x2{2 Hn´1pxq 20 associated with the eigenvalues ρn “ ´pn ´ 1{2q and the Hermite polynomials Hnpxq “ p´1qn ex2 Bne´x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the eigenstate associated with the top eigenvalue ρ “ ρ1 “ ´1{2 is given by the harmonic function hpxq “ ϕ1pxq “ π´1{4 e´x2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (51) The spectral resolution of integral operator P h t px, dyq and its density ph t px, yq with respect to the invariant measure ηh 8pdyq “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='π e´y2 dy are given as in (29) and (30) with L2pηh 8q orthonormal basis defined for any n ě 2 by hn “ p2n´1pn ´ 1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='q´1{2 Hn´1 and ρh n “ ρn ´ ρ1 “ ´pn ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the h-process is given by the Ornstein-Uhlenbeck diffusion dXh t pxq “ B log hpXh t pxqq dt ` dBt “ ´Xh t pxq dt ` dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (52) In the above display, Bt “ Btp0q stands for the one dimensional Brownian motion starting at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The conjugate formula Qtphfq{Qtphq “ P h t pfq ðñ Qtpfq “ eρth P h t pf{hq (53) yields the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any time horizon t ě 0 we have Qtpx, dyq “ 1 a coshptq exp ˆ ´x2 2 pt ˙ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2πpt exp ˆ ´py ´ mtpxqq2 2pt ˙ dy with the mean and variance parameters pmtpxq, ptq defined by mtpxq “ x{coshptq and pt “ tanhptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The proof of the above proposition is a direct consequence of the conjugate formula, thus it is provided in the appendix, on page 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Choosing V pxq “ 1 ` |x|n, for some n ě 1, we readily check that V P C8pEq and QtpV q{V ď vt Qtp1q P C0pEq (54) where vt is a constant depending only on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 The half-harmonic oscillator Consider the case E “s0, 8rĂ E “ R, and let Xtpxq “ Btpxq be a Brownian motion starting at x P R and let Upxq “ x2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the semigroup QrUs t “ Qt defined in (49) coincides with the harmonic oscillator with an infinite barrier at the origin BE “ t0u (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the half-harmonic oscillator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the fact that ex2{2 1 2 B2 e´x2{2 “ Upxq ´ 1{2 we have the conjugate formula Qtpfqpxq “ e´t{2 e´x2{2 E ´ fpYtpxqq eYtpxq2{2 1T Y pxqąt ¯ with the Ornstein-Uhlenbeck diffusion dYtpxq “ ´Ytpxq dt ` dBt and T Y pxq :“ inf tt ě 0 : Ytpxq P BEu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (55) Note that the stochastic flow Ytpxq coincides with the h-process of the harmonic oscillator discussed in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Thus, by reflection arguments we have QY t pfqpxq :“ E ` fpYtpxqq 1T Y pxqąt ˘ “ E ` fpBσtpǫtxqq 1Tpǫtxqąt ˘ “ ż 8 0 fpyq qY t px, yq dy with qY t px, yq :“ prtpx, yq ´ rtpx, ´yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, pǫt, σtq stands for the parameters pǫt, σtq :“ ˜ e´t, c 1 ´ ǫ2 t 2 ¸ and rtpx, yq “ 1 a 2πσ2 t exp ˆ ´ 1 2σ2 t py ´ ǫtxq2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0 and x P E “s0, 8r we have Qtpx, dyq “ sinh py mtpxqq P ` 0 ď Z ď mtpxq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='pt ˘ ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2πpt exp ˆ ´y2 ` mtpxq2 2pt ˙ νpdyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, νpdyq :“ 1r0,8rpyq dy stands for the trace of the Lebesgue measure on the half-line, Z is a centered Gaussian variable with unit variance and pmtpxq, ptq are the mean and variance parameters defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, the total mass function Qtp1qpxq is given by the formula Qtp1qpxq “ 2 e´ x2 2 pt a coshptq ˆ P p0 ď Z ď mtpxq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='ptq P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 22 The proof of the above proposition follows the same lines of arguments as the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' it is provided in the appendix, on page 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Choosing V pxq “ xn ` 1{x, for some n ě 1, we readily check that V P C8pEq and QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (56) The proof of the above estimate follows elementary but lengthly calculations, thus it is provided in the appendix on page 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0, the integral operator Qt has a continuous density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the uniform measure ν on E given by qtpx, yq “ ÿ ně1 eρnt ϕnpxq ϕnpyq with the L2pνq orthonormal basis eigenstates ϕnpxq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2 p22n´1p2n ´ 1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='πq´1{2 e´x2{2 H2n´1pxq associated with the eigenvalues ρn “ ´pp2n ´ 1q ` 1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the eigenstate associated with the top eigenvalue ρ “ ρ1 “ ´3{2 is given for any x Ps0, 8r by the harmonic function hpxq “ ϕ1pxq “ 2π´1{4 x e´x2{2 “ h0pxq H1pxq with the ground state h0 of the harmonic oscillator discussed in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that h coincides with the restriction on s0, 8r of the first excited state of the harmonic- oscillator (negative on s ´ 8, 0s and crossing the origin at x “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The spectral resolution of integral operator P h t px, dyq and its density ph t px, yq with respect to the invariant measure ηh 8pdyq “ 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='π y2 e´y2 1s0,8rpyq dy are given for any x, y Ps0, 8r as in (29) and (30) with L2pηh 8q orthonormal basis defined for any n ě 2 and x Ps0, 8r by the odd Hermite functions hnpxq “ p22np2n ´ 1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='q´1{2 H2n´1pxq{x and ρh n “ ´2pn ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the h-process is given by the diffusion dXh t pxq “ B log hpXh t pxqq dt ` dBt “ ˆ 1 Xh t pxq ´ Xh t pxq ˙ dt ` dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (57) 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 The Dirichlet heat kernel Let Xtpxq “ Btpxq be a Brownian motion starting at x P E :“s0, 1rĂ E :“ R and Tpxq be the first time t ě 0 the process Btpxq P BE :“ t0, 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Choosing U “ 0 in (49), the semigroup QrUs t “ Qt takes the following form Qtpfqpxq :“ EpfpBtpxqq 1Tpxqątq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0, the integral operator Qt has a continuous density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the uniform measure ν on E given by the Dirichlet heat kernel qtpx, yq “ ÿ ně1 eρnt ϕnpxqϕnpyq (58) with the L2pνq orthonormal basis eigenstates ϕnpxq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2 sin pnπxq associated with the eigenvalues ρn “ ´pnπq2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the eigenstate hpxq “ ϕ1pxq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2 sin pπxq associated with the top eigenvalue ρ “ ρ1 “ ´π2{2 is strictly positive except at the boundary t0, 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By removing the boundary, the semigroup P h t of the process evolving in the ground state hpxq on the open interval E :“s0, 1r is a self-adjoint operators on L2pηh 8q with ηh 8pdxq “ h2pxq νpdxq “ 2 sin2 pπxq 1Epxq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, we have the spectral decomposition (29) with the L2pηh 8q orthonormal basis eigenstates hnpxq :“ sin pnπxq{ sin pπxq associated with the eigenvalues λn “ ´π2pn2 ´ 1q{2 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Our next objective is to estimate the density ph t px, yq of the integral operator P h t px, dyq w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' ηh 8 defined in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Recalling that | sin pnyq| ď n| sin pyq|, for any n ě 1 and y P R, for any x P E we have the diagonal estimate ph t px, xq ´ 1 “ ÿ ně2 eρh nt hnpxq2 with hnpxq2 “ ˆsin pnπxq sin pπxq ˙2 ď n2 so that condition (31) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that the function V : x P E ÞÑ V pxq :“ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 2{hpxq P r1, 8r 24 is locally bounded with compact level sets given for any 0 ă ǫ ď 1 by the formulae Kǫ :“ tx Ps0, 1r : V pxq ď 1{ǫu “ tx : sin pπxq ě ǫu Ă E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In any dimension we can use the intrinsic ultracontractivity to produce a Lyapunov function V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let E be a bounded domain of Rn for some n ě 1 and assume that it is a C1,α domain for some α ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Denote by qtpx, yq the Dirichlet heat kernel on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By [57] one has qtpx, yq ď ct dpx, BEqdpy, BEq for some constant ct independent of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Here dpx, BEq denotes the distance from x to the boundary of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Set V pxq “ 1 dpx,BEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The above intrinsic ultracontractivity implies QtpV qpxq “ ż E qtpx, yqV pyqdy ď ct|E| dpx, BEq which in turn gives QtpV q{V ď ct|E|{V 2 P B0pEq, where |E| stands for the volume of the bounded set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 3 Lyapunov design principles The aim of this section is to present some general principles to construct Lyapunov functions for positive semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 provides equivalent formulations of the Lyapunov condition in (1) encountered in the literature in terms of exhausting sequences of compact level sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This section also presents simple ways to design Lyapunov functions for sub-Markov semigroups on normed spaces in terms of their generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 presents some principles to construct Lyapunov functions for positive semigroups dominated by semigroups with known Lyapunov functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 is dedicated to the design of Lyapunov functions for conjugate semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' All the principles discussed in this section are illustrated in Section 5 as well as in Section 6 in the context of conditional diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Foster-Lyapunov conditions For time homogeneous models Qs,s`t :“ Qt, the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (1) takes the form QτpV q{V ď Θτ P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In terms of the compact sets Kǫ :“ tΘτ ě ǫu, the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s Lyapunov condition in (1) yields for any τ ą 0 the estimate QτpV qpxq ď ǫ V pxq ` 1Kǫpxq cǫ (59) for any ǫ ą 0 with the parameter cǫ :“ supKǫpV Θτq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that for any n ě 1 we have QτpV qpxq ď ǫn V pxq ` 1Kǫnpxq cǫn (60) 25 where Kǫn Ă E stands for some increasing sequence of compacts sets and cǫn some finite constants, indexed by a decreasing sequence of parameters ǫn P r0, 1s such that ǫn ÝÑ 0 as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the reverse angle, assume that QτpV q{V is locally lower bounded and lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, condition (60) ensures that QτpV q{V P B0pEq for any τ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Indeed, for any δ ą 0, there exists some n ě 1 such that ǫn ă δ and we have tQτpV q{V ě δu Ă tQτpV q{V ą ǫnu Ă Kǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Since tQτpV q{V ě δu is a closed subset of a compact set it is also compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, whenever (60) is met for some exhausting sequence of compact sets Kǫn, in the sense that for any compact subset K Ă E there exists some n ě 1 such that K Ă Kǫn we have inf K QτpV q{V ě inf Kǫn QτpV q{V ě ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ensures that the function QτpV q{V is necessarily locally lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have QτpV q{V P B0pEq as soon as QτpV q{V is lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that the sub-level set Vprq :“ tV ď ru of the Lyapunov function V P B8pEq and the ǫ-super-level sets Kǫ :“ tΘτ ě ǫu of Θτ P B0pEq are equivalent compact exhausting sequences, in the sense that for any r ě 1 we have Vprq Ă Kǫr Ă Vprǫq with ǫr :“ inf Vprq Θτ and rǫ :“ sup Kǫr V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever E is a locally compact Polish space, the abstract sequence Cn :“ Kǫn in (60) is automatically exhausting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' that is, we have that E “ Yně0Cn with Cn is included in the interior C˝ n`1 of the compact set Cn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To check this claim, observe that for any n ě 1 there exists some mn ě n such that Cn Ă tΘτ ě inf Cn Θu Ă Cmn Ă tΘτ ě inf Cmn Θu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Thus, the exhausting sequence Cn is equivalent to the one defined by the super-level sets of Θτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The rather abstract condition (60) is often presented in the literature as an initial condition to check on a case-by-case basis to analyze the stability property of time homogenous sub-Markov semigroups (see for instance [31, 36], as well as Section 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5 in [22] in the context of Markov semigroups and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We end this section with a brief discussion on condition (60) in the context of the sub-Markov semigroup discussed in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that this semigroup can be turned into a Markov semigroup by sending the killed process into a cemetery state, say ∆, at the killing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this interpretation, functions on E are extended to E∆ “ E Yt∆u by setting fp∆q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More interestingly, whenever E is locally compact its topology coincides with the weak topology induced by C0pEq :“ B0pEqXCbpEq, and inversely (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context a continuous function f vanishes at infinity 26 if and only if its extension to the one point compactification (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Alexandroff compactification) E∆ :“ E Y t∆u (obtained by setting fp∆q “ 0) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For locally compact spaces, we also recall that the one point extension E∆ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever it exists, the generator LU of these sub-Markov semigroups QrUs t are de- fined on domain of functions DpLUq Ă B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' As expected, the analysis of this class of models in terms of generators often requires to develop a sophisticated analysis tak- ing into account the topological structure of the set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To the best of our knowledge, there is no simple sufficient condition to check (60) in terms of these generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The situation is greatly simplified for sub-Markov semigroups with soft obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When E “ E is a locally compact normed space pE, }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' }q we let L be the generator of the flow Xtpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the generator of the sub-Markov semigroup QrUs t is given by LU “ L´U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We further assume that L and LU are defined on some common domain DpLq Ă BpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 ([31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let V, V0 P DpLq be a couple of functions such that V, V0 ě 1 and V pxq ÝÑ}x}Ñ8 8 and V pxq{V0pxq ÝÑ}x}Ñ8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (61) In this situation, condition (60) is satisfied as soon as there exists some finite constant c0 ă 8 such that LUpV0q{V0 ď c0 and LUpV qpxq{V pxq ÝÑ}x}Ñ8 ´8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (62) Note that in this context, the compact sets in (60) are given for some sufficiently large radii rǫ ą 0 by the closed balls: Kǫ “ Bp0, rǫq :“ tx P E : }x} ď rǫu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (63) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Semigroup domination For a given p ě 1 we clearly have V P B8pEq ðñ V p P B8pEq and BV 1{ppEq Ă BV pEq Ă BV ppEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We say that a V -positive semigroup Qs,t is p-dominated by a collection of integral operators Qs,t on BV ppEq and we write Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='p Q as soon as for any non negative function f P BV pEq and any s ď t we have Qs,tpfq ď ct´sppq Qs,tpf pq1{p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To simplify notation, when p “ 1 we write Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Q instead of Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='p Q ùñ @s ď t pQs,tpV q{V qp ď ct´sppqp Qs,tpV pq{V p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields for any τ ą 0 and θτ P B0pEq the Lyapunov estimate Qs,s`τpV pq{V p ď θp τ ùñ Qs,s`τpV q{V ď cτ θτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (64) 27 We illustrate the above domination property with the Langevin diffusion flow X paq t pzq “ pXtpzq, Ytpzqq P pRn ˆ Rnq starting at z “ px, yq P pRn ˆ Rnq and defined by the hypo- elliptic diffusion dXtpzq “ Ytpzq{m dt dYtpzq “ papXtpzqq ´ γXtpzq ´ βYtpzq{mq dt ` σ dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (65) In the above display, σ, γ, β, m ą 0 stands for some parameters and a some Lipschitz function on Rn, with n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that when a “ 0, the flow X p0q t pzq resumes to an hypo-elliptic Ornstein-Ulhenbeck on R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a bounded open connected domain D Ă Rn and set @z P E :“ D ˆ Rn T paqpzq :“ inf !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' t ě 0 : X paq t pzq P BE ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with these objects, the sub-Markov semigroup defined for any f P BbpEq and z “ px, yq P E by Qpaq t pfqpzq :“ E ´ fpX paq t pzqq 1T paqpzqąt ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have sup D a ă 8 ùñ @p ą 1 Qpaq !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='p Qp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (66) The proof of the above assertion is a direct consequence of Girsanov’s theorem and H¨older’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For the convenience of the reader, a detailed proof is provided in the appendix on page 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To emphasize the role of the absorption in sub-Markov semigroups we return to the class of models discussed in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We let Pt be the free evolution Markov semigroup associated with the stochastic flow Xtpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that QrUs t p1q P B0pEq and }QrUs t pV q{V } ă 8 for some t ą 0 and V P B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (67) Applying H¨older’s inequality and choosing Vp :“ V 1{p P B8pEq with p ą 1 we readily check the estimate QrUs t pVpq{Vp ď ctppq QrUs t p1q1´1{p P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (68) The next lemma provides several practical conditions to check the uniform estimate (67) for sub-Markov semigroups associated with soft obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the sub-Markov semigroup discussed in (49) when E “ E is a locally compact normed space pE, }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='}q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that the generators L and LU of the flows Pt and QrUs t are defined on some common domain DpLq Ă BpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, for any V P B8pEq X DpLq and parameter a ą 0 we have LUpV q ď ´aV ` c ùñ @t ě 0 }QrUs t pV q{V } ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (69) Whenever U P B8pEq X DpLq, for any a0 ě 0 and a1 P R we have LpUq ď a0 ` a1U ùñ @t ě 0 }QrUs t pUq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (70) 28 The proof of the above lemma follows essentially the same lines of arguments as the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' thus it is provided in the appendix, on page 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever E “ E and the absorption potential function U is bounded, we have P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' QrUs !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, there is no hope to have that QrUs t p1q P B0pEq for some t ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Nevertheless, for any V P B8pEq and any time horizon t ą 0 we have QrUs t pV q{V P B0pEq ðñ PtpV q{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the design of Lyapunov functions V satisfying (1) or equivalently Foster-Lyapunov conditions of the form (60) is equivalent to the problem of finding a Lyapunov function for the Markov semigroup Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever Pt is stable, in the sense that it has a Lyapunov V P B8pEq such that PtpV q{V P B0pEq for some t ą 0, then the domination property QrUs !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' P yields automatically a Lyapunov function for QrUs t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever Pt is not necessarily stable but we have }PtpV q{V } ă 8 for some t ą 0 and V P B8pEq, applying (68) the domination property QrUs !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' P ensures that for any p ą 1 we have Vp :“ V 1{p P B8pEq and QrUs t p1q P B0pEq ùñ QrUs t pVpq{Vp P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Last, but not least, note that the above discussion extends without difficulties to time varying models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Some conjugacy principles For any given V P B8pEq, observe that for any positive function H, H P B0,V pEq ðñ V H :“ V {H P B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Thus, Qt is a V -positive semigroup on BV pEq if and only if the H-conjugate semigroup QH t pfq :“ QtpfHq{H is a V H-positive semigroup on BV HpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, any semigroup Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' QH dominated by QH yields for any s ě 0 and t ą 0 the Lyapunov estimate Qs,s`tpV Hq{V H ď ct QtpV q{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To get one step further, observe that QtpV q{V “ Qtp1q QtpV q{V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, for any H P B0,V pEq and any V -positive semigroup Qt on BV pEq such Qtp1q P B0pEq and ˇˇˇˇˇˇQt ˇˇˇˇˇˇ V ă 8 we have Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' QH ùñ Qs,s`tpV Hq{V H ď ct Qtp1q P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (71) 29 We illustrate the above comparison principles with an elementary example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let E :“ R and W P B8pRq be some non negative function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the stochastic flow XW t pxq of a one-dimensional Langevin diffusion on E with generator Lpfq “ 1 2 e2WB ` e´2WBf ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (72) We associate with a given open connected interval E Ă E, the sub-Markov semigroup Qt on BbpEq defined by Qtpfqpxq :“ EpfpXW t pxqq 1T W BEpxqątq with T W BEpxq :“ inf ␣ t ě 0 : XW t pxq P BE ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (73) Observe that H :“ e´W ùñ U :“ H´11 2 B2H “ 1 2 ` pBWq2 ´ B2W ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (74) When W “ 0 the flow X0 t pxq “ Btpxq coincides with the Brownian flow Btpxq starting at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Thus, by a change of probability we check that Qt “ QH t with Qtpfqpxq :“ E ´ fpBtpxqq 1T 0 BEpxqąt e´ şt 0 UpBspxqq ds¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (75) Whenever E “s0, 1r the semigroup Qt is dominated by the Dirichlet heat kernel on s0, 1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When E “ R, respectively E “s0, 8r, and Upxq ě c ` ς x2{2, for some c ă 8 and ς ą 0, the semigroup Qt is dominated by the harmonic oscillator, respectively the half-harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' All of these dominating semigroups are completely solvable with Qtp1q P B0pEq and known Lyapunov functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 4 Boundary problems Let pE, dq be a locally compact Polish space with a distinguish complete metric d : px, yq P E2 ÞÑ dpx, yq P R`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We recall that these metric spaces are complete σ- compact and locally compact metric spaces, thus they have the Heine-Borel property, that is each closed and bounded subsets in E are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also recall that a subspace E Ă E is Polish if and only if it is the intersection of a countable collection of open subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The distance from x P E to a measurable subset A Ă E is denoted by dpx, Aq :“ inf tdpx, yq : y P Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also denote by BE :“ E ´E˝ the boundary of some domain (open and connected) E Ă E, where E and E˝ stand for the closure and the interior of a subset E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the further development of the article, χ stands for some decreasing positive function χ on s0, 8r such that for any 0 ă α ă 1 we have lim αÑ0 χpαq “ `8 χpαq ă 1{α and χpαq :“ ż α 0 χpuqdu ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 30 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with χ the function VB P CpEq defined by VB : x P E ÞÑ VBpxq :“ χpdpx, BEqq Ps0, 8r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (76) For instance, we can choose χpuq “ 1{u1´ǫ, for some ǫ Ps0, 1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any r ą 0 the r-sub-level sets of VB are given by the closed subsets VBprq :“ tx P E : VBpxq ď ru “ tx P E : dpx, BEq ě χ´1prqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that VB P C8pEq as soon as E is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Bounded domains Let E Ă E :“ Rn be some bounded domain with locally Lipschitz boundary BE, for some n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a semigroup of integral operators Qtpx, dyq “ qtpx, yq dy (77) having for any t ą 0 a bounded density px, yq P E2 ÞÑ qtpx, yq P r0, 8r w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the trace of the Lebesgue measure νpdyq “ dy on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0 we have VB P C8pEq and }QtpVBq} ď ct ż E χpdpx, BEqq dx ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (78) The proof of the above lemma follows from an elementary change of variable formulae, thus it is provided in the appendix, on page 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The estimate (78) clearly applies to the class of sub-Markov semigroups QrUs t de- fined in (49) for any choice of the absorption potential function, as soon as the semi- group QrUs !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Q is dominated by a collection of integral operators Qtpx, dyq having a bounded density qtpx, yq on E2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the Lebesgue measure on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, when the transition semigroup of the free evolution flow Xtpxq in (49) has a density ptpx, yq for any non negative function f on E and any x P E we have QrUs t pfqpxq ď ż qtpx, yq fpyq dy with qtpx, yq :“ ptpx, yq 1Epyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We summarize the above discussion with the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that QrUs !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Q is dominated by a collection of integral operators Qt satisfying (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Then, QrUs t pVBq{VB ď ct{VB P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 31 The choice of the Lyapunov function V is clearly not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, when E “s0, 1r instead of VB we can choose V pxq :“ 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='x ` 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 ´ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For the Dirichlet heat kernel discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 we can also choose V pxq “ 1{ sin pπxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We emphasize that sub-Markov integral operators on the compact interval E “ r0, 1s with a positive continuous density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the Lebesgue measure on E arise when the free evolution process is reflected at both sides of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context the process is not conditioned by any type of non absorption at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the unit function V “ 1 belongs to B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, sub- Markov integral operators with mixed boundary conditions on the left-closed interval E “ r0, 1r, or respectively on the right-closed interval E “s0, 1s arise when the free evolution process is reflected at the Neumann boundary BNE :“ t0u and non absorbed at the Dirichlet boundary BDE “ t1u, or respectively reflected at BNE :“ t1u and non absorbed at BDE “ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, consider a bounded domain Ω Ă Rn with Lipschitz boundary BΩ “ BDΩYBNΩ consisting of two disjoint connected components BDΩ and BNΩ closed in Rn, and set E :“ Ω Y BNΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the function VBpxq :“ χ pdpx, BDEqq belongs to C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, for any bounded density qtpx, yq on E2 we have the uniform estimate ż E qtpx, yq VBpyq dy ď ct ż E VBpyq dy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The above estimate also holds for the function VBpxq “ χ pdpx, BEqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Unbounded domains When the domain E is not bounded the function VB R B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, one natural way to design a Lyapunov function V P B8pEq is to consider an auxiliary function VE P C8pEq with VEpxq ě 1 for any x P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have V :“ VB ` VE P C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To check this claim, observe that the sub-level sets of VB are given by the closed subsets VBprq :“ tVB ď ru “ tx P E : dpx, BEq ě χ´1prqu Ă E and we have the compact inclusion Vprq :“ tV ď ru Ă VEprq X VBprq with VEprq :“ tx P E : VEpxq ď ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the following easily checked proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0 we have }QtpVBq} _ }QtpVEq} ă 8 ùñ QtpV q{V ď ct{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When Qtp1q P B0pEq we also have }QtpVBq} _ }QtpVEq{VE} ă 8 ùñ QtpV q{V ď ct Qtp1q P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 32 The design of a function VE is rather flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, assume that Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' P is dominated by some Markov integral operators Pt on BbpEq such that }PtpVEq{VE} ă 8 for some VE P B8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have }QtpVEq{VE} ă 8 as well as }VE Qtp1q} ă 8 ùñ }QtpVEq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, when Pt satisfies the sub-Gaussian estimates (39) on E “ Rn we can choose VEpxq :“ 1`}x}k, for some k ě 1, as soon as the function Qtp1qpxq ÝÑ}x}Ñ8 0 faster than }x}´k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When the domain E and its boundary BE are both non necessarily bounded, it may happens that Qtp1q P B0pEq but QtpVBq R BbpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we can use the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume there exists some VE P C8pEq with VEpxq ě 1 for any x P E and such that }QtpVBq{VE} _ }QtpVEq{VE} _ }Qtp1qVE} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Then we have QtpV q{V ď ct{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the following decompositions QtpVBq “ Qtp1qVE QtpVBq{VE and QtpVEq “ Qtp1qVE QtpVEq{VE and applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 we have }QtpVBq} _ }QtpVEq} ă 8 and therefore QtpV q{V ď ct{V P B0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The case Qtp1q R B0pEq can also be handle whenever the pair pVB, VEq can be chosen so that @δ ą 0 VB V δ E P C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (79) For instance we can choose for some v ą 0 and ǫ Ps0, 1r the functions VEpxq :“ exp pv}x}q and χpuq :“ 1{u1´ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that dpx, BEq ď }x} ` dp0, BEq and VBpxq ě χp}x} ` p1 _ dp0, BEqqq and for any m ě 0 and δ ą 0 we have VEpxq ě cvpm, δq p1 ` }x}qpm`1q{δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 33 This implies that VEpxqδVBpxq ě c2 p1 ` }x}qm`1 p}x} ` p1 _ dp0, BEqqq1´ǫ ě c p1 ` }x}qm`ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the fact that VEpxq ě 1 for any x P E, this implies that tx P E : VEpxqδVBpxq ď ru Ă tx P E : c p1 ` }x}qm`ǫ ď ru X tx P E : VBpxq ď ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We conclude that V δ E VB has compact level sets and (79) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a couple of functions pVB, VEq satisfying (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume there exists some parameters t ą 0, δt ą 0 and ǫ ě 0 such that QtpVEq{VE ď ct{V δt E and QtpVBq ď ct V ǫ δt E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (80) In this situation, for any p ą 1 ` ǫ we have V :“ V 1´1{p E V 1{p B P C8pEq as well as Qt pV q {V ď ct{pV δtǫp E V 1{p B q P C0pEq with ǫp :“ p1 ´ p1 ` ǫq{pq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that for any p ą 1 ` ǫ we have VBV p´1 E P C8pEq and therefore V :“ V 1{p B V 1´1{p E P C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, for any ǫ ě 0 we have p79q ùñ VBV pδtǫp E P C8pEq and therefore V 1{p B V δtǫp E P C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, using H¨older’s inequality, we have Qt pV q {V ď pQtpVEq{VEq1´1{p pQtpVBq{VBq1{p ď ctp1qpQtpVBq{pV δtpp´1q E VBqq1{p ď ctp2qp1{pV pδtp1´p1`ǫq{pq E VBqq1{p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The design of a function VE satisfying (80) is rather flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, (80) is automatically satisfied when Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' P is dominated by some Markov integral operators Pt on BbpEq such that PtpVEqpxq{VEpxq ď ctp1q{VEpxqδt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 discusses a variety of Lyapunov functions VE satisfying the above condi- tion for Markov diffusion semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' These Lyapunov functions can also be designed 34 using the domination principles presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance, consider the semigroup Qt :“ Qpaq t associated with the Langevin diffusion flow on a cylinder dis- cussed in (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, combining (64) with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='9, for any v ě 0 and t ą 0 there exists some finite constant δt ą 0 such that VEpxq :“ exp pv|x|q ùñ QtpVEq{VE ď ct{V δt E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Next, we illustrate the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (80) when qt are sub-Gaussian densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' in the sense that for any x, y P E we have qtpx, yq ď ct gtpx, yq with gtpx, yq :“ 1 p2πσ2 t qn{2 exp ˆ ´ 1 2σ2 t }y ´ mtpxq}2 ˙ (81) for some parameter σt ą 0 and some non necessarily bounded function mt on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let ϕ be a Lipschitz function on Rn´1 with uniformly bounded gradient and set E :“ tx “ pxiq1ďiďn P Rn : xn ą ϕpx´nqu with x´n :“ pxiq1ďiăn P Rn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Then the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' condition in (80) is met with ǫ “ 0 for any positive semigroup sat- isfying (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The same property holds when the boundary BE can be decomposed as a finite union of graphs of differentiable functions on Rn´1 with uniformly bounded gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We choose α ą 0 sufficiently small so that for any x P DαpEq :“ tx P E : dpx, BEq ď αu there exists a projection x P BE with dpx, BEq “ }x ´ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let C̟pxq be an interior cone with a given base vertex x “ px´n, ϕpx´nqq P BE and a given half-opening angle ̟ around the axis Apxq :“ tpx´n, xnq : xn ě ϕpx´nqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any x P Apxq there exists a projection px P BC̟pxq on the boundary BC̟pxq with dpx, BC̟pxqq “ dpx, pxq “ cos ´π 2 ´ ̟ ¯ pxn ´ ϕpx´nqq ď dpx, xq On the other hand, for any y P BE we have z :“ py´n, ϕpx´nqq ùñ 0 ď π 2 ´ ̟ ď y yxz and tanpy yxzq “ |ϕpx´nq ´ ϕpy´nq| }x´n ´ y´n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the estimate cos ´π 2 ´ ̟ ¯ ě cos ´ y yxz ¯ “ 1 b 1 ` tan2py yxzq 35 from which we conclude that 0 ď xn ´ ϕpx´nq ď κ dpx, xq with κ :“ a 1 ` }∇ϕ}2 and }∇ϕ} :“ sup yPRn´1 }∇ϕpyq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that ż DαpEq χ pdpy, BEqqqtpx, yq dy ď ctp1q ż DαpEq χ ppyn ´ ϕpy´nqq {κqq exp ˆ ´ 1 2σ2 t ppyn ´ ϕpy´nqq ` pϕpy´nq ´ pmtpxqqnqq2 ˙ ˆ exp ˆ ´ 1 2σ2 t }y´n ´ pmtpxqq´n}2 ˙ dyndy´n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the change of variables z :“ pyn ´ ϕpy´nqq {κ ùñ dyn “ κ dz we find that ż DαpEq χ pdpy, BEqqqtpx, yq dy ď κ ctp1q χpαq ż Rn´1 exp ˆ ´ 1 2σ2 t }y´n ´ pmtpxqq´n}2 ˙ dy´n ď ctp2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, for any α ą 0 we have ż E´DαpEq χ pdpy, BEqq qtpx, yq dy ď χ pαq }Qtp1q} ď ctp3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Smooth boundaries Next, we illustrate the Lyapunov conditions on VB in the context of absolutely con- tinuous sub-Markov semigroup of the form (77) with a bounded density qtpx, yq on a non necessarily bounded domain E Ă Rn with smooth non necessarily bounded C2-boundary with uniformly bounded interior curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 36 We assume that there exists α ą 0 sufficiently small so that every point of the α-offset of BE (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' α-tubular neighborhood) defined by TubαpBEq :“ tx P Rn : dpx, BEq ď αu lies on some normal ray passing through a point on BE and no two normal rays passing through different points of BE intersect in TubαpBEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We let Npzq be the unit normal vector at z P TubαpBEq pointing inward E, and let DrpEq the closed subset defined for any r ď α by DrpEq :“ tx P E : dpx, BEq ď ru and D´rpEq :“ tx P Rn ´ E : dpx, BEq ď ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the inverse of the normal coordinate map F : pz, rq P BE ˆ r´α, αs ÞÑ Fpz, rq “ z ` r Npzq P TubαpBEq (82) is given for any x P TubαpBEq by F ´1pxq “ pprojBEpxq, dαpx, BEqq where projBEpxq stands for the projection of x P TubαpBEq onto BE and dαpx, BEq stands for the signed distance function dαpx, BEq :“ dpx, BEq 1DαpEqpxq ´ dpx, BEq 1D´αpEqpxq P r´α, αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, the inward normal Npxq at any x on the C2 boundary BE is given by ∇dαpx, BEq “ Npxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The Hessian of the signed distance function on the boundary BE gives the Weingarten map Wpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' With this notation at hand, we have ż DαpEq fpdpy, BEqq qtpx, yq dy “ ż α 0 fprq qB t px, rq dr with the level-set density function qB t px, rq :“ ż BEr qtpx, yq σB,rpdyq (83) “ ż BE qt px, z ` rNpzqq |det pI ´ r Wpzqq| σBpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, σB,rpdzq stands for the Riemannian volume measure on the r- extended boundary BEr :“ tx P E : dpx, BEq “ ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 37 Moreover, since E has uniformly bounded interior curvature, for any r ď α we have κBpαq :“ sup |det pI ´ r Wpzqq| ă 8 and κ´ B pαq :“ sup |det pI ` r Wpyqq| ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, the supremum is taken over all z P BE, y P BEr, and r ď α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Several examples of hypersurface boundaries satisfying the above conditions are discussed in Section 7 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' for instance Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We denote by qB t ě qB t the function defined as qB t by replacing qt by qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the fact that QtpVBqpxq ď χpαq ` ż α 0 χprq qB tpx, rq dr we readily check the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any t ą 0 we have sup 0ďrďα sup xPE qB t px, rq ă 8 ùñ QtpVBq ď χpαq Qtp1q ` ctpαq χpαq sup 0ďrďα sup xPE qB t px, rq ă 8 ùñ QtpVBq ď χpαq ` ctpαq χpαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (84) When the boundary BE is bounded, for any t ą 0 we have the estimate }QtpVBq} ď ctpαq ˆ χpαq ` χpαq sup 0ďrďα σB,r pBErq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (85) We end this section with some practical tools to estimate the level-set density functions discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Most of our estimates are based on the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a couple of non negative functions f, g on Rn and some pa- rameter α ą 0 such that sup }u}ďα fpz ` uq ď ιpαq gpzq for some ιpαq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have the uniform estimate sup 0ďrďα ż BEr fpzq σB,rpdzq ď ιpαq κBpαq ż BE g pzq σBpdzq as well as the co-area estimate ż BE fpzq σBpdzq ď 1 α ιpαq κ´ B pαq ż DαpEq gpzq dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The proof of the above lemma is provided in the appendix, on page 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that the level-set density function defined in (83) can be estimated for any 0 ď r ď α by the formula qB t px, rq ď κBpαq ż BE qt px, z ` rNpzqq σBpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 38 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that qtpx, yq ď ̟t gtpx, yq is dominated by some proba- bility density y ÞÑ gtpx, yq on Rn for some t ą 0 and some parameter ̟t ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, we have sup }u}ďα gtpx, y ` uq ď ιtpαq gα,tpx, yq (86) for some probability density y ÞÑ gα,tpx, yq and some ιtpαq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have the uniform density estimates sup 0ďrďα sup xPE qB t px, rq ď ̟t ιtpαqκ´ B pαqκBpαq{α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (87) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By (86) for any 0 ď r ď α we have qB t px, rq ď ̟t κBpαq ż BE gt px, z ` rNpzqq σBpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, we have ż DαpEq gt,αpx, yq dy ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The estimate (87) is now a direct consequence of the co-area estimate stated in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We illustrate the above condition when qt are the sub-Gaussian densities discussed in (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, using the fact that 2a1b ď 1 ǫ}a}2 ` ǫ}b}2 for any 0 ă ǫ ă 1 and }u} ď α we check that ´ 1 2σ2 t }py ` uq ´ mtpxq}2 ď ´p1 ´ ǫq 2σ2 t }y ´ mtpxq}2 ` 1 2σ2 t ˆ1 ǫ ´ 1 ˙ α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, condition (86) is met with the gaussian density gα,tpx, yq :“ e ´ 1 2σtpǫq2 }y´mtpxq}2 p2πσtpǫq2qn{2 with ιtpαq :“ ct e α2{ǫ 2σtpǫq2 and σtpǫq2 :“ σ2 t {p1 ´ ǫq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 5 Riccati type processes 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Positive diffusions Consider the Riccati type diffusion on E “s0, `8r defined for any x P E by dXtpxq “ ` a0 ` a1 Xtpxq ´ b Xtpxq2˘ dt`σ1pXtpxqq dB1 t `σ2pXtpxqq dB2 t , X0pxq “ x 39 for some Brownian motion pB1 t , B2 t q on R2, the diffusion functions σ1pxq :“ ς1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='x σ2pxq :“ ς2 x and the parameters a1 P R a0 ą ς2 1 b ą 0 and ς1, ς2 ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying Itˆo’s formula, we readily check that BtEpXtpxqq ď Ricc pEpXtpxqqq and BtEp1{Xtpxqq ď Ricc´ pEp1{Xtpxqqq with the Riccati drift functions defined by Riccpzq :“ a0 ` a1z ´ bz2 and Ricc´pzq :“ a´ 0 ` a´ 1 z ´ b´z2 (88) with the parameters a´ 0 :“ b a´ 1 :“ pς2 2 ´ a1q and b´ :“ a0 ´ ς2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the Lyapunov function V P B8pEq defined by V pxq :“ x ` 1{x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By well known properties of Riccati flows, for any t ą 0 we have }PtpV q} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a more thorough discussion on this class of one-dimensional Riccati diffusions, we refer to the article [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Matrix valued diffusions Let E and E be the space of pn ˆ nq-positive semi-definite and definite matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Also let λ1pxq ě .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' ě λnpxq denote the ordered eigenvalues of x P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Wt denotes an pn ˆ nq-matrix with independent Brownian entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Also let A be an pn ˆ nq-matrix with real entries and let R, S P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with these objects the E-valued diffusion dXt “ pAXt ` XtA1 ` R ´ XtSXtq dt ` ǫ 2 ” X1{2 t dWt R1{2 ` R1{2 dW1 t X1{2 t ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever ǫ ď 2{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' n ` 1, the diffusion Xt has a unique strong solution that never hits the boundary BE “ E ´ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, the transition semigroup Pt of Xt is strongly Feller and admits a smooth density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the Lebesgue measure on E, thus it is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Furthermore, when ǫ2p1 ` nq{2 ď λnpRq{λ1pRq then the function V pxq “ Trpxq ` Trpx´1q is a Lyapunov function with compact level subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a detailed proof of the above assertion for more general classes of Riccati matrix valued diffusions we refer to [5] (see for instance the stability Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Logistic birth and death process Let Xtpxq be the stochastic flow on E :“ N ´ t0u with generator L defined for any f P BbpEq and x ě 2 by Lpfqpxq “ Jpx, x ´ 1q pfpx ´ 1q ´ fpxqq ` Jpx, x ` 1q pfpx ` 1q ´ fpxqq and for x “ 1 by Lpfqp1q “ Jp1, 2qpfp2q ´ fp1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, the birth and death rates in the above display are given by Jpx, x ` 1q :“ λb x ` υb and Jpx, x ´ 1q :“ λd x ` λl xpx ´ 1q ` υd (89) for some non negative parameters λd, λb, υb, υd ě 0 and λl ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the identity function V : x P E ÞÑ V pxq “ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any x ě 2 we have LpV qpxq “ Jpx, x ` 1q ´ Jpx, x ´ 1q “ pF ˝ V q pxq with the concave function z P R` ÞÑ Fpzq :“ pυb ´ υdq ` pλb ` λl ´ λdq z ´ λl z2 P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (90) Observe that LpV qp1q ´ FpV p1qq “ Jp1, 2q ´ Fp1q “ Jp1, 0q “ υd ` λd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the estimate PtpLpV qqpxq “ Ptp1r2,8r LpV qqpxq ` Ptp1t1u LpV qqpxq “ PtppF ˝ V qqpxq ` Ptp1t1uqpxq pLpV qp1q ´ FpV p1qqq ď FpPtpV qqpxq ` Jp1, 0q from which we check that BtPtpV qpxq ď Ricc pPtpV qpxqq with the Riccati drift function defined in (88) with the parameters a0 :“ υb ` λd a1 :“ λb ` λl ´ λd and b :“ λl ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By well known properties of Riccati flows, for any t ą 0 we conclude that }PtpV q} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 Multivariate birth and death processes We denote by e :“ tei, 1 ď i ď nu the collection of column vector ei on t0, 1un with entries eipjq “ 1i“j and with a slight abuse of notation we denote by 0 the null state in Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Xtpxq be a stochastic flow on E “ Nn ´ t0u with generator L defined by Lpfqpxq :“ ÿ yPE Jpx, yq pfpyq ´ fpxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (91) Let λ, µ, υ, ς be some column vectors and let C, D some pd ˆ dq-matrices with real entries such that for any 1 ď i ď d and any x P E we have Jpx, x ` eiq :“ υi ` xi pλi ` pCxqiq ě 0 and Jpx, x ´ eiq :“ ςi ` xi pµi ` pDxqiq ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also set Jpx, yq “ 0 as soon as |x ´ y| ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We further assume that |υ| ě |ς| B :“ pD ´ Cq ě b I ą 0 for some b ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' and we set a0 :“ |υ| ´ |ς| ě 0 a1 :“ _1ďiďnpλi ´ µiq and for any x P Nn }x} :“ ˜ ÿ 1ďiďn x2 i ¸1{2 ě |x| :“ ÿ 1ďiďn xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the Lyapunov function x P E ÞÑ V pxq “ |x| P N`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that V is locally bounded with finite level sets and for any x P E ´ e we have LpV qpxq “ ÿ 1ďiďn ppυi ` xi pλi ` pCxqiqq ´ pςi ` xi pµi ` pDxqiqqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, we have the formula LpV qpxq “ a0 ` pλ ´ µq1x ´ x1Bx ď a0 ` a1|x| ´ b}x}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (92) On the other hand, for any y “ ej we have LpV qpyq “ ÿ 1ďiďn Jpy, y ` eiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 42 This implies that PtpLpV qq “ Ptp1E´e LpV qq ` Ptp1e LpV qq “ a0 ` a1 PtpV q ´ b PtpV 2q ` ÿ 1ďjďn Ptp1ejq ` LpV qpejq ´ ` a0 ` pλ ´ µq1ej ´ e1 jBej ˘˘ from which we readily check that BtEpV pXtpxqqq ď a` 0 ` a1EpV pXtpxqqq ´ b pEpV pXtpxqqqq2 with a` 0 :“ a0 ` ÿ 1ďjďd ˜ ÿ 1ďiďd Jpej, ej ` eiq ´ ` |υ| ´ |ς| ` pλ ´ µq1ej ´ e1 jpD ´ Cqej ˘ ¸ “ a0 ` ÿ 1ďjďd ` |ς| ` µ1ej ` e1 jDej ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We conclude that }PtpV q} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The semigroup analysis discussed above can be extended without difficulties to more general process on countable spaces models satisfying condition (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The extension to time varying models can also be handle using a more refined analysis on time varying Riccati equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also mention, that the case |υ| “ 0 “ |ς| coincides with the competitive and multivariate Lotka-Volterra birth and death process discussed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 6 Some conditional diffusions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Coupled harmonic oscillators Consider the Rn-valued diffusion (33) with pbpxq, σpxqq “ pAx, Σq, for some non neces- sarily stable drift matrix A and some diffusion matrix Σ with appropriate dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We associate with a given semi-definite positive pn ˆ nq matrix S ě 0 the potential function Upxq :“ 1 2 x1Sx and we set R “ ΣΣ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (93) We assume that the pairs pA, R1{2q and pA1, S1{2q are both controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Qt “ QrUs t be the sub-Markov semigroup defined in (49) on the Euclidean space E “ E “ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' As shown in [13], the leading-triple pρ, h, η8q discussed in (24) is given by ρ “ ´TrpRq8q{2 “ ´Trpp8Sq{2 hpxq “ exp p´x1q8x{2q and η8pdxq “ exp p´x1p´1 8 x{2q a detp2πp8q dx, (94) 43 with the positive fixed points p8 and q8 of the dual algebraic Riccati matrix equation Ap8 ` p8A1 ` R ´ p8Sp8 “ 0 and A1q8 ` q8A ` S ´ q8Rq8 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the h-process, denoted pXh t pxqqtě0 and defined by the stochastic dif- ferential equation dXh t pxq “ AhXh t pxq dt ` Σ dBt with Ah :“ A ´ R q8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (95) Our controllability conditions ensures that Ah is a stable matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that Xh t pxq is an Rn-valued Gaussian random variable with mean mh t pxq and covariance matrix ph t P Rnˆn given for any t ą 0 by mh t pxq “ exp ` Aht ˘ x and ph t “ ż t 0 exp ` Ahs ˘ R exp ` pAhq1s ˘ ds ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the explicit formula P h t px, dyq “ 1 a detp2πph t q exp ˆ ´1 2py ´ mh t pxqq1pph t q´1py ´ mh t pxqq ˙ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Moreover the invariant measure ηh 8 “ ηh 8Ph t is unique and given by ηh 8pdxq “ 1 a detp2πph 8q exp ˆ ´1 2y1pph 8q´1y ˙ dy with the limiting covariance matrix ph 8 :“ ż 8 0 exp ` Ahs ˘ Σ2 exp ` pAhq1s ˘ ds “ pp´1 8 ` q8q´1 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any time horizon t ě 0 and any measurable function F on the set Cpr0, ts, Rnq of continuous paths from r0, ts into Rn we have the path space exponential change of measure Feynman-Kac formula E ˆ FpXtpxqq exp ˆż t 0 UspXspxqq ds ˙˙ “ eρt hpxq E ` FpXh t pxqq{hpXh t pxqq ˘ with the historical processes Xtpxq :“ pXspxqq0ďsďt, Xh t pxq :“ pXh s pxqq0ďsďt and UspXspxqq :“ UspXspxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the conjugate formulae Qtpfq “ eρt h P h t pf{hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We denote by pmtpxq, ptq P pRnˆRnˆnq the mean and covariance parameters satisfying the linear evolution and the Riccati matrix differential equations $ & % Btmtpxq “ pA ´ ptSq mtpxq Btpt “ Apt ` ptA1 ` Σ2 ´ ptSpt with pm0pxq, p0q “ px, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (96) The next proposition provides an explicit description of these semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 44 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 ([13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any time horizon t ą 0 we have pt ą 0 and Qtpx, dyq “ 1 a detp2πptq exp ˆ ´1 2py ´ mtpxqq1p´1 t py ´ mtpxqq ˙ dy (97) as well as ´2 log Qtp1qpxq “ x1 ˆż t 0 F 1 sSFs ds ˙ x ` ż t 0 TrpSpsq ds with the fundamental matrix semigroup Ft starting at F0 “ I given by BtFt “ pA ´ ptSq Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that the normalized Markov operator Qt satisfies (39) and (40) with the parameters ct “ 1 a detp2πptq , σ2 t “ λmaxpptq and ǫτ “ |eτpA´p8Sq| ÝÑ 0 as τ Ñ 8 (98) for some matrix norm |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' assertion is a direct consequence of the Floquet representation theorem presented in [3] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1) and the fact that pA´ p8Sq is a stable matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='9 for any v ě 0 and t ą 0 there also exists some finite constant δt ą 0 such that V pxq :“ exp pv|x|q ùñ QtpV q{V ď ct{V δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1, for any k ě 0 and t ě 0 it is also readily checked that V pxq :“ p1 ` }x}qk ùñ }QtpV q{V } ă 8 and }QtpV q} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 Half-harmonic linear diffusions For one dimensional models, the coupled harmonic oscillator discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 resumes to one dimensional linear diffusion dXtpxq “ a Xtpxq dt ` dBt and the potential Upxq “ ςx2{2 (99) for some parameters ς ą 0 and a P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We set β :“ a ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' a2 ` ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the leading pair pρ, hq “ pρ1, ϕ1q is given by ρ “ ´β{2 and hpxq “ ppβ ´ aq{πq1{4 exp ` ´βx2{2 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (100) The quasi-invariant measure is therefore given by η8pdxq “ c ς 2πβ exp ` ´ςx2{p2βq ˘ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 45 Therefore, the h-process resumes to the Ornstein-Uhlenbeck diffusion dXh t pxq “ ´b Xh t pxq dt ` dBt (101) with the invariant measure ηh 8pdxq :“ c b π exp ` ´b x2˘ dx with b :“ pβ ´ aq “ a a2 ` ς ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that any Ornstein-Uhlenbeck process can be seen as the h-process associated with a non absorbed (possibly transient) linear diffusion evolving in some quadratic potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 is also satisfied with the mean and variance pa- rameters $ & % Btmtpxq “ pa ´ ptςq mtpxq Btpt “ 2apt ` 1 ´ ςp2 t with pm0pxq, p0q “ px, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (102) We also have ´2 ς log Qtp1qpxq “ pt ` χt x2 (103) with χt :“ ż t 0 exp ˆ ´2 ż s 0 pa ´ puςqdu ˙ ds and pt :“ ż t 0 psds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The half-harmonic semigroup associated with the flow Xtpxq is defined for any x P E :“s0, 8r and f P BbpEq by the formulae Qtpfqpxq :“ E ˆ fpXtpxqq 1Tpxqąt exp " ´ ż t 0 UpXspxqq ds ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (104) In the above display, Tpxq stands for the hitting time of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In terms of the h-process of the flow in the harmonic potential (101) we also have the conjugate formula Qtpfqpxq “ etρ e´βx2{2 E ´ fpYtpxqq eβYtpxq2{2 1T Y pxqąt ¯ (105) with the parameters pρ, βq defined in (100) and the Ornstein-Uhlenbeck diffusion flow defined by dYtpxq “ ´b Ytpxq dt ` dBt with b :“ pβ ´ aq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, T Y pxq stands for the hitting time of the origin by the flow Ytpxq starting at x ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Arguing as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 we check that Qtpx, dyq “ sinh py mtpxqq exp ` ´ ς 2 pχt x2 ` ptq ˘ ˆ c 2 πpt exp ˆ ´y2 ` mtpxq2 2pt ˙ 1s0,8rpyq dy 46 with the parameters pmtpxq, ptq and pχt, ptq defined in (102) and (103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Arguing as in (56), choosing the Lyapunov function V pxq “ xn ` 1{x, for some n ě 1, we readily check that V P C8pEq and QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (106) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Linear diffusions in some domains Consider the one-dimensional stochastic flow Ytpxq of an Ornstein-Uhlenbeck dYtpxq “ ´b Ytpxq dt ` dBt for some b ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, Bt is a one-dimensional Brownian motion starting at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For a given x P E :“s0, 8r, we let T Y pxq be the hitting time of the origin by the flow Ytpxq starting at x ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the semigroup QY t pfqpxq :“ EpfpYtpxqq 1T Y pxqątq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Choosing pa, ς, β, ρq “ p0, b2, b, ´b{2q in (100), formula (105) takes the form QY t pfqpxq “ e´ρt Hpxq´1QtpfHqpxq with Hpxq “ exp ` ´bx2{2 ˘ with the semigroup Qt defined in (104) with Upxq “ b2x{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any given n ě 1 we have V pxq :“ xn ` 1{x ùñ V P C8pEq and V H :“ V {H P C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using (106) we conclude that V H P C8pEq and QY t pV Hq{V H “ e´ρt QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The long time behavior of the positive semigroup QY t is also studied in [46], and more recently in [55] in terms of the tangent of the h-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, consider the Rn-valued diffusion flow Xtpxq defined in (33) with pbpxq, σpxq “ pAx, Σq, for some matrices pA, Σq with appropriate dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Assume that R :“ ΣΣ1 is positive semi-definite and the pair of matrices pA, R1{2q are control- lable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the Markov semigroup Pt of the stochastic flow Xtpxq satisfies the sub-Gaussian estimate (81) for some parameters pσt, mtpxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a domain E Ă Rn with C2-boundary with uniformly bounded interior curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any given x P E, let Qt be the sub-Markov semigroup Qtpfqpxq :“ EpfpXtpxqq 1Tpxqątq with Tpxq :“ inf tt ě 0 : Xtpxq P BEu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (107) We clearly have QtpVBq ď PtpVBq, with the function VB defined in (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When E is non necessarily bounded but its boundary BE is bounded we known from (85) that }QtpVBq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For non necessarily bounded boundaries the sub-gaussian property (81) ensures that }QtpVBq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 47 When E is bounded, applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 (see also Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3) we have VB P C8pEq and QtpVBq{VB ď ct{VB P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For unbounded domains we need to ensure that A is stable so that (44) is satisfied for some norm |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='| on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='10 for any t ą 0 there exists some δt ą 0 such that VEpxq :“ exp pv|x|q ùñ QtpVEq{VE ď ct{V δt E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='6 with ǫ “ 0, for any p ą 1 we conclude that Vp :“ V 1´1{p E V 1{p B P C8pEq and Qt pVpq {Vp ď ct Θp,t (108) with the function Θp,t :“ 1{pV δtp1´1{pq E V 1{p B q P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 Langevin diffusions in some domains Consider the semigroup Qt of the one-dimensional Langevin diffusion defined in (73) with E “s0, 8r and a quadratic confinement potential Wpxq “ x2{2 ùñ Hpxq :“ e´W pxq “ e´x2{2 and U :“ 1 2 ` x2 ` 1 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this case, the semigroup Qt defined in (75) coincides with the semigroup of the half-harmonic oscillator discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By (56) for any n ě 1 we have V pxq :“ xn ` 1{x ùñ QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that V Hpxq :“ V pxq{Hpxq “ xn ex2{2 ` ex2{2 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (109) Using (75) we conclude that V H P C8pEq and QtpV Hq{V H “ QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, consider the case E “s0, 8r with at least a quadratic confinement potential U, in the sense that Upxq “ 1 2 ` pBWq2 ´ B2W ˘ pxq ě U2pxq :“ c ` ς x2{2 for some ς ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, Qt !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' QrU2s is dominated by the semigroup QrU2s of the half-harmonic oscillator discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Arguing as in (109) we have H :“ e´W V H :“ V {H P C8pEq and QtpV Hq{V H ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 48 For instance, whenever the confinement potential W is chosen so that Wpxq ě ǫ0 log x ` W1pxq for some 0 ď ǫ0 ă 1 and some function W1 ě 1 such that W1pxq ÝÑxÑ8 8 we have H “ e´W ùñ V Hpxq :“ V pxq{Hpxq “ xn eW pxq ` eW pxq x ě xn eW pxq ` eW1pxq x1´ǫ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using (75) we conclude that V H P C8pEq and QtpV Hq{V H “ QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We illustrate the above result, with the logistic diffusion discussed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the generalized Feller diffusion dYtpxq :“ ` 2a Ytpxq ´ p8b{σ2q Ytpxq2˘ dt ` σ a Ytpxq dBt starting at x P E :“s0, 8r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, Bt is a one dimensional Brownian motion starting at the origin and a, b, σ ą 0 some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that Xtpxq :“ p2{σq a Ytpxq ùñ dXtpxq “ ´BW pXtpxqq dt ` dBt with the potential function BWpxq “ 1 2x ´ a x ` b x3 with a, b ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Thus, choosing Wpxq “ 1 2 log x ` bx4 4 ´ ax2 2 we readily check that V Hpxq :“ eW pxqpxn ` 1{xq “ pxn´1{2 ` 1{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='xq eb x4 4 ´a x2 2 ùñ V H P C8pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, consider the Langevin diffusion flow Xtpzq “ pXtpzq, Ytpzqq P pRn ˆ Rnq starting at z “ px, yq P pRn ˆ Rnq and defined by the hypo-elliptic diffusion (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We further assume that supD a ă 8 for some bounded open connected domain D Ă Rn with C2-boundary, and for any z P E :“ D ˆ Rn and f P BbpEq we set Qtpfqpzq :“ E ` fpXtpzqq 1T paqpzqąt ˘ with Tpzq :“ inf tt ě 0 : Xtpzq P BDu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We know from (66) that for any q ą 1 we have Q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='q Q is q-dominated by the sub- Markov semigroup Qt associated with the Ornstein-Uhlenbeck diffusion on E defined in (107), with the matrices pA, Σq defined in (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In terms of the functions pVp, Θp,tq defined in (108), combining (64) with (108) for any p, q ą 1 we conclude that QtpVp,qq{Vp,q ď ctpp, qq Θp,q,t with the collection of Lyapunov functions Vp,q :“ V 1{q p P C8pEq and the function Θp,q,t :“ Θ1{q p,t P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 49 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5 Coupled oscillators in some domains Consider the Rn-valued diffusion Xtpxq and the quadratic potential function U dis- cussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1, for some n ě 2 and set E :“s0, 8rˆRn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let Qt be the semigroup defined for any f P BbpEq and x P E by the formulae Qtpfqpxq :“ E ˆ fpXtpxqq 1Tpxqąt exp ˆ ´ ż t 0 UpXspxqqds ˙˙ (110) with the quadratic function U in (93) and the exit time Tpxq given by Tpxq :“ inf tt ě 0 : Xtpxq P BEu with BE “ t0u ˆ Rn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In terms of the h-process Ytpxq :“ Xh t pxq associated with the leading pair pρ, hq defined in (95) we also have the conjugate formula Qtpfq “ eρt h QY t pf{hq with QY t pfqpxq :“ E ` fpYtpxqq 1T Y pxqąt ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, T Y pxq stands for the boundary hitting time T Y pxq :“ inf tt ě 0 : Ytpxq P BEu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' When n “ 2, the linear diffusion Xtpxq associated to the matrices A1,2 “ A2,1 “ A2,2 “ 0 and A1,2 “ 1 and Σ1,1 “ Σ1,2 “ Σ2,1 “ 0 and Σ2,2 “ 1 coincides with the integrated Wiener process model discussed in [45, 35, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In a seminal article [48], McKean obtained the joint distribution of the pair pTpxq, X2 Tpxqq in the absence of soft absorption, that is when U “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To the best of our knowledge, an explicit descrip- tion of the distribution of this pair and the corresponding sub-Markov semigroup is unknown in more general situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that for any x P E and any non negative function f P BbpRnq we have Qtpfqpxq ď Qtpfqpxq :“ eρt hpxq E pfpYtpxqq{hpYtpxqqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The semigroup Qt defined above coincides with the semigroup of the coupled harmonic oscillator discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We know from (98) that Qt satisfies the sub- Gaussian estimates (39) with ct “ 1 a detp2πptq and σ2 t “ λmaxpptq with the solution pt of the Riccati-matrix equation (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 for any k ě 1 we have VEpxq :“ 1 ` }x}k ùñ }QtpVEq{VE} ă 8 ùñ QtpVEq ď ct Qtp1qVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Recalling that Qtp1qpxq tends to 0 exponentially fast as }x} Ñ 8, this implies that @t ą 0 }QtpVEq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 50 On the other hand for any y “ py1, y´1q P E “ ps0, 8rˆRn´1q, the distance to the boundary is given by dpy, BEq “ y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In terms of the function VB defined in (76) his implies that QtpVBqpxq ď ż Qtpx, dyq 1s0,1rpy1q χpy1q ` χp1q Qtp1qpxq from which we check that }QtpVBq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 we conclude that V :“ VB ` VE P C8pEq and QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The same analysis applies by replacing the half line E1 by the unit interval E1 :“s0, 1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the boundary is given by the two infinite potential walls BE “ pt0u ˆ Rn´1q Y pt1u ˆ Rn´1q and dpx, BEq “ x1 ^ p1 ´ x1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, consider a domain E Ă Rn with C2-boundary with uniformly bounded interior curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the sub-Gaussian property (81) ensures that }QtpVBq} ă 8 and therefore }QtpVBq} ď }QtpVBq} “ }Qtp1qQtpVBq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4, we conclude that V :“ VB ` VE P C8pEq and and QtpV q{V ď ct{V P C0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 7 Some hypersurface boundaries 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 Defining functions and charts Consider a smooth function y P Rn´1 ÞÑ ϕpyq P R with non empty and connected level set, for some n ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a domain E in Rn with a smooth boundary BE “ ϕ´1pt0uq defined as the null level set of the function x “ pxiq1ďiďn P Rn ÞÑ ϕpxq :“ ϕpx´nq ´ xn with x´n :“ pxiq1ďiăn P Rn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the column vectors ∇ϕpx´nq :“ pBxiϕpx´nqq1ďiăn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the unit normal vector Npxq at x P BE is given by the column vectors Npxq “ ∇ϕpxq }∇ϕpxq} “ 1 a 1 ` }∇ϕpx´nq}2 ˆ ∇ϕpx´nq ´1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that the vector Npxq is the outward-pointing normal direction to E as soon as E “ ϕ´1 ps ´ 8, 0rq and the inward-pointing normal direction to E when E “ ϕ´1 ps0, `8rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 51 Consider the column vectors ei :“ p1ipjqq1ďiăn, with 1 ď i ă n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the pn ´ 1q tangential column vectors Tipxq at x P BE are given for any 1 ď i ă n by the column vectors Tipxq :“ ˆ ei Bxiϕpx´nq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The inner product gpxq on the tangent space TxpBEq (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the first fundamental form on BE) is given by the Gramian matrix gpxq “ pTipxq1Tjpxqq1ďi,jăn “ TpxqTpxq1 with Tpxq1 :“ pT1pxq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Tn´1pxqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the matrix formula gpxq “ ` I, ∇ϕpx´nq ˘ ˆ I ∇ϕpx´nq1 ˙ “ I ` ∇ϕpx´nq∇ϕpx´nq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the projection projTxpBEq on the tangent space TxpBEq is defined for any column vector V “ pV iq1ďiďn P Rn by projTxpBEqpV q :“ pT1pxq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Tn´1pxqq gpxq´1 ¨ ˚ ˝ T1pxq1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Tn´1pxq1 ˛ ‹‚ ¨ ˚ ˝ V 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' V n ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In matrix notation, the projection of m column vectors Vi P Rn, with i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , mu and any m ě 1 takes the synthetic form projTxpBEqpV1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Vmq “ ` Tpxq1gpxq´1Tpxq ˘ pV1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Vmq “ ` projTxpBEqpV1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , projTxpBEqpVmq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Equivalently, if gpxqi,j denotes the pi, jq-entry of the inverse matrix gpxq´1, the pro- jection of a column vector V P Rn onto TxpBEq is defined by projTxpBEqpV q “ ÿ 1ďi,jăn gpxqi,j pTjpxq1V q Tipxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 The shape matrix Consider the Monge parametrization ψ : θ “ pθiq1ďiăn P S :“ Rn´1 ÞÑ ψpθq “ ˆ θ ϕpθq ˙ P BE Ă Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (111) In this chart, the tangent vectors and the normal unit vector at x “ ψpθq are given for any 1 ď i ă n by T ψ i pθq :“ Bθiψpθq “ Ti pψpθqq P TxpBEq and Nψpθq :“ Npψpθqq P TK x pBEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 52 For any 1 ď i, j ă n we have pBθiψpθqq1 Nψpθq “ 0 ùñ Ωpψpθqqi,j :“ ` Bθi,θjψpθq ˘1 Nψpθq “ ´ pBθiψpθqq1 BθjNψpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that for x “ ψpθq, BθiNψpθq “ ÿ 1ďkďn pBxkNq pxq Bθiψkpθq “ p∇Npxqq1 Bθiψpθq from which we check that for any 1 ď i, j ă n the coefficients of the second funda- mental form can be computed as follows: Ωpxqi,j “ ´ pBθiψpθqq1 p∇Npxqq1 Bθiψpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We set ` BNψpθq ˘1 :“ ` Bθ1Nψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1Nψpθq ˘ P ` TψpθqpBEq ˘n´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, for any x “ ψpθq we have the matrix formulation Ωpxq :“ ´Bψpθq ` BNψpθq ˘1 “ ´` Bθi,θjψpθq ˘1 Npxq ¯ 1ďi,jăn “ ´ ∇2ϕpθq a 1 ` }∇ϕpθq}2 with ∇2ϕpθq :“ ` Bθi,θjϕpθq ˘ 1ďi,jăn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also readily check the matrix formulation of the Weingarten’s equations ` BNψpθq ˘1 “ ` pBψpθqq1 gpψpθqq´1˘ pBψpθqq ` BNψpθq ˘1 “ ´ pBψpθqq1 W pxq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, Wpxq stands for the shape matrix (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the Weingarten map or the mixed second fundamental form) defined by Wpxq :“ gpxq´1Ωpxq “ ´ 1 a 1 ` }∇ϕpx´nq}2 pI ` ∇ϕpx´nq∇ϕpx´nq1q´1 ∇2ϕpx´nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We summarize the above discussion in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any 1 ď i ă n we have the Weingarten’s equations BθiNψpθq “ ´ ÿ 1ďkăn W pψpθqqk,i Bθkψpθq P TψpθqpBEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For n “ 2 we have x P R ÞÑ ϕpxq “ ϕpxq ´ x, so that the boundary BE “ ϕ´1pt0uq coincides with the graph of the function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the metric and Weingarten map at x P BE “ tx “ px1, x2q P R2 : x2 “ ϕpx1qu take the form gpxq “ 1 ` }Bϕpx1q}2 and Wpxq “ ´ 1 p1 ` }Bϕpx1q}2q3{2 B2ϕpx1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 53 Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For n “ 3, the boundary BE is given by the surface in R3 defined BE :“ tx “ pxiq1ďiď3 P R3 : x3 “ ϕpx1, x2qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The Monge parametrization is given by ψ : θ “ pθ1, θ2q P R2 ÞÑ ψpθq “ ¨ ˝ θ1 θ2 ϕpθ1, θ2q ˛ ‚P BE Ă R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the tangent vectors at x P BE are given by T1pxq “ ¨ ˝ 1 0 Bx1ϕpxq ˛ ‚ and T2pxq “ ¨ ˝ 0 1 Bx2ϕpxq ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, whenever E “ tx P R3 : ϕpx1, x2q ď x3u the outward pointing unit normal at x P BE is given by Npxq “ 1 a 1 ` pBx1ϕpxqq2 ` pBx2ϕpxqq2 ¨ ˝ Bx1ϕpxq Bx2ϕpxq ´1 ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The inner product gpxq is easily computed and given by gpxq “ ˆ 1 ` pBx1ϕpxqq2 pBx1ϕpxqqpBx2ϕpxqq pBx1ϕpxqqpBx2ϕpxqq 1 ` pBx2ϕpxqq2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The inverse metric is given by gpxq´1 “ 1 detpgpxqq ˆ 1 ` pBx2ϕpxqq2 ´pBx1ϕpxqqpBx2ϕpxqq ´pBx1ϕpxqqpBx2ϕpxqq 1 ` pBx1ϕpxqq2 ˙ with detpgpxqq “ 1 ` pBx1ϕpxqq2 ` pBx2ϕpxqq2 “ 1 ` }∇ϕpxq}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The second fundamental form is also given by Ωpxq “ ´ 1 a 1 ` }∇ϕpxq}2 ˆ B2 x1ϕpxq Bx1,x2ϕpxq Bx1,x2ϕpxq B2 x2ϕpxq ˙ and the Weingarten map is defined by Wpxq “ ´ 1 p1 ` }∇ϕpxq}2q3{2 ˆ ¨ ˝ p1 ` pBx2ϕpxqq2qB2 x1ϕpxq ´ pBx1ϕpxqqpBx2ϕpxqqBx1,x2ϕpxq p1 ` pBx2ϕpxqq2qBx1,x2ϕpxq ´ ´pBx1ϕpxqqpBx2ϕpxqqB2 x2ϕpxq ´pBx1ϕpxqqpBx2ϕpxqqB2 x1ϕpxq ` p1 ` pBx1ϕpxqq2qBx1,x2ϕpxq ´pBx1ϕpxqqpBx2ϕpxqqBx1,x2ϕpxq ` p1 ` pBx1ϕpxqq2qB2 x2ϕpxq ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 54 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='3 Surface and volume forms The surface form σB on the boundary BE expressed in the chart ψ introduced in (111) is given by ` σB ˝ ψ´1˘ pdθq “ a det pgpψpθqqq dθ with the Gramian of the coordinate chart gpψpθqq :“ Gram ` Bθ1ψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθq ˘ :“ pBψpθqq pBψpθqq1 “ I ` ∇ϕpθqp∇ϕpθqq1 with the coordinates tangent vectors Bψpθq “ T ψpθq :“ Tpψpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' To check this claim recall that the surface area spaced by the column vectors Bψpθq1 :“ ` Bθ1ψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθq ˘ is equal to the volume of the parallelepided generated by the column vectors pBψpθq1, Npψpθqqq :“ ` Bθ1ψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθq, Npψpθqq ˘ which is given by the determinant of the column vectors, so that ` σB ˝ ψ´1˘ pdθq “ |det pBψpθq1, Npψpθqqq | dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, we have ˆ Bψpθq Npψpθqq1 ˙ ` Bψpθq1, Npψpθqq ˘ “ ¨ ˚ ˚ ˚ ˝ pBθ1ψpθqq1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' pBθ1ψpθqq1 Npψpθqq1 ˛ ‹‹‹‚ ` Bθ1ψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθ1ψpθq, Npψpθqq ˘ “ ˆ pBψpθqq pBψpθqq1 0n´1,1 01,n´1 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that |det pBψpθq1, Npψpθqqq | “ b |det ` pBψpθq1, Npψpθqqq1 pBψpθq1, Npψpθqqq ˘ | “ b det ` pBψpθqq pBψpθqq1˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using the determinant perturbation formula w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' rank-one matrices det pI ` uv1q “ 1 ` v1u which is valid for any column vectors u, v P Rn we check that det pI ` ∇ϕpθq∇ϕpθq1q “ 1 ` }∇ϕpθq}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the formula ` σB ˝ ψ´1˘ pdθq “ a 1 ` }∇ϕpθq}2 dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 55 The mapping F defined in (82) can also be rewritten as a chart ψ on DrpEq defined for any pθ, uq P pS ˆ r0, rsq defined by ψpθ, uq :“ Fpψpθq, uq “ ψpθq ` u Npψpθqq P DrpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The Jacobian matrix of ψ is given by Jacpψqpθ, uq “ ` Bθ1ψpθ, uq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθ, uq, Npψpθqq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='1 we have Bθiψpθ, uq “ Bθiψpθq ` u BθiNψpθq “ Bθiψpθq ´ u ÿ 1ďkăn Bθkψpθq W pψpθqqk,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the formula ` Bθ1ψpθ, uq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθ, uq ˘ “ ` Bθ1ψpθ, uq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθq ˘ pI ´ u Wpψpθqqq from which we check that |det ` Jacpψqpθ, uq ˘ | “ a detpgpψpθqq |det pI ´ u Wpψpθqqq| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that ψpθ, 0q “ ψpθq, and for any given u ă r, the mapping θ ÞÑ ψpθ, uq is a chart on BEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For the convenience of the reader, a more detailed proof of the next proposition is provided in the appendix on page 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any u ď r, the surface form σB,u on the boundary BEu expressed in the chart θ P S ÞÑ ψpθ, uq :“ Fpψpθq, uq is given by the formula ` σB,u ˝ ψp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=', uq´1˘ pdθq “ |det pI ´ u Wpψpθqqq| ` σB ˝ ψ´1˘ pdθq with |det pI ´ u Wpψpθqqq| “ |det ˜ I ` u a 1 ` }∇ϕpθq}2 pI ` ∇ϕpθq∇ϕpθq1q´1 ∇2ϕpθq ¸ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, the volume form σDrpEq on DrpEq expressed in the chart ψ is given by ´ σDrpEq ˝ ψ ´1¯ pdpθ, uqq “ |det pI ´ u Wpψpθqqq| ` σB ˝ ψ´1˘ pdθq du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Using Jacobi’s formula for the derivative of determinants, we also have Bu log det pI ´ u Wpxqq “ ´Tr ` pI ´ u Wpxqq´1 Wpxq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The level-set density function defined in (83) expressed in the chart ψ is given by the formula qB t px, rq “ ż S qt px, ψpθq ` r Npψpθqqq |det pI ´ r Wpψpθqqq| a detpgpψpθqqq dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 Boundary decompositions For some given coordinate index k P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , nu and x “ pxiq1ďiďn P Rn we set x´k :“ pxiqiPI with I :“ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , nu ´ tku We further assume that BE “ tx P Rn : x´k P S and ϕpx´kq “ xku “ ϕ´1pt0uq is defined as the null level set of some global defining function of the form ϕ : x P tpxiq1ďiďn P Rn : x´k P Su ÞÑ ϕpxq :“ ϕpx´kq ´ xk P R for some open domain S Ă Rn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5 (Cylindrical boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Let 1 ď k ď n1 and n “ n1 ` n2 for some n1 ą 1 and n2 ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider a domain S of the form S “ p pS ˆ Rn2q with pS Ă Rn1´1 and assume that @y P Rn1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' y´k P pS and @z P Rn2 we have ϕpy´k, zq :“ pϕpy´kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the set BE is a cylindrical boundary given by the formula BE “ B pE ˆ Rn2 with B pE :“ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' y P Rn1 : y´k P pS and pϕpy´kq “ yk ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, the coordinates of the outward normal by Njpxq “ ǫ a 1 ` }∇ϕpx´kq}2 ` 1Ipjq Bxjϕpx´kq ` 1kpjq p´1q ˘ with the orientation parameter ǫ “ 1 when E “ ϕ´1ps ´ 8, 0rq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' and ǫ “ ´1 when E “ ϕ´1ps0, `8rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, the entries T j i pxq of the tangent vectors Tipxq indexed by i P I are given for any 1 ď j ď n by T j i pxq “ 1ipjq ` 1kpjq Bxiϕpx´kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the pn ˆ pn ´ 1qq-matrix Tpxq1 :“ pT1pxq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Tk´1pxq, Tk`1pxq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Tnpxqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the inner product gpxq on the tangent space TxpBEq is given by the ppn ´ 1q ˆ pn ´ 1qq-square Gramian matrix gpxq “ TpxqTpxq1 “ I ` ∇ϕpx´kq∇ϕpx´kq1 57 with the gradient column vector ∇ϕpx´kq :“ pBxiϕ px´kqqiPI “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ Bx1ϕ px´kq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Bxk´1ϕ px´kq Bxk`1ϕ px´kq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Bxnϕ px´kq ˛ ‹‹‹‹‹‹‹‚ P Rn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We check this claim using the fact that for any i1, i2 P I we have Ti1pxq1Ti2pxq “ ÿ 1ďjďn ` 1i1pjq ` 1kpjq Bxi1ϕpx´kq ˘ ` 1i2pjq ` 1kpjq Bxi2ϕpx´kq ˘ “ 1i1“i2 ` Bxi1ϕpx´kq Bxi2ϕpx´kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The parametrization of the hyper surface BE is now given by the chart function ψ : θ “ pθiqiPI P S ÞÑ ψpθq P BE with @1 ď j ď n ψjpθq :“ 1Ipjq θj ` 1kpjq ϕpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any 1 ď j ď n and i1, i2 P I observe that Bθi1ψpθq “ T ψ i1pθq :“ Ti1pψpθqq and Bθi1,θi2 ψjpθq “ 1kιpjq Bθi1,θi2ϕpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that ´ Bθi1,θi2 ψpθq ¯1 Npψpθqq “ ´ǫ p∇2ϕpθqqi1,i2 a 1 ` }∇ϕpx´kq}2 with ∇2ϕpθq :“ ` Bθi1,θi2ϕpθq ˘ pi1,i2qPI2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We set pBψpθqq1 :“ Tpψpθqq1 and Nψpθq :“ Npψpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, we also have ` BNψpθq ˘1 :“ ` Bθ1Nψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθkι´1Nψpθq, Bθkι`1Nψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , BθnNψpθq ˘ “ ´Wpψpθqq :“ ´gpψpθqq´1Ωpψpθqq with Ωpψpθqq “ ´ǫ ∇2ϕpθq a 1 ` }∇ϕpθq}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For the cylindrical boundary discussed in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='5, the inner prod- uct and the Weingarten map on the boundary B pE are given for any y P B pE by the matrices pgpyq “ Ipn1´1,n1´1q ` ∇pϕpy´kq∇pϕpy´kq1 and x Wpyq :“ ǫ pgpyq´1 ∇2 pϕpy´kq a 1 ` }∇pϕpy´kq}2 58 with the gradient column vector and the Hessian matrix given by ∇pϕpy´kq :“ pByi pϕpy´kqqiPpI ∇2 pϕpy´kq :“ ` Byi1,yi´2 pϕpy´kq ˘ i1,i2PpI with pI :“ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , n1u ´ tku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that detppgpyqq “ 1 ` }∇pϕpy´kq}2 and ∇2ϕpy´k, zq “ ˆ ∇2 pϕpy´kq 0pn1´1,n2q 0pn2,n1´1q 0pn2,n2q ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this case, the inner product and the Weingarten map on the boundary BE are given for any point x “ py, zq P pB pE ˆ Rn2q by the matrices gpxq “ ˆ pgpyq 0pn1´1,n2q 0pn2,n1´1q Ipn2,n2q ˙ and Wpxq “ ˆ x Wpyq 0pn1´1,n2q 0pn2,n1´1q 0pn2,n2q ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that the above matrices are bounded (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' any matrix norm) as soon as B pE is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' More generally, assume that the boundary BE Ă YιPJ Opιq Ă Rn admits a finite covering by open connected subsets Opιq Ă Rn indexed by some finite set J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, there exists some local defining smooth functions ϕι with non vanishing gradients on Opιq such that BEpιq :“ BE X Opιq “ ϕ´1 ι pt0uq and Epιq :“ E X Opιq “ ϕ´1 ι ps0, 8rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Up to shrinking the set Opιq, by the implicit function theorem there is no loss of generality to assume that the defining functions are given by ϕι : x “ pxiq1ďiďn P Opιq ÞÑ ϕιpxq “ ϕιpx´kιq ´ xkι for some parameter 1 ď kι ď n and some smooth function ϕι on some ball Spιq Ă Rn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We set Iι :“ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , nu ´ tkιu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, the parametrization of the hyper surface BEpιq is now given by the smooth homeomorphism ψι : θ “ pθiqiPIι P Spιq ÞÑ ψιpθq P BEpιq with ψj ι pθq :“ 1Iιpjq θj ` 1kιpjq ϕιpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (112) The first and second fundamental forms on Tx pBEpιqq as well as the Weingarten map at x P BEpιq are given by gιpxq “ I ` ∇ϕιpx´kιq∇ϕpx´kιq1 Ωιpxq “ ´ ∇2ϕιpx´kιq a 1 ` }∇ϕιpx´kιq}2 and Wιpxq :“ gιpxq´1Ωιpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 59 The atlas A “ pψι, SιqιPJ represents a collection of local coordinate systems of the boundary BE “ YιPJ BEpιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this situation, the surface form on BE and the volume form σDrpEq on DrpEq expressed in the atlas A are defined by the formulae σA B pdθq :“ ÿ ιPJ πι pψιpθqq 1Spιqpθq a 1 ` }∇ϕιpθq}2 dθ σA DrpEqpdpθ, uqq :“ ÿ ιPJ πι pψιpθqq 1Spιqpθq |det pI ´ u Wιpψιpθqqq| a 1 ` }∇ϕιpθq}2 du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, πι : BE ÞÑ r0, 1s stands for some partition of unity subordinate to the open cover of the boundary induced by the atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that the metric in the graph model discussed in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 is not necessarily bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, we can also use for any a ă a` ă b´ ă b a covering of the form Op0q “sa, brˆR Op´1q “sb´, `8rˆR and Op1q “s ´ 8, a`rˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For instance when ϕpzq “ z2 and pa, a`, b´, bq “ p´2, ´1, 1, 2q we have BEp0q “ tpx1, x2q Ps ´ 2, 2rˆs4, 8r : x2 “ ϕ0px1qu BEp1q “ tpx1, x2q Ps ´ 8, ´1rˆs1, `8r : x1 “ ϕ1px2qu BEp´1q “ tpx1, x2q Ps1, 8rˆs1, `8r : x1 “ ϕ´1px2qu with the functions ϕ0pzq “ z2 and @ǫ P t´1, 1u ϕǫpzq “ ´ǫ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Whenever E is the sub-graph of ϕ, the parameter ǫ P t´1, 1u plays the role of the orientation and the outward pointing unit normal vector at x P BEp0q and y P BEpǫq are given by N0pxq “ 1 a 1 ` 4x2 1 ˆ 2x1 ´1 ˙ and Nǫpyq “ ǫ a 1 ` 1{p4y2q ˆ ´1 ´ǫ{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4y2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The tangent vectors at x P BEp0q and at y P BEpǫq are defined by T0pxq “ ˆ 1 2x1 ˙ and Tǫpyq “ ˆ ´ǫ{p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4y2q 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The above sub-graphs can be described with 3 charts tψ0, ψ`1, ψ´1u defined for any ǫ P t´1, 1u by ψ0 : θ Ps ´ 2, 2rÞÑ ψ0pθq “ ˆ θ θ2 ˙ and ψǫ : θ Ps1, 8rÞÑ ψǫpθq :“ ˆ ´ǫ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ θ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 60 In this situation, the tangent vectors are given by Bθψ0pθq “ T0pψ0pθqq “ ˆ 1 2θ ˙ and Bθψǫpθq “ Tǫ pψǫpθqq “ ˆ ´ǫ{p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 4θq 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this context, for any θ Ps ´ 2, 2r we have gpψ0pθqq “ 1 ` 4θ2 and Wpψ0pθqq “ ´2 ` 1 ` 4θ2˘´3{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, for any θ Ps1, 8r we have gpψǫpθqq “ 1 ` 1{p4θq and Wpψǫpθqq “ ´2ǫ p1 ` 4θq´3{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that the metric expressed in the chart tψ0, ψ`1, ψ´1u is defined in terms of bounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Consider the hyperbolic paraboloid boundary BE “ tpy1, y2, y3q P R3 : y3 “ y2 1 ` y2 2u “ BEp0q Y BEp1, 1q Y BEp1, ´1q Y BEp2, 1q Y BEp2, ´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, BEp0q and BEpi, ǫq with i P t1, 2u and ǫ P t´1, 1u stands for the partition defined for any ǫ P t´1, 1u by BEp0q :“ ty P R3 : py1, y2q P S0 y3 “ ϕ0py1, y2q :“ y2 1 ` y2 2u BEp1, ǫq :“ ty P R3 : py1, y3q P S y2 “ ϕ1,ǫpy1, y3q :“ ǫ b y3 ´ y2 1u BEp2, ǫq :“ ty P R3 : py2, y3q P S y1 “ ϕ2,ǫpy1, y2q :“ ǫ b y3 ´ y2 2u with the sets S0 :“ tpy1, y2q P R2 : y2 1 ` y2 2 ă 2u S :“ tpy2, y3q P R2 : y3 ą 1 & |y2| ă a 3y3{4u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the truncated boundary BEp0q we use a single chart defined by ψ0 : θ “ pθ1, θ2q P S0 ÞÑ ψ0pθq “ ¨ ˝ θ1 θ2 θ2 1 ` θ2 2 ˛ ‚P BEp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On BEp1, ǫq we use the chart defined by ψ1,ǫ : θ “ pθ1, θ3q P S ÞÑ ψ1,ǫpθq “ ¨ ˝ θ1 ǫ a θ3 ´ θ2 1 θ3 ˛ ‚P BEp1, ǫq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 61 Finally, on BEp2q we use the chart defined by ψ2,ǫ : θ “ pθ2, θ3q P S ÞÑ ψ2,ǫpθq “ ¨ ˝ ǫ a θ3 ´ θ2 2 θ2 θ3 ˛ ‚P BEp2, ǫq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any θ “ pθ1, θ2q P S0 we have Bθ1ψ0pθq “ ¨ ˝ 1 0 2θ1 ˛ ‚ and Bθ2ψ0pθq “ ¨ ˝ 0 1 2θ2 ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this chart, the metric is given by gpψ0pθqq “ ˆ 1 ` 4θ2 1 4θ1θ2 4θ1θ2 1 ` 4θ2 2 ˙ and gpψ0pθqq´1 “ 1 1 ` 4pθ2 1 ` θ2 2q ˆ 1 ` 4θ2 2 ´4θ1θ2 ´4θ1θ2 1 ` 4θ2 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, the outward pointing unit normal at ψ0pθq P BEp0q is given by N0 pψ0pθqq “ 1 a 1 ` 4pθ2 1 ` θ2 2q ¨ ˝ 2θ1 2θ2 ´1 ˛ ‚ and Ω0 pψ0pθqq “ 1 a 1 ` 4pθ2 1 ` θ2 2q ˆ ´2 0 0 ´2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any θ “ pθ1, θ3q P S we have Bθ1ψ1,ǫpθq “ ¨ ˚ ˝ 1 ´ǫθ1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 1 0 ˛ ‹‚ and Bθ3ψ1,ǫpθq “ ¨ ˚ ˝ 0 ǫ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 1 1 ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this chart, the metric is given by gpψ1,ǫpθqq “ ˜ 1 ` θ2 1 θ3´θ2 1 ´ θ1 2pθ3´θ2 1q ´ θ1 2pθ3´θ2 1q 1 ` 1 4pθ3´θ2 1q ¸ and gpψ1,ǫpθqq´1 “ 1 1 ` θ2 1 θ3´θ2 1 ` 1 4pθ3´θ2 1q ˜ 1 ` 1 4pθ3´θ2 1q θ1 2pθ3´θ2 1q θ1 2pθ3´θ2 1q 1 ` θ2 1 θ3´θ2 1 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, the outward pointing unit normal at ψ1,ǫpθq P BEp1, ǫq is given by N1,ǫ pψ1,ǫpθqq “ ´ǫ b 1 ` θ2 1 θ3´θ2 1 ` 1 4pθ3´θ2 1q ¨ ˚ ˝ ´ǫθ1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 1 ´1 ǫ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 1 ˛ ‹‚ and Ω1,ǫ pψ1,ǫpθqq “ ´ǫ b 1 ` θ2 1 θ3´θ2 1 ` 1 4pθ3´θ2 1q ˜ ǫθ3 pθ3´θ2 1q3{2 ´ǫθ1 pθ3´θ2 1q3{2 ´ǫθ1 pθ3´θ2 1q3{2 ´ ǫ 4pθ3´θ2 1q3{2 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Finally, for any θ “ pθ2, θ3q P S we have Bθ2ψ2,ǫpθq “ ¨ ˚ ˝ ´ǫθ2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 2 1 0 ˛ ‹‚ and Bθ3ψ2,ǫpθq “ ¨ ˚ ˝ ǫ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 2 0 1 ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this chart, the metric and the outward pointing unit normal at ψ2,ǫpθq P BEp2, ǫq are given by gpψ2,ǫpθqq “ ˜ 1 ` θ2 2 θ3´θ2 2 ´ θ2 2pθ3´θ2 2q ´ θ2 2pθ3´θ2 2q 1 ` 1 4pθ3´θ2 2q ¸ and N2,ǫ pψ2,ǫpθqq “ ´ ǫ b 1 ` θ2 2 θ3´θ2 2 ` 1 4pθ3´θ2 2q ¨ ˚ ˝ ´1 ´ǫθ2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 2 ǫ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' θ3´θ2 2 ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 62 Appendix Proof of (9) We have Ps,s`tpV q ď V ` ż s`t s p´aPs,upV q ` cq du “ V ` ct ´ a ż t 0 Ps,s`upV q du and ż t 0 Ps,s`upV q du “ tPs,s`tpV q ´ ż t 0 uPs,s`upLs`upV qq du ě tPs,s`tpV q ´ ct2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Combining the above estimates, we readily check that Ps,s`tpV q ď p1 ` atq´1V ` ct 1 ` at{2 1 ` at ď p1 ` atq´1V ` ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='7 We have the following almost sure estimate }∇Xτpxq}2 ď e´λτ, where }A}2 stands for the spectral norm of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields for any x, y P Rn the almost sure estimate }Xτpxq ´ Xτpyq} ď e´λτ }x ´ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (113) Applying the above to y “ 0 we find that PτpV qpxq ď PτpV qp0q V pxq1´δ with δ “ 1 ´ e´λτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Next, we check that P X τ pV qp0q ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We have Xup0q “ ż u 0 pbp0q ds ` σ dBsq ` ż u 0 ż 1 0 ∇bpǫXsp0qq1 Xsp0q dǫ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that }Xup0q} ď β pu ` }Bu}q ` β ż u 0 }Xsp0q} ds with β :“ σ _ }bp0q} _ }∇b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying Gr¨onwall lemma we check that }Xup0q} law ď β pu ` }Bu}q ` β2 ż u 0 ps ` }Bs}q eβpu´sq ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 63 On the other hand, we have ż u 0 }Bs} ds “ u ż 1 0 }Bus} ds law “ u3{2 ż 1 0 }Bs} ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the rather crude estimate }Xup0q} law ď β ` u ` u1{2 }B1} ˘ ` β2 u2{2 ` β2eβu u3{2 ż 1 0 }Bs} ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any a ě 0 by Jensen’s inequality E ´ ea ş1 0 }Bs} ds¯ ď ż 1 0 E ` ea}Bs}˘ ds ď ea2r{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' It is now an elementary exercise to check that Epev}Xτ p0q}q ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='8 Consider the function ftpxq :“ exp ˆ 2ǫ ˆ e´αt Wpxq ´ β 1 ´ e´αt α ˙˙ ùñ ´Bt log ftpxq “ 2ǫ e´αt pαW ` βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, we check that Bxiftpxq{ftpxq “ 2ǫ e´αt BxiW Bxi,xjftpxq{ftpxq “ 2ǫ e´αt ` 2ǫ e´αt BxiWBxjW ` Bxi,xjW ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that pLpftq ´ Btftq {ft “ 2ǫ e´αt ` pαW ` βq ` LpWq ` ǫ e´αt ΓLpW, Wq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Combining the above with (37) we find that pLpftqpxq ´ Btftpxqq ď ´2 ǫ2 e´αt ` 1 ´ e´αt˘ ΓLpW, Wq ftpxq ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the interpolation formula E pf0pXtpxqqq ´ ftpxq “ ż t 0 E pBsft´spXspxqqq ds ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We check (38) after some elementary manipulations, thus there are skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 64 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='11 Notice that Xh t pxq law “ ǫt x ` σt Z “ Bσt pǫt xq with ǫt :“ e´t and σt :“ c 1 ´ ǫ2 t 2 and some centered Gaussian random variable Z with unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The conjugate formula (53) yields the integral operator equation Qtpx, dyq “ e´t{2 e´x2{2 1 a 2πσ2 t exp ˆ ´py ´ ǫtxq2 2σ2 t ` y2 2 ˙ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Observe that ´py ´ ǫtxq2 σ2 t ` y2 “ ´ 1 pt ˆ y ´ ǫt 1 ´ σ2 t x ˙2 ` x2 ǫ2 t 1 ´ σ2 t with pt :“ 1 ´ ǫ2 t 1 ` ǫ2 t “ tanhptq ðñ Btpt “ 1 ´ p2 t with p0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' (114) We check this claim using the fact that 1 σ2 t “ 2 1 ´ ǫ2 t “ 1 ` 1 ` ǫ2 t 1 ´ ǫ2 t “ 1 ` 1 pt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, we have 1 ´ σ2 t ǫt “ coshptq and Bt log coshptq “ pt “ tanhptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that ż t 0 ps ds “ log coshptq and ǫt 1 ´ σ2 t “ 1 coshptq “ exp ˆ ´ ż t 0 psds ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We also have 1 ´ σ2 t “ 1 ´ 1 ´ ǫ2 t 2 “ 1 ` ǫ2 t 2 ùñ 1 ´ ǫ2 t 1 ´ σ2 t “ 1 ´ ǫ2 t 1 ` ǫ2 t “ pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that Qtp1qpxq “ e´t{2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='pt σt exp ˆ ´x2 2 pt ˙ “ e´t{2 hpxq P h t p1{hqpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that e´t{2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='pt σt “ d ǫt 1 ´ ǫ2 t 1 ` ǫ2 t 2 1 ´ ǫ2 t “ d 2 1{ǫt ` ǫt “ 1 a coshptq and Btmtpxq “ ´pt mtpxq and Btpt “ 1 ´ p2 t with pm0pxq, p0q “ px, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 65 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='12 Notice that et{2 ex2{2 Qtpx, dyq “ 1 a 2πσ2 t ˆ exp ˆ ´py ´ ǫtxq2 2σ2 t ` y2 2 ˙ ´ exp ˆ ´py ` ǫtxq2 2σ2 t ` y2 2 ˙˙ 1r0,8rpyq dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that Qtp1qpxq “ e´ x2 2 tanhptq a coshptq ˆ ż 8 0 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2πpt ˆ exp ˆ ´py ´ mtpxqq2 2pt ˙ ´ exp ˆ ´py ` mtpxqq2 2pt ˙˙ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We conclude that Qtp1qpxq “ e´ x2 2 tanhptq a coshptq ˆ P p´mtpxq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='pt ď Z ď mtpxq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='ptq “ 2 e´ x2 2 tanhptq a coshptq ˆ P ˜ 0 ď Z ď x a sinhptq coshptq ¸ ÝÑ 0 as x Ñ 8 or x Ñ 0 or as t Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the above display, Z stands for some centered Gaussian random variable with unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Note that we have used the fact that mtpxq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='pt “ x coshptq a tanhptq “ x a sinhptq coshptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, we have Qtpx, dyq “ 1 P ´ 0 ď Z ď x{ a sinhptq coshptq ¯ ˆ 1 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2πpt ˆ exp ˆ ´py ´ mtpxqq2 2pt ˙ ´ exp ˆ ´py ` mtpxqq2 2pt ˙˙ 1r0,8rpyq dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 66 Proof of (48) The generator of the process (47) is defined by Lpfqpq, pq “ β p m Bf Bq ´ β ˆBW Bq ` σ2 2 p m ˙ Bf Bp ` σ2 2 B2f Bp2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Recalling that 2pq ď p2 ` q2, we prove that V pq, pq ď 1 2 ˆ 1 m ` ǫ ˙ p2 ` ǫ 2 ˆσ2 2 ` 1 ˙ q2 ` Wpqq ď C‹pǫq ` 1 ` p2 ` q2 ` Wpqq ˘ with C‹pǫq :“ max "1 2 ˆ 1 m ` ǫ ˙ , ǫ 2 ˆσ2 2 ` 1 ˙ , 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, we have LpV q “ β p m ˆBW Bq ` ǫ σ2 2 q ` ǫ p ˙ ´β ˆBW Bq ` σ2 2 p m ˙ ´ p m ` ǫ q ¯ ` σ2 2m “ ´β „ 1 m ˆ σ2 2m ´ ǫ ˙ p2 ` ǫ q BW Bq \uf6be ` σ2 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Under our assumptions, this implies that for any |q| ě r we have LpV q ď ´β „ 1 m ˆ σ2 2m ´ ǫ ˙ p2 ` ǫ δ ` Wpqq ` q2˘\uf6be ` σ2 2m ď ´C‹pǫ, δq ` 1 ` p2 ` q2 ` Wpqq ˘ ` cmpǫ, δq with C‹pǫ, δq :“ β min "ˆ 1 m ˆ σ2 2m ´ ǫ ˙ , ǫ δ ˙* and cmpǫ, δq :“ C‹pǫ, δq ` σ2 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We conclude that for any |q| ą r, pV ´1LpV qqpq, pq ď ´C‹pǫ, δq p1 ` p2 ` q2 ` Wpqqq ´ cmpǫ, δq V pq, pq ď ´C‹pǫ, δq p1 ` p2 ` q2 ` Wpqqq ´ cmpǫ, δq C‹pǫq p1 ` p2 ` q2 ` Wpqqq “ ´C‹pǫ, δq C‹pǫq ` cmpǫ, δq C‹pǫq 1 1 ` p2 ` q2 ` Wpqq ď ´ „C‹pǫ, δq C‹pǫq ´ cmpǫ, δq C‹pǫq 1 1 ` p2 ` q2 \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 67 We choose r sufficiently large to satisfy |p| ą r or |q| ą r ñ C‹pǫ, δq C‹pǫq ´ cmpǫ, δq C‹pǫq 1 p2 ` q2 ě C‹pǫ, δq C‹pǫq ´ cmpǫ, δq C‹pǫq 1 r2 ě a :“ C‹pǫ, δq 2C‹pǫq ą 0, and we set Kr :“ tpq, pq P R2 : |p| _ |q| ď ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, we have LpV q ď ´aV 1E´Kr ` sup Kr LpV q ď ´aV ` c with c :“ sup Kr LpV q ` a sup Kr V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof of (56) Observe that for any 0 ă y ď 1 and z P E “s0, 8r we have sinh pyzq ď y sinh pzq and sinh pzq ď 1 2 ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that ż 8 0 Qtpx, dyq 1 y ď sinh pmtpxqq e´ x2 2 ppt`e´2t{ptq a coshptq ˆ ż 8 0 c 2 πpt exp ˆ ´ y2 2pt ˙ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' from which we check that ż 8 0 Qtpx, dyq 1 y 1s0,1spyq ď exp ´ ´ ´ x2 2 ppt ` e´2t pt q ´ e´tx ¯¯ 2 a coshptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, for any n ě 1 we have ż 8 0 Qtpx, dyq yn ď 1 2 1 a coshptq c 2 πpt exp ˆ ´ǫ2 tx2 2pt ´ x2 2 pt ˙ ż 8 0 yn exp ˆ yǫtx ´ y2 2pt ˙ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Notice that yǫtx ´ y2 2pt “ ´ 1 2pt py ´ ǫtxptq2 ` x2 2 ǫ2 t pt so that ż 8 0 Qtpx, dyq yn ď 1 2 1 a coshptq c 2 πpt ˆ exp ˆ ´x2 2 ˆ p1 ´ ǫ2 tq pt ` ǫ2 t pt ˙˙ ż 8 0 yn exp ˆ ´ 1 2pt py ´ ǫtxptq2 ˙ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 68 For any n ě 1, we conclude that V pxq :“ xn ` 1{x ùñ V P C8pEq and }QtpV q} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 To simplify notation, we write Qt instead of QrUs t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any V P B8pEq X DpLq we have QtpV q “ V ` ż t 0 QspLpV q ´ UV q ds ď V ` ż t 0 r´a QspV q ` c Qsp1qs ds “ V ` c ż t 0 Qsp1q ds ´ a ż t 0 QspV qds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, through integration by parts we have ż t 0 QspV qds “ rs QspV qst 0 ´ ż t 0 s d dsQspWq ds “ t QtpV q ´ ż t 0 s QspLpV q ´ UV looooomooooon ďc q ds ě t QtpV q ´ c ż t 0 s Qsp1qds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This implies that QtpV q ď V ` c ż t 0 Qsp1q ds ´ a ˆ t QtpV q ´ c ż t 0 s Qsp1qds ˙ from which we conclude that QtpV q ď V 1 ` at ` c ż t 0 Qsp1qds ùñ QtpV q ď V 1 ` at ` ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Now, we come to the proof of (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We have the forward evolution equation given for any f P DpLq by BtQtpfq “ QtpLUpfqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Applying the above to f “ U we readily check that BtQtpUq ď a0 ` a1 QtpUq ´ QtpU2q ď a0 ` a1 QtpUq ´ pQtpUqq2{Qtp1q from which we find the Riccati estimates BtQtpUq ď a0 ` a1 QtpUq ´ pQtpUqq2 ùñ @t ą 0 }QtpUq} ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 69 Proof of (66) By Girsanov theorem we have Qpaq t pfqpzq “ E ` fpX 0 t pzqq Ztpzq 1T 0pzqąt ˘ with the exponential martingale Ztpzq “ exp ˆ 1 σ ż t 0 apXspzqq1 dBu ´ 1 2σ2 ż t s }apXspzqq}2 du ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' By H¨older’s inequality, for any non negative function f on E, any z P E and any conjugate parameters p, q ą 1 with 1{p ` 1{q “ 1 we have Qpaq t pfqpzq ď E ` Ztpzqq 1T 0pzqąt ˘1{q Qpaq t pf pqpzq1{p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand, we have E ` Ztpzqq 1T 0pzqąt ˘ “ E ˆ Ztpzq exp ˆqpq ´ 1q 2σ2 ż t s }apXspzqq}2 du ˙ 1T 0pzqąt ˙ ď ctppq :“ exp ˆ pt 2ppp ´ 1qσq2 sup D a ˙ with the exponential martingale Ztpzq “ exp ˆ q σ ż t 0 apXspzqq1 dBu ´ q2 2σ2 ż t s }apXspzqq}2 du ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of roof of (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='2 For any z P BE there exists some open ball Bpz, rq Ă Rn with r ą 0 and some C1-mapping g from Rn´1 into R such that E X Bpz, rq “ tx P Bpz, rq : xn ă gpx´nqu BE X Bpz, rq “ tx P Bpz, rq : xn “ gpx´nqu with x´n :“ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , xn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We make the change of variables Epz, rq :“ E X Bpz, rq ÞÑ ςpxq :“ px´n, xn ´ gpx´nqq P Opz, rq :“ ςpEpz, rqq Ă pRn´1 ˆ R`q with Jacobian ∇ςpxq “ ˆ Ipn´1qˆpn´1q ´∇gpx´nq 0 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 70 Observe that ς : x P E0pz, rq :“ pBE X Bpz, rqq ùñ ςpxq “ px´n, 0q P O0pz, rq :“ ςpE0pz, rqq Ă pRn´1 ˆ t0uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' The inverse is given by y P Opz, rq ÞÑ ς´1pyq “ py´n, yn ` gpy´nqq P Epz, rq ùñ ∇ς´1pyq “ ˆ Ipn´1qˆpn´1q ∇gpy´nq 0 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' On the other hand we have }ςpxq ´ ςpxq} “ ` }x´n ´ x´n}2 ` p|xn ´ xn| ` |gpx´nq ´ gpx´nq|q2˘1{2 ď ` }x´n ´ x´n}2 ` 2|xn ´ xn|2 ` 2}∇g}2}x´n ´ x´n}2˘1{2 ď cpgq }x ´ x} with cpgq :“ a 2 _ p1 ` 2}∇g}2q ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, we have }ς´1pyq ´ ς´1pyq} ď cpgq }y ´ y} so that 1 cpgq }y ´ y} ď }ς´1pyq ´ ς´1pyq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any x P Epz, rq and x P E0pz, rq we have ςpxq P O0pz, rq and }x ´ x} “ }ς´1pςpxqq ´ ς´1pςpxqq} ě 1 cpgq }ςpxq ´ ςpxq} ě 1 cpgq |ςpxqn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Taking the infimum of all x P E0pz, rq this implies that dpx, E0pz, rqq ě 1 cpgq |ςpxqn| and dpς´1pyq, E0pz, rqq ě 1 cpgq |yn| for any x P Epz, rq and y P Opz, rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We conclude that ż Epz,rq χ pdpx, E0pz, rqqq dx “ ż Opz,rq χ ` dpς´1pyq, E0pz, rqq ˘ |det ` ς´1pyq ˘ | dy ď 1 cpgq sup yPOpz,rq |det ` ς´1pyq ˘ | ż Opz,rq χpynq dy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We end the proof of the lemma by covering BE by finitely many boundary coordinates patches pEpzi, riq, giq1ďiďn, for some zi P BE, ri ą 0 and some local defining functions gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 71 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='9 Using the change of variable formulae ż BEr fpzq σB,rpdzq “ ż BE f pz ` rNpzqq |det pI ´ r Wpzqq | σBpdzq and ż BE fpzq σBpdzq “ ż BEr f pz ´ rNpzqq |det pI ` r Wpzqq | σB,rpdzq we check that ż BEr fpzq σB,rpdzq ď κBpαq ż BE f pz ` rNpzqqq σBpdzq and ż BE fpzq σBpdzq ď κ´ B pαq ż BEr f pz ´ rNpzqq σB,rpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the estimate ż BEr fpzq σB,rpdzq ď ιpαq κBpαq ż BE gpzq σBpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In the same vein, we have ż BE fpzq σBpdzq ď ιpαqκ´ B pαq ż BEr g pzq σB,rpdzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Integrating w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' the parameter r P r0, αs we check the co-area estimate α ż BE fpzq σBpdzq ď ιpαqκ´ B pαq ż α 0 dr ż BEr gpzq σB,rpdzq “ ιpαq κ´ B pαq ż DαpEq gpzq dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='4 For any given θ :“ pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , θnq P pRn´1 ˆ r0, rsq we set θ´n :“ pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , θn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In this notation, we have ψ : θ P pRn´1 ˆ r0, rsq ÞÑ ψpθq :“ Fpψpθ´nq, θnq “ ψpθ´nq ` θn Npψpθ´nqq P DrpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 72 The volume form σDrpEq on DrpEq expressed in the chart ψ is given by ´ σDrpEq ˝ ψ ´1¯ pdθq “ |det ` Jacpψqpθq ˘ | “ c det ´` Bψpθq ˘ ` Bψpθq ˘1¯ dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Arguing as above, we have ` Bψpθq ˘1 “ ˆ´ Bθ´nψpθq ¯1 , Bθnψpθq ˙ P ´ TψpθqpDrpEqq ¯n with the tangent vectors ´ Bθ´nψpθq ¯1 :“ ´ Bθ1ψpθq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' , Bθn´1ψpθq ¯ and Bθnψpθq “ Npψpθ´nqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' In addition, we have ´ Bθ´nψpθq ¯1 “ ` Bψpθ´nq ˘1 ` θn ` BpNpψpθ´nqqq ˘1 “ ` Bψpθ´nq ˘1 ` I ´ θn Wpψpθ´nqq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This yields the formula ´ Bθ´nψpθq ¯ ´ Bθ´nψpθq ¯1 “ gpψpθ´nqq ` I ´ θn Wpψpθ´nqq ˘2 from which we check that ` Bψpθq ˘ ` Bψpθq ˘1 “ ˆ gpψpθ´nq ` I ´ θn Wpψpθ´nqq ˘2 0n´1 0 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' We conclude that c det ´` Bψpθq ˘ ` Bψpθq ˘1¯ “ b detpgpψpθ´nqq ˇˇdet ` I ´ θn Wpψpθ´nqq ˘ˇˇ and therefore ´ σDrpEq ˝ ψ ´1¯ pdθq “ ˇˇdet ` I ´ θn Wpψpθ´nqq ˘ˇˇ dθn ` σB,0 ˝ ψ´1˘ pdθ´nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' For any given θn “ u P r0, rs, the volume form σB,u on the boundary BEu expressed in the boundary chart ψp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=', uq : θ P Rn´1 ÞÑ ψpθ, uq “ Fpψpθq, uq P BEu is given by ` σB,u ˝ ψp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=', uq´1˘ pdθq “ |det pI ´ u Wpψpθqqq| pσB,0 ˝ ψ´1q pdθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 73 References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Aliabad, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Azarpanah and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' Namdari.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 3, 1322– 1355 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} +page_content=' 79' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE1T4oBgHgl3EQf3AU7/content/2301.03484v1.pdf'} diff --git a/ttE4T4oBgHgl3EQfWAzx/vector_store/index.pkl b/ttE4T4oBgHgl3EQfWAzx/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b3951f551af27a59641cf0b16b92445daadbe684 --- /dev/null +++ b/ttE4T4oBgHgl3EQfWAzx/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b8e9deaa4d62feb7c6cbf00d14d1990658c3fc5b3ee3c02cbdb02d07236e7d6a +size 300994 diff --git a/xNE2T4oBgHgl3EQfhAd2/content/tmp_files/2301.03943v1.pdf.txt b/xNE2T4oBgHgl3EQfhAd2/content/tmp_files/2301.03943v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef61d4aebba2eda4ffb19195b5c99dd4a1f401e3 --- /dev/null +++ b/xNE2T4oBgHgl3EQfhAd2/content/tmp_files/2301.03943v1.pdf.txt @@ -0,0 +1,2262 @@ +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +1 +Rethinking Smart Contract Fuzzing: Fuzzing With Invocation +Ordering and Important Branch Revisiting +Zhenguang Liu, Peng Qian, Jiaxu Yang, Lingfeng Liu, Xiaojun Xu, Qinming He, and Xiaosong Zhang +Abstract—Blockchain smart contracts have given rise to a +variety of interesting and compelling applications and emerged +as a revolutionary force for the Internet. Smart contracts from +various fields now hold over one trillion dollars worth of virtual +coins, attracting numerous attacks. Quite a few practitioners +have devoted themselves to developing tools for detecting bugs +in smart contracts. One line of efforts revolve around static +analysis techniques, which heavily suffer from high false positive +rates. Another line of works concentrate on fuzzing techniques. +Unfortunately, current fuzzing approaches for smart contracts +tend to conduct fuzzing starting from the initial state of the +contract, which expends too much energy revolving around the +initial state of the contract and thus is usually unable to unearth +bugs triggered by other states. Moreover, most existing methods +treat each branch equally, failing to take care of the branches +that are rare or more likely to possess bugs. This might lead to +resources wasted on normal branches. +In this paper, we try to tackle these challenges from three +aspects: (1) In generating function invocation sequences, we +explicitly consider data dependencies between functions to fa- +cilitate exploring richer states. We further prolong a function +invocation sequence S1 by appending a new sequence S2, so that +the appended sequence S2 can start fuzzing from states that +are different from the initial state. (2) We incorporate a branch +distance-based measure to evolve test cases iteratively towards +a target branch. (3) We engage a branch search algorithm to +discover rare and vulnerable branches, and design an energy +allocation mechanism to take care of exercising these crucial +branches. We implement IR-Fuzz and extensively evaluate it +over 12K real-world contracts. Empirical results show that: (i) +IR-Fuzz achieves 28% higher branch coverage than state-of-the- +art fuzzing approaches, (ii) IR-Fuzz detects more vulnerabilities +and increases the average accuracy of vulnerability detection by +7% over current methods, and (iii) IR-Fuzz is fast, generating +an average of 350 test cases per second. Our implementation +and dataset are released at https://github.com/Messi-Q/IR-Fuzz, +hoping to facilitate future research. +Index Terms—Fuzzing, smart contract, vulnerability detection, +blockchain, sequence generation, seed evolution. +I. INTRODUCTION +This work was supported in part by the National Key R&D Program of +China under Grant 2021YFB2700500 and 2021YFB2700501, in part by the +Key R&D Program of Zhejiang Province under Grant 2022C01086, and in part +by the Scientific Research Fund of Zhejiang Provincial Education Department +under Grant Y202250832. (Corresponding author: Peng Qian.) +Zhenguang Liu, Peng Qian, and Qinming He are with College of Com- +puter Science and Technology, Zhejiang University, Hangzhou 310058, +China +(e-mail: +liuzhenguang2008@gmail.com; +messi.qp711@gmail.com; +hqm@zju.edu.cn). +Jiaxu Yang, Lingfeng Liu, and Xiaojun Xu are with School of Computer +and Information Engineering, Zhejiang Gongshang University, Hangzhou +310018, China (e-mail: yjx.00@foxmail.com; liulingfengxx@gmail.com; +xuxj2022@gmail.com). +Xiaosong Zhang is with the Center for Cyber Security, University of +Electronic Science and Technology of China, Chengdu 611731, China (e- +mail: johnsonzxs@uestc.edu.cn). +S +MART contracts are programs executing on top of a +blockchain system [1]. A smart contract encodes prede- +fined contract terms into runnable code. Due to the immutable +nature of blockchain, once a smart contract is deployed on the +blockchain, its defined rules will be strictly followed during +execution. Smart contracts make the automatic execution of +contract terms possible, giving rise to a variety of decentralized +applications [2], [3]. +Notably, not all blockchains support smart contracts. +Ethereum, one of the most prominent blockchains enabling +the execution of smart contracts, has attracted widespread +attention worldwide. So far, tens of millions of contracts have +been deployed on Ethereum [4], enabling a broad spectrum +of applications, including wallet [5], crowdfunding [6], supply +chain [7], and cross-industry finance [8]. Smart contracts from +various fields now hold over one thousand billion dollars worth +of virtual coins, and the number of contracts is still increasing +rapidly [2]. Smart contracts have long been appealing targets +for attackers since they manipulate so many digital assets. +Specifically, the source code of a Ethereum smart contract +will be compiled into bytecode and executed on Ethereum +Virtual Machine [9]. Like traditional programs, smart con- +tracts may contain vulnerabilities. Therefore, it is important +to identify potential vulnerabilities in smart contracts, ideally +before their deployments. Malicious attackers may exploit the +bugs in smart contracts to gain illegal profits. Recently, there +was an increasing number of security vulnerability incidents +in smart contracts [10], [11], leading to enormous financial +losses. One infamous example was the reentrancy attack, i.e., +attackers stole more than $130 million worth of digital assets, +exploiting the reentrancy vulnerability in the Cream.Finance +contract [12]. This case is not isolated, e.g., a delegatecall bug +accidentally triggered resulted in freezing over $280 million +worth of Ether in the Parity Multisig Wallet contract [13]. +Obviously, conducting security vetting on smart contracts to +avoid exposing vulnerabilities to attackers is much coveted. +Fueled by the maturity of static analysis techniques such as +formal verification [14] and symbolic execution [15], smart +contract vulnerability detection has undergone considerable +progress in the past few years. These methods, however, +inherently suffer from high false positive rates since they do +not actually execute each path. Recent efforts resort to fuzzing +techniques [16]–[18], which have the merits of producing neg- +ligible false positives in discovering software vulnerabilities. +This can be attributed to the fact that fuzzers usually generate +test cases to exercise a branch, and report vulnerabilities only +when they detect abnormal results during fuzzing. +After scrutinizing existing released fuzzers for smart con- +tracts, such as [16], [18]–[23], we obtain the following obser- +arXiv:2301.03943v1 [cs.PL] 10 Jan 2023 + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +2 +vations. (1) Current fuzzers (e.g., sFuzz [16] and Harvey [21]) +tend to generate function invocation sequences randomly, +overlooking the data dependencies (such as read and write +dependencies) between functions. More importantly, a smart +contract may transition through many different states during its +lifecycle [21]. For example, every bet in a gambling contract +will change the contract state. However, current methods +generally conduct fuzzing starting from the initial state of the +contract, which actually expends too much energy revolving +around the initial state of the contract and is incapable of +unearthing bugs triggered by complex states. (2) Most current +approaches fail to take into account the distance between +test cases and branch conditions in seed mutation, resulting +in generating seeds that have low probabilities to enter a +target branch. (3) Existing fuzzers (e.g., ILF [22] and Con- +fuzzius [23]) often treat program branches equally. As a result, +fuzzers might waste too many resources in fuzzing normal +branches and are unable to dive deep into crucial branches +that are rare or more likely to have bugs. +To tackle these challenges, we propose IR-Fuzz, a fully +automatic Fuzzing framework equipped with Invocation or- +dering and impoRtant branch revisiting, for detecting security +vulnerabilities in Ethereum smart contracts. In particular, IR- +Fuzz consists of three key components. +Sequence Generation. +Usually, there are multiple func- +tions within a contract, we introduce a function-invocation- +sequence generation strategy, which consists of function in- +vocation ordering and sequence prolongation. Specifically, +we build a data flow analyzer to capture the data flow +dependencies of global variables and then define a rule to +compute the order priority of each function call, inferring the +ordered function invocation sequence. Further, we introduce a +prolongation technique to extend the sequence, enforcing the +fuzzer to tap into unprecedented states. +Seed Optimization. +We also present a seed optimiza- +tion paradigm, which drives the fuzzer to generate branch- +condition-aware test cases. In practice, we employ a branch +distance-based measure to select test cases according to how +far a test case is from satisfying the condition (e.g., x==10) +of a just-missed branch1. Intuitively, the test case has a higher +probability to enter the just-missed branch as the distance +decreases. In this way, IR-Fuzz iteratively evolves test cases +to get increasingly closer to satisfying the branch conditions, +which boosts its ability to find a high-quality test case and +achieve a higher branch coverage. +Energy Allocation. +Finally, we design an energy allo- +cation mechanism that takes into account the significance +of a branch. Technically, we first propose a branch search +algorithm to pick out rare branches and branches that are +likely to have vulnerabilities. Then, we formulate a customized +energy schedule and develop two rules to guide fuzzing +energy allocation. As such, IR-Fuzz can flexibly assign fuzzing +resources to more important branches, which increases the +overall fuzzing efficiency by a large margin (4.9x faster than +sFuzz [16]) and further improves branch coverage. +1A just-missed branch stands for the unexplored if-branch or then-branch +of a conditional statement (such as if and require) or a recurrent statement +(such as for and while). +We implement IR-Fuzz and extensively evaluate this system +over 12K real-world smart contracts. Experimental results +show that: (i) IR-Fuzz achieves high average branch cover- +age by up to 90%, yielding a 28% improvement compared +with state-of-the-art fuzzing approaches. (ii) IR-Fuzz identifies +more vulnerabilities and increases the average accuracy of +vulnerability detection by 7% over current methods. (iii) IR- +Fuzz generates an average of 350 test cases per second, in most +cases orders-of-magnitude faster than conventional fuzzers. +Our key contributions can be summarized as follows: +• We design and implement a novel framework IR-Fuzz for +smart contract fuzzing, which consists of three key com- +ponents, i.e., function invocation sequence prolongation, +branch-distance-driven seed optimization, and branch- +importance-aware energy allocation. +• Within the framework, we present a sequence generation +strategy to infer high-quality function invocation se- +quences, steering fuzzing to explore unprecedented states. +Further, we introduce a seed optimization paradigm that +incorporates a branch distance-based measure to select +and evolve test cases towards new branches. Finally, we +develop a branch search algorithm to discover rare and +vulnerable branches, and propose an energy allocation +mechanism to concentrate on these critical branches. +• We evaluate IR-Fuzz over large-scale real world smart +contracts, and empirical results show that the proposed +techniques are indeed useful in achieving high branch +coverages. IR-Fuzz surpasses state-of-the-art fuzzers and +overall provides interesting insights. As a side contribu- +tion, we construct a large benchmark dataset for evaluat- +ing smart contract fuzzing approaches. Our implementa- +tion and dataset are released, hoping to inspire others. +II. RELATED WORK +A. Smart Contract Vulnerability Detection +Since blockchain endows smart contracts with unalterable +nature, there is no way to patch the vulnerabilities of a smart +contract without forking the blockchain (almost an impossible +mission), regardless of how much money the contract holds +or how popular it is [2], [24]–[26]. Therefore, it is critical +to conduct security vetting for smart contracts, especially +before their deployments. Early works for smart contract +vulnerability detection employ formal verification techniques. +For example, [14] introduces a framework to compile smart +contracts to EVM bytecode and then put them into an existing +verification system. [27] proposes a formal model and verifies +smart contracts using the Isabelle/HOL tool. Further, [28] +and [29] define the formal semantics of contracts using the F* +framework and the K framework, respectively. Although these +frameworks provide strong formal verification guarantees, they +are still semi-automated and yield high false positives. Another +stream of works rely on symbolic execution methods, such +as Oyente [15], Slither [30], and Securify [31]. Oyente is +one of the pioneering works to perform symbolic execution +on smart contracts, which checks bugs based on expert- +defined rules. [30] converts smart contracts into intermediate + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +3 +representations and conducts static analysis to detect vulnera- +bilities. Whereas symbolic execution is a powerful technique +for discovering bugs, it still suffers from the inherent problem +of symbolic execution path explosion. +Recent efforts resort to using fuzzing techniques for smart +contract vulnerability detection. ContractFuzzer [20] is the first +to apply fuzzing techniques to smart contracts and identi- +fies vulnerabilities by monitoring runtime behaviors during +fuzzing. ReGuard [32] and Harvey [21] are dedicated to +generating a number of test cases that cover as many paths as +possible to trigger a vulnerability. ILF [22] and sFuzz [16] aim +to design a feedback-based seed mutation strategy. Despite the +practicality of fuzzing techniques, existing fuzzers still have +difficulties in achieving high coverage and fuzzing efficiency. +Instead, our work alleviates the issues by carefully designing +a sequence generation strategy, a seed optimization paradigm, +and an energy allocation mechanism. +B. Greybox Fuzzing +Fuzzing techniques have been proven as an effective way +to discover software vulnerabilities. According to how much +information is available about the program under test [33], +fuzzing techniques can be cast into three categories: whitebox, +blackbox, and greybox [34]–[36]. Put succinctly, blackbox test- +ing conducts fuzzing without knowing any internal structure +of the target program. In contrast, whitebox testing performs +fuzzing while having full knowledge about the internal archi- +tecture of the target program. Greybox fuzzing stands in the +middle of blackbox fuzzing and whitebox fuzzing, where we +have partial knowledge of the internal structure of the target +program. Particularly, greybox fuzzing can be further divided +into two groups. +One spectrum of works [37], [38] aim at covering as many +paths or branches as possible, expecting to reveal a bug in +the program, namely coverage-guided greybox fuzzing. AFL, +one of the most well-known fuzzers, employs the lightweight +instrumentation technique and genetic algorithm to improve +coverage [39]. Some other researchers [16], [40] increase +code coverage by smartly selecting and mutating test cases. +Typically, these methods improve coverage by generating +as many new test cases as possible to traverse previously +uncovered program paths. Another spectrum of works [35], +[37], [41] are designed to direct greybox fuzzing towards +a set of specific target locations, termed targeted greybox +fuzzing. There is a number of greybox fuzzers that focus on +specific program locations, e.g., low-frequency or uncovered +branches. For example, [19] utilizes a power schedule to +collect feedback information and steer fuzzing towards target +locations. AFLGo [35] calculates the distance between entry +points and buggy code in the control flow graph, guiding +seed mutation to cover the target locations. Overall, targeted +greybox fuzzing generates test cases to reach certain target +locations, attempting to further trigger a bug. +III. MOTIVATING EXAMPLE +As a motivating example, we present a real world smart +contract GuessNum, which implements a gambling game on +contract GuessNum { +mapping(address => uint256) userBalance; +uint256 prizePool; +constructor () payable { prizePool = msg.value; } +/* Initialize the prize pool */ +function guess(uint256 num) payable external { +uint256 luckyNum = uint256(keccak256(abi.encodePacked( +block.difficulty, now))); +/* Generate a lucky number */ +luckyNum = luckyNum % 100; +prizePool = SafeAdd(prizePool, msg.value); +/* Put funds into the prize pool */ +if (num == luckyNum && msg.value == 50 finney) { +/* If a player guessesit, he obtains 40 times the participant fee */ +userBalance[msg.sender] = SafeAdd(userBalance[msg.sender], msg.value*40); +} +} +function getReward() payable external { +if (userBalance[msg.sender] < prizePool && userBalance[msg.sender] > 0) { +msg.sender.call.value(userBalance[msg.sender])(); /* Reentrancy bug */ +prizePool = SafeSub(prizePool, userBalance[msg.sender]); +userBalance[msg.sender] = 0; +} +} +} +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +Fig. 1 A real-world smart contract written in Solidity. +Ethereum [4]. Fig. 1 shows the simplified code2 of GuessNum, +which is written in Solidity. The contract realizes a guess +number game that users play by submitting their guesses along +with participation fees. The fee is fixed to 50 finney each time +and is poured into the prize pool, i.e., prizePool. Function +constructor() runs only once when the contract is created, and +it puts the funds of the contract’s owner into the prize pool. +A player who wants to submit a guess can invoke function +guess(), which compares the received guess number with the +randomly generated lucky number, i.e., luckyNum (line 11). If +guess number exactly matches the luckyNum, the player will +obtain 40 times the participation fee in return (line 13). Players +can get rewards by calling function getReward(). +Vulnerability. +This GuessNum contract, unfortunately, +suffers from a classical reentrancy vulnerability. From line +18 of Fig. 1, we observe that function getReward() invokes +call.value to transfer money to the user. However, due to +the default settings of smart contracts, the transfer operation +will automatically trigger the fallback function of the recipient +contract. Therefore, an attacker may set a malicious second- +time invocation to getReward() in his fallback function for +stealing extra money. Since getReward() waits for the first-time +transfer to finish, the balance of the attacker is not reduced yet +(i.e., the user balance reduction operation at line 19 is behind +call.value and is not executed yet). Function getReward() thus +may wrongly believe that the attacker still has enough balance +and transfers money to the attacker again. +Limitation of Existing Fuzzers. +Interestingly, this simple +smart contract reveals three key challenges for most existing +fuzzers to expose vulnerabilities. (1) The order of function +invocations is critical. We observe that if the conditions at +line 11 are not satisfied (namely the then-branch at line 13 is +not reached), then the second condition at line 17 will never +be met either (because userbalance[msg.sender] is equal to +2Address: 0xd5e94f6350c7911015dd120b0b006420b6e85a58 + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +4 +Sequence Generation +Source Code +.Json File +Compile +Data Flow Analysis +Seed Evolution +Sequence Prolongation +Seed Selection and +Mutation +Bug Report +Fuzzing Execution Results +Branch Analysis +Customized Energy +Schedule +Source Code +Analysis +Energy Allocation +Vulnerability Analysis & Report +- Function Invocation + Sequence +- New Test Cases +Branch Distance +Measure Technique +- High-Quality +Test Suite +Target Location +- Rare Branches +- Vulnerable Branches +- Feedback Result +Security Vulnerability Patterns + - TSInvocation + - TSContaminate + - TSRandom + - ... +- Initial Test Cases +Order +Fig. 2 A high-level overview of IR-Fuzz. IR-Fuzz has four main components, including (1) Sequence Generation, (2) +Seed Optimization, (3) Energy Allocation, and (4) Vulnerability Analysis and Report. +0). As such, the fuzzer will be unable to reach the then- +branch at lines 18–20. (2) Generating a test case to satisfy +the second condition at line 11 (msg.value == 50 finney) is +difficult. More specifically, the variable msg.value has a size +of 32 bytes. Thus, when we generate a random value for +msg.value in fuzzing, we have only +1 +2256 probability to obtain +the value 50 to meet the condition. Indeed, existing fuzzers like +AFL [39] are shown to have difficulties in electing a test case +to enter the then-branch at line 13. (3) Vulnerable branches that +may contain vulnerabilities only take a small fraction of the +program. For example, lines 18–20 are vulnerable code which +only exists in one branch. Current fuzzers [22], [23] tend +to treat each branch equally, which may fail to discover the +vulnerability due to insufficient fuzzing resource allocation. +Fuzzing Policy. +We embrace three key designs in IR- +Fuzz to tackle the challenges. (1) IR-Fuzz leverages the +variable read and write dependencies between functions to +generate the ordered function invocation sequence. It further +extends the ordered sequence with another ordered sequence to +explore more complex states. Specifically, this contract (Fig. 1) +has two global variables, i.e., userBalance and prizePool, +which both appear in functions guess() and getReward(). By +analyzing read and write dependencies of the global variable +userBalance, IR-Fuzz recognizes that function getReward() +depends on function guess(), awaring that guess() should be +called before getReward(). Consequently, IR-Fuzz generates +the function invocation sequence as: guess()→getReward(). (2) +IR-Fuzz adopts a branch distance-based schema to select test +cases according to how far a test case is from satisfying the +condition of a just-missed branch. For example, the distance of +reaching the just-missed branch (i.e., then-branch at line 13) +is calculated as |msg.value - 50| since the branch condition +at line 11 is msg.value == 50. Intuitively, the test case has +a higher probability to enter the just-missed branch as the +distance decreases. With the guidance of distance measure, IR- +Fuzz iteratively evolves test cases to get increasingly closer to +satisfying the branch condition at line 11. (3) IR-Fuzz engages +a branch search algorithm to pick out vulnerable (e.g., then- +branch at lines 18–20) and rare branches, and then formulates +an energy schedule to expend more fuzzing resources on these +important branches. In our experiments, after only 26s, IR- +Fuzz generates a test case to reach the then-branch at line 18 +and exposes the reentrancy vulnerability. +IV. METHOD +Overview. +The overall architecture of IR-Fuzz is outlined +in Fig. 2. Generally, IR-Fuzz consists of four key components: +• Sequence Generation: Given that a contract might contain +multiple functions, to explore their possible function in- +vocation sequences, IR-Fuzz first analyzes the data flow +dependencies between functions. Then, it defines a rule to +compute the order priority of each function, and generates +a function invocation sequence that successively calls the +ordered functions. Further, IR-Fuzz adopts a prolongation +technique to extend the sequence, driving the fuzzer to dive +into deeper states. +• Seed Evolution: To guide seed mutation so that the gen- +erated cases could reach a target branch, IR-Fuzz utilizes +a branch distance-based measure to select and evolve test +cases iteratively according to how far a test case is from +satisfying the condition of the target branch. +• Energy Allocation: To further take care of the rare branches +and branches that are likely to have vulnerabilities, IR-Fuzz +introduces a branch search algorithm to analyze exercised +branches and picks out those important branches. Then, IR- +Fuzz formulates a customized energy schedule and utilizes +two rules to flexibly guide fuzzing energy allocation towards +these critical branches. +• Vulnerability Analysis and Report: IR-Fuzz analyzes the +generated logs and refers to vulnerability-specific patterns +to discover vulnerabilities. Bug reports are generated for +further manual inspections. +In what follows, we will elaborate on the details of these +components one by one. +A. Sequence Generation +Presented with the multiple functions of a smart contract, +existing methods tend to generate a function invocation se- +quence by randomly picking a function each time. Scrutinizing +12K real-world smart contracts, we empirically observe that +the state of a smart contract is often captured by the state +of its global variables, and different functions do share and +operate differently on the global variables. Some functions + +三IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +5 +perform ‘read’ operations on the variables while some other +functions may perform ‘write’ operations on the variables. +Generating function invocation sequence randomly ignores +such connections between functions. The functions that per- +form ‘write’ operations could change the state of the contract +while the functions that perform only ‘read’ operations are +unable to change the state. Therefore, we assign higher order +priority to functions that perform ‘write’ operations so that we +may explore a broader contract state space and cover more +branches. It is worth mentioning that initializing a variable +will not affect the method. +Sequence Ordering. +Motivated by this, we propose to +model the global variable read and write dependencies be- +tween functions. Specifically, we first characterize the source +code of a smart contract into an abstract syntax tree [42], +from which we extract variable access operations such as +assignments and comparisons. Then, we leverage a data flow +analyzer to capture the read and write dependencies of global +variables between functions. Afterwards, we calculate the +order priority (OP) of each function and sort them according to +their OPs. Finally, we generate the ordered function invocation +sequence, i.e., the invocation sequence that calls the sorted +functions successively. +Formally, we denote the set of functions in a smart contract +as F = {F1, F2, ..., FN}, and global variables that appear +in function Fi as V = {vi1, vi2, ..., viM}, where M is the +number of global variables appear in Fi and vik represents +the k-th global variable in Fi. Specifically, each variable vik +has a unique identifier (i.e., variable name), which is denoted +as vID +ik . To further indicate the operation that Fi exerted on vik, +we use vop +ik = 1 to represent that Fi performs read operation +on vik, and vop +ik = 0 to denote that Fi conducts write operation. +Now, we define the order priority of functions as below. +Rule 1: Read & Write Dependency. +When a global +variable appears in two different functions, the function that +executes write operation on the variable should be called +earlier than the function that executes read operation. Put +differently, given global variables vik in Fi and vjn in Fj +(where vID +ik = vID +jn ), we suggest that Fi should have a higher +order priority (OPFi > OPFj) when vop +ik = 0 and vop +jn = 1. +Guided by this, we may compute the order priority of each +function by converting this rule into the following formula, +which sums up the analysis on read & write dependencies +(Rule 1) of all global variables in different functions. +OPi = +Mi +� +k=1 +Mj +� +p=1 +vop +jp(1 − vop +ik ) · cmp(vik, vjp) +1 ≤ i, j ≤ N +&& +i ̸= j +(1) +where N is the number of functions in the contract. Mi and +Mj denotes the number of the appearance of global variables +in Fi and Fj, respectively. Notably, cmp(vik, vjp) compares +the identifier of global variables in two different functions, +which is given by: +cmp(vik, vjp) = +� +1, +vID +ik = vID +jp +0, +vID +ik ̸= vID +jp +(2) +where vik denotes k-th variable of Fi and vjp represents p-th +variable of Fj. +Mathematically, in Eq.(1), a function accumulates one order +priority score only if it writes on a global variable and +the variable is read by another function. In this context, +the functions that conduct write operations on more global +variables get higher priority and are put in the front of +the function invocation sequence. This drives the fuzzer to +exercise more on the functions that could change the states +and boost the fuzzing by encouraging it to encounter more +states and reach more branches. Interestingly, our experimental +results show that branch coverage is significantly improved +with such sequence ordering (see §V-E). Here, we take the +contract of Fig. 1 as an example to illustrate the order priority +calculation of each function. From Fig. 1, we can observe +that this contract has two global variables, i.e., userBalance +and prizePool. Function guess() performs a read operation +and a write operation on the two variables, respectively. +We represent the variables that appear in function guess() +as Vguess = {vguess1, vguess2, vguess3, vguess4}, where vguess1 = +vguess2 = prizePool, vop +guess1 = 1, vop +guess2 = 0, and vguess3 = +vguess4 = userBalance, vop +guess3 = 1, vop +guess4 = 0. Meanwhile, +cmp(userBalance, prizePool) = 0, while cmp(userBalance, +userBalance) = cmp(prizePool, prizePool) = 1. Similarly, the +global variables appear in getReward() are denoted as VgetReward += {vgetReward1, vgetReward2, ..., vgetReward8}, where vop +getReward1 = ... += vop +getReward6 = 1, and vop +getReward7 = vop +getReward8 = 0. As such, +according to Eq.(1), we calculate the order priority of function +guess() as below. +OPguess =vop +getReward1(1 − vop +guess4) + vop +getReward2(1 − vop +guess2) ++vop +getReward3(1 − vop +guess4) + vop +getReward4(1 − vop +guess4) ++vop +getReward5(1 − vop +guess2) + vop +getReward6(1 − vop +guess4) +=6 +(3) +For function getReward(), we calculate its order priority as: +OPgetReward =vop +guess1(1 − vop +getReward7) ++vop +guess3(1 − vop +getReward8) +=2 +(4) +According to the order priority calculation, we can see that the +order priority of calling function guess() is greater than that +of function getReward(). Therefore, IR-Fuzz finally generates +the function invocation sequence as: guess() → getReward(). +Sequence Prolongation. +Another important insight is that +a smart contract might go through many different states dur- +ing its lifecycle. However, current methods typically conduct +fuzzing starting from the initial state of the contract, which +expends too much energy revolving around the initial state +and is usually incapable of unearthing bugs triggered by other +states. These facts inspire us to explore richer starting states +via sequence prolongation. Particularly, we first exercise the +ordered function invocation sequence S using various test +cases, which result in different states of the contract. We +then engage in the invocation sequence S again but execute +S starting from these different states, i.e., appending a new +sequence S after S. +For instance, presented with the crowdfunding contract in +Fig. 3, we observe that covering the if-branch at line 21 for +traditional fuzzers is difficult, which requires at least twice + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +6 +contract CrowdFunding { +uint256 goal; +uint256 raised; +uint256 phase; +address beneficiary; +constructor () { +beneficiary = msg.sender; +goal = 300 ether; +raised = 0; +phase = 0; +} +/* 0: Active 1: Finished */ +function donate(uint256 donations) payable public { +/* Check if the crowdfunding goal is reached */ +if (raised < goal) { raised += donations; } +else { phase = 1; } +} +function withdraw() public { +/* The crowdfunding goal has been reached */ +if (phase == 1) { +beneficiary.delegatecall(bytes4(keccak256(“transfer(uint256)”)), raised); } +} +} +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +Fig. 3 An example contract for illustrating how the +sequence prolongation technique is used in IR-Fuzz. +invocations of function donate(). Particularly, this contract +implements a simple crowdfunding project that allows users +to donate money by calling donate(). The constructor() sets +the goal of crowdfunding as 300 Ether (line 9), and the +raised money is initialized to 0 (i.e., raised = 0 at line 10). +The status of the crowdfunding process is initialized to 0 +(i.e., phase = 0 at line 11), which represents unaccomplished. +According to Rule 1, we realize that the invocation to func- +tion donate() should have a higher order priority than that +of function withdraw(). As a result, the function invocation +sequence is generated as: donate()→ withdraw(). However, +such a sequence fails to satisfy the branch condition at line +21 because call function donate() once cannot enter the else- +branch at line 16 to set phase = 1. To reach the else-branch +at line 16, function donate() needs to be called at least twice. +Namely, the required amount of money for the crowdfunding +is accomplished at the first call (i.e., raised ≥ goal), and +the second call enters the else-branch at line 16. Therefore, +to explore deeper states, we propose to prolong the function +invocation sequence. +In contrast to most fuzzing methods that conventionally +call each function only once in the invocation sequence, we +further extend the ordered function invocation sequence S by +appending the same invocation sequence to S, namely S → S, +such that the second sequence starts its execution from a +different state rather than the initial state. To explore different +starting states for the second sequence, the first sequence is +presented with different sets of parameters, which lead to +different inner statuses. Technically, we first try to sufficiently +exercise sequence S with various different input parameters. +A different set of input parameters to S translates to a variant +of S, denoted as Sj. Then, we concatenate two sequences Si +and Sj that have different input parameters and exercise the +new concatenated sequence. The following rule formulates the +Algorithm 1: SEED ITERATIVE OPTIMIZATION +1 currentTestSuite ← ∅; +2 currentTestCase ← initialTestCase(); +3 while ¬Terminated() do +4 +Let BtestCase be covered branches by +currentTestCase; +5 +Let Bmiss be just-missed branches in +currentTestSuite; +6 +for br ∈ BtestCase do +7 +if br is new branch then +8 +currentTestSuite.ADD(currentTestCase); +9 +for br ∈ Bmiss do +10 +for seed ∈ currentTestSuite && +seed ̸= currentTestCase do +11 +dist_1 ← distance(currentTestCase, br); +12 +dist_2 ← distance(seed, br); +13 +if dist_1 < dist_2 then +14 +currentTestSuite.REMOVE(seed); +15 +currentTestSuite.ADD(currentTestCase); +16 +energy ← 0; +17 +MutationEnergy ← AssignMutationEnergy(); +18 +while energy < MutationEnergy do +19 +testCase ← SelectInput(currentTestSuite); +20 +newCase ← Mutation(testCase); +21 +if ¬RepeatCheck(currentTestSuite, newCase) +&& ¬V alidityCheck(newCase) then +22 +currentTestCase.ADD(newCase); +23 +log ← FuzzInput(newCase); +24 +energy ← UpdateEnergy(log, energy); +25 return currentTestSuite: A set of high-quality test cases +sequence pair selection to generate a new prolonged sequence, +which constrains that the input parameters of Si and Sj should +be quite different. +Rule 2: Sequence Pair Selection. +(1) When the number +of required function input parameters in the sequence is no +less than 2, we select the sequence pairs iff at least one input +paramter is different between the two sequences. For example, +given three sequences S1: F1(x1) → F2(), S2: F1(x2) → +F2(), and S3: F1(x2) → F2(), pairs P1(S1, S2) and P2(S1, +S3) are selected. In contrast, pair P3(S2, S3) is not selected +since S2 and S3 have the same parameters. (2) When the +number of required function input parameters is larger than 2, +we select the sequence pairs iff at least two input parameters +are different between the two sequences. For example, given +three sequences S1: F1(x1, y1) → F2(z1), S2: F1(x1, y2) → +F2(z1), and S3: F1(x2, y2) → F2(z2), pairs P1(S1, S3) and +P2(S2, S3) are selected. +Upon each sequence pair selection, we obtain a prolonged +function invocation sequence. For the crowdfunding contract +demonstrated in Fig. 3, IR-Fuzz first generates two sequences, +e.g., S1: donate(300) → withdraw() and S2: donate(200) → +withdraw(). According to Rule 2, IR-Fuzz combines S1 and S2 +as a new prolonged sequence S: donate(300) → withdraw()→ +donate(200) → withdraw(). Since the goal of crowdfunding is +300, the first call to donate(300) in S satisfies the condition +of the if-branch at line 15, and the second call to donate(200) +reaches the else-branch at line 16 (i.e., set phase = 1). The +new sequence thus can reach the then-branch at line 22 on the + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +7 +second call to withdraw() and expose the potential Ether frozen +vulnerability. Promisingly, with the prolongation technique, +IR-Fuzz greatly expands the scope of explored states and +branches. Note that the sequence prolongation technique yields +little impact on the overhead of the fuzzer. +B. Seed Evolution +To fuzz a function invocation sequence, the most intuitive +and direct way is to generate test cases randomly. Despite +its simplicity, this strategy is not favorable for reaching unex- +plored conditional branches due to its random nature. IR-Fuzz, +instead, incorporates a seed evolution paradigm to refine test +cases iteratively. The seed evolution framework is summarized +in Algorithm 1. First, IR-Fuzz initializes an empty test suite +and a set of test cases (lines 1-2). Then, the first loop from +lines 6 to 8 performs seed selection. Whenever a test case +covers a new branch (i.e., any branch not covered by test +cases in the test suite), it is added to the suite. Next, the loop +from lines 9 to 15 evolves test cases iteratively. Particularly, +we propose a branch distance-based measure to select those +test cases which are closer to satisfying the conditions of new +branches. Thereafter, the loop from lines 18 to 24 executes +seed mutation, in which function Mutation() generates the +mutated test cases based on the test cases selected from the +suite (lines 19-20). Then, we adopt seed verification strategies +to guarantee the validity of mutated test cases (line 21). This +mutation process continues until a mutation energy upper- +bound is reached (line 18). Finally, a new test suite that +contains a set of high-quality test cases is shaped. In what +follows, we present the technical details of seed selection and +seed mutation, respectively. +Seed Selection. +In IR-Fuzz, we first try a classical seed +selection strategy. That is, IR-Fuzz monitors the execution of +test cases and records the branches that each test case traverses. +A test case is added into the test suite as long as it covers a new +branch, i.e., a branch which is not covered by any test case in +the suite. Empirically, our experimental results show that this +strategy could reveal a number of branches. However, it is +still quite inefficient in reaching those complex branches with +strict conditions. For example, the probability of satisfying +the second condition (msg.value == 50 finney) at line 11 of +Fig. 1 is +1 +2256 , which is extremely low. To meet such strict +branch conditions, we design a novel seed selection strategy. +Inspired by [16], we adopt a distance function dist(T, br) +to compute a branch distance indicating how far a test case +is from covering a just-missed branch (i.e., uncovered then- +branch). More specifically, let br be a just-missed branch, +which is not covered by any test case T. We suppose that br +is a branch of condition C. Note that C can be either x==k, +x! = k, x ≤ k, x < k, x ≥ k or x > k, where x and k are +variables or constants. The function dist(T, br) is given by: +dist(T, br) = +� +� +� +� +� +� +� +� +� +|x − k|, +if C is x == k +1, +if C is x ! = k +max(x − k, 0) +if C is x ≤ k or x < k +max(k − x, 0) +if C is x ≥ k or x > k +(5) +where x and k are extracted from the stack information +recorded by IR-Fuzz. Intuitively, dist(T, br) is defined such +Algorithm 2: BRANCH SEARCHING +Input: Program P, Test case case, Vulnerable statements T +Output: Brare and Bvulnerable +1 br ← FuzzRun(P, case); +2 Brare ← ∅; +// Rare Branches +3 Bvulnerable ← ∅; +// Vulnerable Branches +4 R ← 0; +5 i ← 0; +6 while i < |br| do +7 +if IsConditionInstruction(i, Cb) then +8 +R ← R + 1, bpre ← br[0...i + 1]; +9 +c, state ← StateInference(bpre); +10 +if V ulnerableStatementReached(P, state, c, T ) +then +11 +Bvulnerable.ADD(br); +12 +i ← i + 1; +13 if R ⩾ 2 then +14 +Brare.ADD(br); +that the closer a test case T is from satisfying the condition +of branch br, the smaller the distance is. For example, a test +case with msg.value = 100 is closer to satisfying the condition +msg.value == 50 than a test case with msg.value = 10,000. +For each just-missed branch, IR-Fuzz selects a test case has +the smallest dist(T, br). With the feedback of the branch +distance measurement, IR-Fuzz can quickly approach complex +branches guarded by strict conditions, improving the overall +branch coverage. Note that all selected test cases are added to +the test suite and transferred to the seed mutation phase for +generating new test cases. +Seed Mutation. +Seed mutation plays an important role +in enriching the test cases. In IR-Fuzz, we refer to several +mutation strategies from AFL and introduce new ones tailored +for smart contracts. Particularly, we preferentially mutate those +test cases with smaller branch distances. +Given a test case encoded in the form of a bit vector, +sFuzz [16] engages a set of mutation operators to generate +new test cases, such as bit flipping, interest value insertion, and +key-value insertion. IR-Fuzz additionally ensures the generated +test cases are valid by advocating two principles. (1) IR-Fuzz +checks the validity and integrity of the mutated test cases by +using a bit verification approach, which sets random bits of +a given seed to random values while keeping other bits of +the seed unchanged. Thereafter, IR-Fuzz saves a new test case +as a new seed based on whether the new test case detects a +new branch. Moreover, IR-Fuzz will discard invalid test cases +which lead to fuzzing crashes or bring much overhead to the +fuzzer. (2) IR-Fuzz removes duplicate test cases by comparing +the mutated cases with the test cases in the test suite. +In practice, we also apply multiple heuristics to save the +mutation energy of IR-Fuzz. For example, any test case in the +test suite that does not discover a new branch after a round of +mutation will be assigned with low priority in the mutation. +IR-Fuzz updates energy according to the generated logs during +fuzzing (lines 23-24 in Algorithm 1). The process of seed +mutation continues until the mutation energy upper-bound is +reached. Each seed is assigned with a priority score which +measures its ability to detect new branches after mutations. + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +8 +contract Cheer { +uint256 num = 0; +function random(uint256 x, uint256 y, uint256 z) returns (uint256) { +if (x % 2 == 0) { +num = 256; +while (x != z) { +num = SafeMul(x, z); +if (x > z) { z = SafeAdd(z, z); } +else { x = SafeAdd(x, x); } +} +if (y % 2 == 0) { +num = uint256(keccak256(abi.encodePacked( +block.number, now))); /* Block number dependency */ +} +} else { +num = 3; } +return num; +} +} +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +Fig. 4 An example contract for illustrating how IR-Fuzz +allocates energy to the target branches flexibly. +C. Energy Allocation +In this subsection, we introduce how IR-Fuzz performs +fuzzing energy allocation. Recall that most existing fuzzers +treat program branches equally, ignoring the fact that vulner- +able code usually takes a tiny fraction of the entire code [18], +[43]. As a result, conventional fuzzers may waste massive +resources in fuzzing normal branches instead of rare branches +and branches that are more likely to possess bugs. To tackle +this problem, we design an energy allocation mechanism, +guiding IR-Fuzz to assign fuzzing resources towards these +important branches. Specifically, this mechanism consists of +two modules: branch analysis and energy schedule. +Branch Analysis. +We remark that the first challenge is +how to pick out the important branches. To address this, we +introduce a branch search algorithm to analyze all branches +discovered during fuzzing and focus on two types of branches: +rare and vulnerable, which are defined as follows. +Definition 1 (Branch): +Given a path p exercised by a test +case in smart contract S, we say that br is a prefix subpath +of p if br is a subpath of p and br begins with the same +starting point as p. Further, br is a branch of p if br is a prefix +subpath of p and br ends with the if-branch or then-branch +of a conditional or recurrent statement (e.g., if, require, for, +while) in S. +Definition 2 (Rare Branch): +We consider a branch br is +a rare branch if br contains at least two nested conditional +statements (e.g, two nested if). Each rare branch is associated +with a rarity factor R, which is set to the number of nested +conditional statements (at the end of this branch). +Definition 3 (Vulnerable Branch): +Given a branch br +and a set of vulnerable statements T +that may introduce +bugs (e.g., block.number and call.value), we say that br is +a vulnerable branch when br contains a vulnerable statement +t ∈ T . +After empirically scrutinizing real-world smart contracts, +we found that over 50% (65% and 86% in our experiments) +of bugs are located in rare and vulnerable branches. To +pick out the two types of branches, we design a branch +analysis algorithm shown in Algorithm 2. Technically, IR-Fuzz +employs the abstract interpreter A to pick out the important +branches. First, A analyzes all branches Br discovered during +fuzzing (line 1). Then, the loop from lines 6 to 12 checks +whether there exists a branch br that reaches the vulnerable +statement t ∈ T and computes the rarity factor R for each +branch. Afterwards, A adds the branches with R ⩾ 2 into +Brare (lines 13-14). Finally, A obtains a set of rare branches +Brare and vulnerable branches Bvulnerable. +Energy Schedule. +We remark that the second challenge is +how to assign resources to these important branches. Towards +this aim, we formulate a customized energy schedule Ω to +manage the fuzzing energy allocation. This schedule adopts +two rules for rare and vulnerable branches, respectively. +Rule 3: Energy Allocation for Rare Branches. +Given +branches br1 with R1 and br2 with R2 in a same path p, +where Ri is rarity factor and R1 < R2. To facilitate that +rare branches with higher rarity factors are more sufficiently +fuzzed, Ω assigns energy ∇1∗E to br1 and ∇2∗E to br2 where +∇1 < ∇2. +Rule 4: Energy Allocation for Vulnerable Branches. +Given a branch br1 in a path p1 and a branch br2 in a path +p2, Ω assigns α∗E energy to br1 (α > 1) and E energy to br2 +when R1 = R2 and br1 ∈ Bvulnerable. Coefficient α controls +the preference degree for vulnerable branches. +We use the example of Fig. 4 to show how IR-Fuzz works +with the energy allocation mechanism. Specifically, IR-Fuzz +first analyzes all discovered branches and picks out br at +line 12 as the vulnerable branch since it contains a vulner- +able statement (i.e., block.number) at line 13. Then, IR-Fuzz +calculates the rarity factor R of this branch, where R = 2. +As a result, IR-Fuzz assigns energy (α + ∇) ∗ E to branch +br, according to Rule 3 and Rule 4. In our experiments, an +interesting observation is that state-of-the-art tools like sFuzz +cannot reach the if-branch at line 12 since they waste too much +energy in fuzzing the while branch at line 7. In contrast, IR- +Fuzz successfully covers this branch and exposes the block +number dependency vulnerability at line 13 in 8s on average +(between 4s and 12s in 5 runs). +Moreover, IR-Fuzz also utilizes the feedback of energy +allocation to guide seed mutation. For example, the test cases +which cover the vulnerable branches will be selected and +fuzzed first. Inspiringly, with the energy allocation mechanism, +IR-Fuzz can flexibly assign fuzzing resources to the rare and +vulnerable branches, increasing overall fuzzing efficiency and +branch coverage. +D. Vulnerability Analysis and Report +In this stage, IR-Fuzz turns to vulnerability analysis and +report generation, which reveals vulnerabilities in smart con- +tracts and generates a detailed bug report for further manual +confirmation. In particular, we investigate previous works +(e.g., [2], [20], [44]) and define the specific patterns for eight +types of vulnerabilities, namely timestamp dependency, block +number dependency, dangerous delegatecall, Ether frozen, + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +9 +Execution & Analyze +Execution & Analyze +Smart Contract + Generate compiled JSON file +Seed selection with +branch-distance-based feedback + Branch energy allocation +Seed mutation with +energy-allocation-based feedback +Bug reports +Data flow +dependency +Function invocation +sequence +Seed Selection & Mutation + Vulnerability analysis and report +Seed evolution + Generate function invocation sequence +b1 +b2 +bn +1.{ +2. bytecode:{ +3. ... +4. }； +5. abi:{ +6. ... +7. }; +8. ... +9.} +1 +1 +4 +1 +... +b1 +b2 +bn +2 +3 +2 +k-1 +k +Rare +Vulnerable +b1 : e1 + +b2 : e2 + ... +bn : en +bx: emax +sx(seed of bx) +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +k +k-1 +... +Energy schedule +Branch +Energy +b1 +e1 +... +bn +en +Abstract syntax +tree +Compile +guess() +getReward() +Test Suite +Log information +Vulnerability-specific +patterns +Log +- CALL +- ... + Branch analysis +Order +Test Case +Test Case +Fig. 5 The workflow of IR-Fuzz to reveal the reentrancy vulnerability in the real-world contract of Fig. 1. +unchecked external call, reentrancy, integer overflow, and dan- +gerous Ether strict equality. We have implemented a pattern +analyzer to handle these patterns. IR-Fuzz analyzes the fuzzing +results and reveals bugs with the assistance of the pattern +analyzer. In the following, we show an example of how IR- +Fuzz exposes a reentrancy vulnerability using patterns. +Vulnerability Pattern Example. +Reentrancy vulnerability +is considered as an invocation to call.value that can call +back to itself through a chain of calls. That is, the in- +vocation of call.value is successfully re-entered to perform +unexpected repeat money transfers. Specifically, we design +two patterns to expose the reentrancy vulnerability. The first +pattern CALLValueInvocation checks if there exists an in- +vocation to call.value in the contract. The second pattern +RepeatedCallValue concerns whether a specific function with +call.value invocation is called repeatedly during fuzzing. IR- +Fuzz reports that a function has a reentrancy vulnerability +if it fulfills the combined pattern: CALLValueInvocation ∧ +RepeatedCallValue. +E. IR-Fuzz Workflow Illustration with an Example +In this subsection, we take the contract of Fig. 1 as an +example to show the workflow of IR-Fuzz on revealing a +reentrancy vulnerability. The workflow consists of six steps, +illustrated in Fig. 5. Given the contract with source code as +input, IR-Fuzz first compiles the source code to the JSON +file, which consists of EVM bytecode and application binary +interface (ABI) (‚ in Fig. 5). Second, IR-Fuzz extracts the +abstract syntax tree and captures the data flow dependencies +of global variables. By analyzing these dependencies, IR- +Fuzz infers the function invocation sequence as: guess() → +getReward() (ƒ in Fig. 5). Thirdly, IR-Fuzz generates test +cases for the two function calls and adds high-quality test cases +into the test suite based on the branch distance-based measure. +As such, IR-Fuzz effectively generates test cases to cover new +branches („ in Fig. 5). Furthermore, IR-Fuzz performs the +branch analysis to pick out the important branches. Then, it +customizes an energy schedule to assign fuzzing resources (… +in Fig. 5). IR-Fuzz utilizes the feedback of energy allocation +to guide seed selection and mutation. After several rounds +of fuzzing, IR-Fuzz reaches the then-branch at line 18 and +triggers the execution of the transfer function (i.e., call.value). +We develop an attack contract generator, which simulates an +attack contract that calls the transfer function getReward() +again. IR-Fuzz records the instructions into a log file († +in Fig. 5). Finally, IR-Fuzz analyzes the log and determines +whether the call.value was called multiple times, exposing the +reentrancy vulnerability with the assistance of vulnerability- +specific patterns (‡ in Fig. 5). Besides fuzzing the ordered +invocation sequence, IR-Fuzz further prolongs the sequence +to explore other complex states. +V. EXPERIMENTS +In this section, we conduct extensive experiments to evaluate +IR-Fuzz, seeking to address the following research questions. +• RQ1: Can IR-Fuzz effectively detect contract vulnerabili- +ties? How is its performance against state-of-the-art tools? +• RQ2: Does IR-Fuzz achieve higher branch coverage than +existing methods? +• RQ3: How efficient is IR-Fuzz in fuzzing smart contracts +and generating test cases compared with other fuzzers? +• RQ4: How much do different components of IR-Fuzz +contribute to its performance in branch coverage and vul- +nerability detection accuracy? +We first introduce the experimental settings, then proceed to +answer the above questions. We also present a case study to +allow for a better understanding of the proposed approach. +A. Experimental Setup +Implementation. +IR-Fuzz in total contains 9K+ lines of +C++ code, which is released for public use at https://github. +com/Messi-Q/IR-Fuzz . We implemented IR-Fuzz on the basis +of sFuzz [16] (a state-of-the-art smart contract fuzzer). +Baselines. +In the experiments, we include seven open- +source methods that either have a high number of citations + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +10 +TABLE I Summary of vulnerability types supported by +state-of-the-art methods. TP is short for timestamp de- +pendency; BN represents block number dependency; DG +represents dangerous delegatecall; EF represents Ether +frozen; UC represents unchecked external call; RE rep- +resents reentrancy; OF represents integer overflow; SE +represents dangerous Ether strict equality. +Methods +Vulnerability Type +#Citation or #GitHub Stars +Publication +TP +BN +DG +EF +UC +RE +OF +SE +Oyente [15] +✓ +✓ +✓ +1,780 citations +CCS’16 +Osiris [45] +✓ +✓ +✓ +182 citations +ACSAC’18 +Securify [31] +✓ +✓ +✓ +604 citations +CCS’18 +ILF [22] +✓ +✓ +✓ +✓ +105 citations +CCS’19 +sFuzz [16] +✓ +✓ +✓ +✓ +✓ +✓ +✓ +91 citations +ICSE’20 +Mythril [46] +✓ +✓ +✓ +✓ +2,900 GitHub stars +White Paper +ConFuzzius [23] +✓ +✓ +✓ +✓ +✓ +✓ +45 GitHub stars +EuroS&P’21 +IR-Fuzz +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +– +– +or receive many stars in GitHub. The methods are summa- +rized in Table I, where we illustrate the vulnerability types +that they can detect, their numbers of citations or GitHub +stars, and their publication information. For fuzzing tools, +we select ConFuzzius [23], ILF [22], and sFuzz [16], which +achieve state-of-the-art performance and support at least four +vulnerability types on smart contracts. For static analysis +tools, we select Mythril [46], Oyente [15], Osiris [45], and +Securify [31], which are well-known vulnerability checkers for +smart contracts. We compare IR-Fuzz with them in terms of +branch coverage, effectiveness, and efficiency. All experiments +are conducted on a computer equipped with an Intel Core i9 +CPU at 3.3GHz, a GPU at 2080Ti, and 64GB Memory. Each +experiment is repeated ten times, we report the average results. +Dataset. +We obtain the dataset by crawling Etherscan [47] +verified contracts, which are real-world smart contracts de- +ployed on Ethereum Mainnet. In practice, we removed 5,074 +duplicate contracts by comparing the hash of the contract bi- +nary code. Our final dataset contains a total 12,515 smart con- +tacts that have source code. As listed in Table I, we focus on +eight types of vulnerabilities in the dataset, namely timestamp +dependency (TP), block number dependency (BN), dangerous +delegatecall (DG), Ether frozen (EF), unchecked external call +(UC), reentrancy (RE), integer overflow (OF), and dangerous +Ether strict equality (SE). We deployed all smart contacts of +the dataset to a local Ethereum test network for experiments. +For the ground truth labels of smart contracts, we define +vulnerability-specific patterns for each kind of vulnerability to +give a preliminary label and then manually check whether a +smart contract in the dataset indeed has a certain vulnerability. +In particular, using the defined vulnerability-specific patterns +(e.g., keyword matching), we could find smart contracts that +may have vulnerabilities and save our time on labeling those +contracts that are safe (e.g., a contract with no ‘call.value’ +invocation will not have reentrancy vulnerabilities). +B. Effectiveness (RQ1) +First, we benchmark IR-Fuzz against existing vulnerability +detection methods. We count the number of smart contracts +that have vulnerabilities and are identified by each method, +and present the accuracy, true positives, and false positives of +each method. +Comparing IR-Fuzz to State-of-the-arts. +We first com- +pare IR-Fuzz to other fuzzers and existing static analysis +TABLE II Accuracy comparison (%) on different methods, +including static analysis tools, fuzzers, and IR-Fuzz. ‘n/a’ +denotes that a tool cannot detect the specific vulnerability. +Methods +Vulnerability Type (Accuracy) +TP +BN +DG +FE +UC +RE +OF +SE +Mythril [46] +n/a +89.97 +70.95 +n/a +n/a +95.09 +89.87 +n/a +Oyente [15] +86.86 +n/a +n/a +n/a +n/a +94.61 +74.76 +n/a +Osiris [45] +86.56 +n/a +n/a +n/a +n/a +93.28 +74.80 +n/a +Securify [31] +n/a +n/a +n/a +79.42 +91.24 +91.52 +n/a +n/a +ILF [22] +n/a +87.53 +80.99 +78.65 +94.71 +n/a +n/a +n/a +sFuzz [16] +87.25 +88.37 +83.33 +83.85 +94.26 +95.20 +89.98 +n/a +ConFuzzius [23] +n/a +87.70 +80.47 +78.91 +94.68 +93.33 +77.35 +n/a +IR-Fuzz +90.25 +94.18 +95.33 +95.05 +98.10 +98.77 +98.79 +99.73 +TABLE III True and false positives of each method in +identifying the eight types of smart contract vulnerabilities. +Methods +Vulnerability Type (True / False Positives) +Total +TP +BN +DG +FE +UC +RE +OF +SE +Mythril [46] +n/a +4/63 +20/20 +n/a +n/a +0/62 +10/245 +n/a +34 +Oyente [15] +12/6 +n/a +n/a +n/a +n/a +8/87 +16/637 +n/a +36 +Osiris [45] +4/5 +n/a +n/a +n/a +n/a +12/139 +12/632 +n/a +28 +Securify [31] +n/a +n/a +n/a +0/0 +7/208 +4/194 +n/a +n/a +11 +ILF [22] +n/a +0/103 +8/4 +0/3 +5/82 +n/a +n/a +n/a +13 +sFuzz [16] +23/8 +20/108 +20/7 +20/3 +7/100 +10/68 +3/235 +n/a +103 +ConFuzzius [23] +n/a +20/120 +4/2 +0/2 +8/86 +6/131 +10/565 +n/a +48 +IR-Fuzz +92/5 +26/3 +58/0 +65/0 +83/36 +95/20 +21/10 +45/0 +485 +tools. Quantitative experimental results of each method are +summarized in Table II. From the table, we obtain the fol- +lowing observations. (1) Compared with other methods, IR- +Fuzz is able to identify more vulnerabilities. Inspiringly, IR- +Fuzz has achieved a high accuracy (more than 90%) on +all eight types of vulnerabilities. (2) IR-Fuzz consistently +outperforms state-of-the-art methods by a large margin in +detecting each type of vulnerability. For example, for Ether +frozen vulnerability (EF), IR-Fuzz gains 15.63% and 16.14% +accuracy improvements over Securify and ConFuzzius. These +strong empirical evidences suggest the great potential of IR- +Fuzz to identify smart contract vulnerabilities. We attribute +its superior performance to the key modules proposed, i.e., +sequence generation, seed optimization, and energy allocation, +which boost the capability of IR-Fuzz in improving branch +coverage and hunting vulnerabilities. (3) Promisingly, IR- +Fuzz discovers a new kind of smart contract vulnerability, +i.e., dangerous Ether strict equality (SE). To the best of our +knowledge, this vulnerability cannot yet be detected by current +automatic tools. We also present an illustrative case study on +how our method detects this vulnerability in §V-F. +Analysis of True and False Positives. +To further eval- +uate the effectiveness of IR-Fuzz, we examine the identified +vulnerable contracts to see whether they are true positives or +not. Table III demonstrates the number of vulnerable contracts +discovered by each method, as well as the numbers of true +positives and false positives of each method. (1) From Ta- +ble III, we observe that existing methods have not yet obtained +a high true positive rate on the eight types of vulnerabilities. +For example, for unchecked external call vulnerability (UC), +Securify and sFuzz generate 7 true positives, while ConFuzzius +and ILF obtain 8 and 2 true positives, respectively. This +is mainly due to the reason that conventional tools ignore +handling exceptions for the return value of external calls. +(2) Moreover, we also find that existing methods have high +false positives. For block number dependency vulnerability +(BN), fuzzing tools sFuzz, ConFuzzius, and ILF produce + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +11 +0 +10 +20 +30 +40 +Time (s) +0 +20 +40 +60 +80 +100 +Branch Coverage (%) +IR-Fuzz +sFuzz +ILF +ConFuzzius +(a) Branch coverage of different meth- +ods on small contracts +0 +10 +20 +30 +40 +Time (s) +0 +20 +40 +60 +80 +100 +Branch Coverage (%) +IR-Fuzz +sFuzz +ILF +ConFuzzius +(b) Branch coverage of different meth- +ods on large contracts +Fig. 6 Curves comparison: the tendency of branch coverage +over time on different fuzzers. +over 100 false positives. For integer overflow vulnerability +(OF), 632, 637, and 565 false positives are reported by +Osiris, Oyente, and ConFuzzius, respectively. The high false +positives of these methods may stem from two facts: (i) Most +methods tend to detect vulnerabilities using a few simple but +imprecise patterns, e.g., identifying block number vulnerability +by crudely checking whether there is a block.number statement +in the function; (ii) Many tools conservatively assume that all +arithmetic operations returning a negative value are vulnerable, +resulting in high false positives. +IR-Fuzz reports more true positives than other methods. For +example, for timestamp dependency vulnerability (TP), IR- +Fuzz generates 92 true positives, 88, 80, and 69 more than +Osiris, Oyente, and sFuzz, respectively. In total, IR-Fuzz finds +vulnerabilities in 485 contracts, roughly 4.7 times more than +sFuzz, which ranks the second. For reentrancy vulnerability +(RE), IR-Fuzz produces 95 true positives, which significantly +outperforms the state-of-the-art tool Osiris. More importantly, +IR-Fuzz can precisely detect a new kind of vulnerability (SE) +without reporting any false positives. We attribute the good +performance of IR-Fuzz to the fact that it integrates the three +presented new techniques, which are able to supplement each +other for precise bug detection. In summary, IR-Fuzz can +effectively identify various vulnerabilities in smart contracts, +surpassing existing static analysis tools and fuzzers by a large +margin. +C. Branch Coverage (RQ2) +We now present evaluation results on branch coverage of +IR-Fuzz. We measure the number of distinct branches covered +by the generated test cases in the test suite. Moreover, to +examine the branch coverage on contracts with different sizes, +we follow the settings of previous work [22] and split the +dataset into 1,885 large contracts (≥3,600 instructions) and +10,630 small ones (<3,600 instructions). +We compare with other fuzzers (i.e., sFuzz, ILF, and Con- +Fuzzius). Particularly, we visualize the comparison results on +small contracts in Fig. 6(a) and on large contracts in Fig. 6(b), +respectively. We plot the tendency of branch coverage over +time. It can be seen that IR-Fuzz consistently outperforms +other fuzzers. Quantitatively, IR-Fuzz achieves 90.10% cover- +age on small contracts, 28.20%, 20.10%, and 10.10% higher +than sFuzz, ILF, and ConFuzzius, respectively. On large con- +tracts, IR-Fuzz achieves 19.20%, 14.00%, and 9.10% higher +IR-Fuzz +Mythril +ConFuzziusOyente +ILF +Osirs +sFuzz +Securify +Tool +0 +50 +100 +150 +200 +250 +300 +Time (s) +21.30 +30.06 +33.26 +33.50 +49.88 +53.41 +103.52 +273.30 +(a) Average execution time +20 +40 +60 +80 +100 +120 +Time (S) +0 +5 +10 +15 +20 +25 +30 +35 +40 +Test Cases (K) +sFuzz +IR-Fuzz +(b) Number of generated test cases +Fig. 7 Visual comparison of efficiency on different tools. +coverage, respectively. Moreover, we also observe that IR-Fuzz +reaches the highest coverage with less time required than other +fuzzers. On average, IR-Fuzz spent only 10s to achieve the +highest coverage (i.e., 90.10% on small contracts and 79.10% +on large contracts), while the other three fuzzers spent 18s, +16s, 13s, respectively. +We conjecture that the advantages of IR-Fuzz in achieving +high branch coverage come from three aspects. First, IR-Fuzz +generates the high-quality function invocation sequence by +adopting a dependency-aware sequence generation strategy, +enforcing the fuzzer to tap into richer states. Second, IR- +Fuzz employs a branch distance-based measure to iteratively +optimize the generated test cases, steering fuzzing towards +covering new branches. Thirdly, IR-Fuzz takes into account +the significance of rare branches and branches that are likely +to have vulnerabilities, and designs an energy allocation mech- +anism to flexibly guide fuzzing energy allocation towards these +critical branches. Moreover, IR-Fuzz utilizes the feedback +results generated by the energy allocation mechanism to guide +seed mutation, which further increases branch coverage. +D. Efficiency (RQ3) +In this subsection, we systematically examine the efficiency +of IR-Fuzz and compare it against other methods. +First, we conduct experiments to measure the overhead of +IR-Fuzz by calculating the average execution time on each +contract. We run IR-Fuzz on the whole dataset, revealing that it +spends 21.30s per contract on average. Fig. 7(a) compares IR- +Fuzz to other methods in terms of the average execution time. +From the figure, we observe that IR-Fuzz is significantly more +efficient than others. Particularly, its average execution time is +251s and 82.22s faster than Securify and sFuzz, respectively. +We believe the reasons for the much faster speed of IR-Fuzz +are as follows. (1) IR-Fuzz can quickly infer the ordered +function invocation sequence, accelerating fuzzing execution. +(2) IR-Fuzz adopts the branch distance-based measure to boost +its efficiency in generating test cases, which requires much +fewer mutations to reach a target branch. (3) IR-Fuzz leverages +the energy allocation mechanism to flexibly assign fuzzing +resources, saving overall fuzzing time. +Next, we further measure the efficiency of IR-Fuzz by +counting how many test cases are generated over time. Specif- +ically, each contract is run for 120 seconds in the experiment. +We show the average statistics in Fig. 7(b), where the x- +axis represents how long a contract is fuzzed, and the y-axis +denotes the number of test cases generated during fuzzing. + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +12 +TABLE IV Accuracy and coverage comparison (%) be- +tween IR-Fuzz and its variants. +Method +Vulnerability Type (Accuracy) +Coverage +TP +BN +DG +FE +UC +RE +OF +SE +IR-Fuzz-WSG +90.01 +93.38 +94.48 +92.73 +96.70 +94.80 +93.86 +96.42 +62.03 +IR-Fuzz-WDM +89.94 +93.56 +93.31 +92.15 +96.26 +94.51 +92.02 +94.12 +69.89 +IR-Fuzz-WEA +89.22 +91.00 +91.86 +91.00 +96.04 +92.49 +91.60 +95.82 +42.63 +IR-Fuzz +90.05 +93.79 +95.06 +94.48 +98.03 +98.73 +98.73 +99.73 +85.65 +From Fig. 7(b), we can learn that (1) IR-Fuzz significantly +generates more test cases than sFuzz within the same time +interval. On average, IR-Fuzz generates approximately 350 +test cases per second, 290 more than sFuzz; (2) The number of +test cases generated by IR-Fuzz has increased rapidly over time +while the process is slow in sFuzz. These evidences reveal that +IR-Fuzz can efficiently generate test cases for fuzzing smart +contracts. +E. Ablation Study (RQ4) +By default, IR-Fuzz adopts the proposed sequence gener- +ation strategy to generate the function invocation sequence. +It is interesting to see the effect of removing this strategy. +Moreover, IR-Fuzz utilizes a branch distance-based measure +to select and evolve test cases iteratively. We are curious +about how much this method contributes to the performance +of IR-Fuzz. Finally, IR-Fuzz introduces an energy allocation +mechanism to flexibly guide fuzzing resource allocation. It +is useful to evaluate the contributions of this mechanism by +removing it from IR-Fuzz as well. In what follows, we conduct +experiments to study the three components, respectively. +Study on Sequence Generation Strategy. +We removed +the sequence generation strategy (introduced in §IV-A) from +IR-Fuzz and replaced it with a random sequence construction +method. This variant is denoted as IR-Fuzz-WSG, where WSG +is short for without sequence generation strategy. Quantitative +results are summarized in Table IV. We can observe that the +performance of IR-Fuzz is significantly better than IR-Fuzz- +WSG. For example, on the reentrancy detection task, IR-Fuzz +achieves 3.97% and 23.62% improvement in terms of accuracy +and branch coverage, respectively. +Study +on +Branch +Distance-based +Seed +Evolution +Paradigm. +To evaluate the effect of the branch distance- +based seed evolution paradigm, we analyze the performance of +IR-Fuzz with and without it, respectively. Towards this aim, +we modify IR-Fuzz by removing this mechanism, utilizing +random test case generation. This variant is denoted as IR- +Fuzz-WDM, where WDM is short for without the distance +measure mechanism. The empirical findings are demonstrated +in Table IV, where we can observe that the accuracy and +branch coverage of IR-Fuzz-WDM are lower than IR-Fuzz +by an average of 2.03% and 15.76% on the eight types +of vulnerabilities. This reveals that incorporating the branch +distance-based measure is necessary and critical to improve +the performance of IR-Fuzz. +Study on Energy Allocation Mechanism. +We further +investigate the impact of the energy allocation mechanism +in IR-Fuzz. Specifically, we remove this mechanism while +replacing it with assigning fuzzing energy equally to every +contract Gamble { +uint256 private number; +uint256 phase; +address winner; +constructor (uint256 num) { +require(num < 100); +number = num; +phase = 0; } +/* 0: guess 1: start a new game */ +function guess(uint256 fee) payable external { +require (phase == 0 && fee == 10 finney); +if (address(this).balance == number * 10 finney) { +/* Ether strict equality */ +winner = msg.sender; +phase = 1; } +} +function newGame(uint256 num) external { +require(phase == 1 && msg.sender == winner); +winner.transfer(address(this).balance); +require(num < 100); +number = num; +phase = 0; } +} +} +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +Fig. 8 An example contract where IR-Fuzz detects a new +kind of vulnerability, i.e., dangerous Ether strict equality. +branch. This new variant is termed as IR-Fuzz-WEA, namely +IR-Fuzz without an energy allocation mechanism. The com- +parison results are presented in Table IV, where all eight types +of vulnerabilities are involved. We can clearly see that the +accuracy and branch coverage of IR-Fuzz-WEA are lower than +IR-Fuzz by an average of 4.87% and 43.02%. This suggests +that the energy allocation mechanism contributes to significant +performance gains in IR-Fuzz. +F. Case Study +We now present a case study on a new vulnerability (i.e., +dangerous Ether strict equality). To our knowledge, existing +investigated methods cannot expose this vulnerability yet. +Fig. 8 shows a simplified example that implements a gambling +game. A user can join the game by transferring participation +fees with 10 finney. If a user is the number-th participant, he +will become the winner of the game (line 14). The winner can +obtain the whole balance of the contract by calling newGame() +and starting a next round of the game. However, if the contract +owner had pre-stored some Ethers in the contract, the balance +of the contract will never be equal to the sum of users’ +participation fees (namely, the branch condition at line 14 will +never be satisfied). This indicates that there will be no winner +in the game, and the participation fees in the contract will be +permanently frozen. +We empirically checked this contract using existing tools +and manually inspected their generated reports. Unfortunately, +the dangerous Ether strict equality vulnerability cannot yet be +detected by these methods. In contrast, IR-Fuzz successfully +identifies this vulnerability. Specifically, T1: IR-Fuzz infers +the function invocation sequence as: guess()→ newGame() +and generates a test case to cover the requirement at line 13. +T2: Record the instruction BALANCE when the fuzzing process + +IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY +13 +reaches line 14. T3: Check if BALANCE is followed by the +jump or compare instructions. T4: IR-Fuzz finds that line 14 is +reachable and the vulnerability-specific patterns of dangerous +Ether strict equality are triggered, outputting that the contract +has such a vulnerability. +VI. DISCUSSION +In this section, we discuss the limitations of IR-Fuzz and +potential future improvements. +Sequence Generation Analysis. +IR-Fuzz generates the +ordered function invocation sequence with the guidance of +the order priority computation rules mentioned in §IV-A. We +calculate the order priority of function calls in the sequence +by analyzing the data flow dependencies of global variables. +In the case that several functions perform frequent write and +read operations on global variables, the calculation of function +order priority may bring a certain amount of computation +overhead. +Seed Mutation Optimization. +IR-Fuzz refers to several +seed mutation strategies adopted from AFL, usually using bit +manipulation techniques, e.g., bit flipping. However, such a +method still bears the problem of generating repetitive and +invalid test cases. Moreover, arbitrarily mutating bits of a +test input may ignore certain critical parts of the input that +should not be mutated, reducing the probability of hitting +the branches guarded by strict conditions. Therefore, in the +subsequent work, we may focus on enabling the fuzzer not to +mutate these crucial parts of a test case, making the fuzzing +trigger deep and complex states. +VII. CONCLUSION +In this paper, we present IR-Fuzz, a fully automatic fuzzing +framework equipped with invocation ordering and crucial +branch revisiting, to detect vulnerabilities in smart contracts. +Specifically, we propose a sequence generation strategy con- +sisting of invocation ordering and prolongation to generate +the high-quality function invocation sequence, enforcing the +fuzzer to trigger complex and deep states. Furthermore, we +design a seed optimization paradigm that engages a branch +distance-based measure to evolve test cases iteratively to- +wards a target branch, alleviating the randomness of test +case generation. Finally, we develop an energy allocation +mechanism to flexibly guide fuzzing resource allocation to- +wards rare and vulnerable branches, improving the overall +fuzzing efficiency and branch coverage. Experimental results +demonstrate that IR-Fuzz significantly surpasses state-of-the- +art fuzzing approaches by a large margin. 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Williams, “Can traditional fault prediction models be used +for vulnerability prediction?” Empirical Software Engineering, vol. 18, +no. 1, pp. 25–59, 2013. +[44] Z. Liu, P. Qian, X. Wang, L. Zhu, Q. He, and S. Ji, “Smart contract +vulnerability detection: From pure neural network to interpretable graph +feature and expert pattern fusion,” in IJCAI, 2021, pp. 2751–2759. +[45] C. F. Torres, J. Schütte, and R. State, “Osiris: Hunting for integer bugs in +ethereum smart contracts,” in Proceedings of the 34th Annual Computer +Security Applications Conference, 2018, pp. 664–676. +[46] B. Mueller, “A framework for bug hunting on the ethereum blockchain,” +Webiste, 2017, https://github.com/ConsenSys/mythril. +[47] “Etherscan,” Website, https://etherscan.io/. + diff --git a/xNE2T4oBgHgl3EQfhAd2/content/tmp_files/load_file.txt b/xNE2T4oBgHgl3EQfhAd2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1bcd48f429e588f5379d6715f0149bcfbdc037c --- /dev/null +++ b/xNE2T4oBgHgl3EQfhAd2/content/tmp_files/load_file.txt @@ -0,0 +1,1424 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf,len=1423 +page_content='IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 1 Rethinking Smart Contract Fuzzing: Fuzzing With Invocation Ordering and Important Branch Revisiting Zhenguang Liu, Peng Qian, Jiaxu Yang, Lingfeng Liu, Xiaojun Xu, Qinming He, and Xiaosong Zhang Abstract—Blockchain smart contracts have given rise to a variety of interesting and compelling applications and emerged as a revolutionary force for the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Smart contracts from various fields now hold over one trillion dollars worth of virtual coins, attracting numerous attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Quite a few practitioners have devoted themselves to developing tools for detecting bugs in smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' One line of efforts revolve around static analysis techniques, which heavily suffer from high false positive rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Another line of works concentrate on fuzzing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Unfortunately, current fuzzing approaches for smart contracts tend to conduct fuzzing starting from the initial state of the contract, which expends too much energy revolving around the initial state of the contract and thus is usually unable to unearth bugs triggered by other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, most existing methods treat each branch equally, failing to take care of the branches that are rare or more likely to possess bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This might lead to resources wasted on normal branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In this paper, we try to tackle these challenges from three aspects: (1) In generating function invocation sequences, we explicitly consider data dependencies between functions to fa- cilitate exploring richer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We further prolong a function invocation sequence S1 by appending a new sequence S2, so that the appended sequence S2 can start fuzzing from states that are different from the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) We incorporate a branch distance-based measure to evolve test cases iteratively towards a target branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (3) We engage a branch search algorithm to discover rare and vulnerable branches, and design an energy allocation mechanism to take care of exercising these crucial branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We implement IR-Fuzz and extensively evaluate it over 12K real-world contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Empirical results show that: (i) IR-Fuzz achieves 28% higher branch coverage than state-of-the- art fuzzing approaches, (ii) IR-Fuzz detects more vulnerabilities and increases the average accuracy of vulnerability detection by 7% over current methods, and (iii) IR-Fuzz is fast, generating an average of 350 test cases per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Our implementation and dataset are released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='com/Messi-Q/IR-Fuzz, hoping to facilitate future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Index Terms—Fuzzing, smart contract, vulnerability detection, blockchain, sequence generation, seed evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' INTRODUCTION This work was supported in part by the National Key R&D Program of China under Grant 2021YFB2700500 and 2021YFB2700501, in part by the Key R&D Program of Zhejiang Province under Grant 2022C01086, and in part by the Scientific Research Fund of Zhejiang Provincial Education Department under Grant Y202250832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (Corresponding author: Peng Qian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=') Zhenguang Liu, Peng Qian, and Qinming He are with College of Com- puter Science and Technology, Zhejiang University, Hangzhou 310058, China (e-mail: liuzhenguang2008@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' messi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='qp711@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' hqm@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Jiaxu Yang, Lingfeng Liu, and Xiaojun Xu are with School of Computer and Information Engineering, Zhejiang Gongshang University, Hangzhou 310018, China (e-mail: yjx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='00@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' liulingfengxx@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' xuxj2022@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Xiaosong Zhang is with the Center for Cyber Security, University of Electronic Science and Technology of China, Chengdu 611731, China (e- mail: johnsonzxs@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' S MART contracts are programs executing on top of a blockchain system [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A smart contract encodes prede- fined contract terms into runnable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Due to the immutable nature of blockchain, once a smart contract is deployed on the blockchain, its defined rules will be strictly followed during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Smart contracts make the automatic execution of contract terms possible, giving rise to a variety of decentralized applications [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Notably, not all blockchains support smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Ethereum, one of the most prominent blockchains enabling the execution of smart contracts, has attracted widespread attention worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' So far, tens of millions of contracts have been deployed on Ethereum [4], enabling a broad spectrum of applications, including wallet [5], crowdfunding [6], supply chain [7], and cross-industry finance [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Smart contracts from various fields now hold over one thousand billion dollars worth of virtual coins, and the number of contracts is still increasing rapidly [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Smart contracts have long been appealing targets for attackers since they manipulate so many digital assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, the source code of a Ethereum smart contract will be compiled into bytecode and executed on Ethereum Virtual Machine [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Like traditional programs, smart con- tracts may contain vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, it is important to identify potential vulnerabilities in smart contracts, ideally before their deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Malicious attackers may exploit the bugs in smart contracts to gain illegal profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Recently, there was an increasing number of security vulnerability incidents in smart contracts [10], [11], leading to enormous financial losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' One infamous example was the reentrancy attack, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', attackers stole more than $130 million worth of digital assets, exploiting the reentrancy vulnerability in the Cream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Finance contract [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This case is not isolated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', a delegatecall bug accidentally triggered resulted in freezing over $280 million worth of Ether in the Parity Multisig Wallet contract [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Obviously, conducting security vetting on smart contracts to avoid exposing vulnerabilities to attackers is much coveted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Fueled by the maturity of static analysis techniques such as formal verification [14] and symbolic execution [15], smart contract vulnerability detection has undergone considerable progress in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' These methods, however, inherently suffer from high false positive rates since they do not actually execute each path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Recent efforts resort to fuzzing techniques [16]–[18], which have the merits of producing neg- ligible false positives in discovering software vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This can be attributed to the fact that fuzzers usually generate test cases to exercise a branch, and report vulnerabilities only when they detect abnormal results during fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' After scrutinizing existing released fuzzers for smart con- tracts, such as [16], [18]–[23], we obtain the following obser- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='03943v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='PL] 10 Jan 2023 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 2 vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) Current fuzzers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', sFuzz [16] and Harvey [21]) tend to generate function invocation sequences randomly, overlooking the data dependencies (such as read and write dependencies) between functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' More importantly, a smart contract may transition through many different states during its lifecycle [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, every bet in a gambling contract will change the contract state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, current methods generally conduct fuzzing starting from the initial state of the contract, which actually expends too much energy revolving around the initial state of the contract and is incapable of unearthing bugs triggered by complex states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) Most current approaches fail to take into account the distance between test cases and branch conditions in seed mutation, resulting in generating seeds that have low probabilities to enter a target branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (3) Existing fuzzers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', ILF [22] and Con- fuzzius [23]) often treat program branches equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As a result, fuzzers might waste too many resources in fuzzing normal branches and are unable to dive deep into crucial branches that are rare or more likely to have bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To tackle these challenges, we propose IR-Fuzz, a fully automatic Fuzzing framework equipped with Invocation or- dering and impoRtant branch revisiting, for detecting security vulnerabilities in Ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In particular, IR- Fuzz consists of three key components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Sequence Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Usually, there are multiple func- tions within a contract, we introduce a function-invocation- sequence generation strategy, which consists of function in- vocation ordering and sequence prolongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, we build a data flow analyzer to capture the data flow dependencies of global variables and then define a rule to compute the order priority of each function call, inferring the ordered function invocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Further, we introduce a prolongation technique to extend the sequence, enforcing the fuzzer to tap into unprecedented states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We also present a seed optimiza- tion paradigm, which drives the fuzzer to generate branch- condition-aware test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In practice, we employ a branch distance-based measure to select test cases according to how far a test case is from satisfying the condition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', x==10) of a just-missed branch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Intuitively, the test case has a higher probability to enter the just-missed branch as the distance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In this way, IR-Fuzz iteratively evolves test cases to get increasingly closer to satisfying the branch conditions, which boosts its ability to find a high-quality test case and achieve a higher branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Energy Allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, we design an energy allo- cation mechanism that takes into account the significance of a branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Technically, we first propose a branch search algorithm to pick out rare branches and branches that are likely to have vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, we formulate a customized energy schedule and develop two rules to guide fuzzing energy allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As such, IR-Fuzz can flexibly assign fuzzing resources to more important branches, which increases the overall fuzzing efficiency by a large margin (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='9x faster than sFuzz [16]) and further improves branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1A just-missed branch stands for the unexplored if-branch or then-branch of a conditional statement (such as if and require) or a recurrent statement (such as for and while).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We implement IR-Fuzz and extensively evaluate this system over 12K real-world smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Experimental results show that: (i) IR-Fuzz achieves high average branch cover- age by up to 90%, yielding a 28% improvement compared with state-of-the-art fuzzing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (ii) IR-Fuzz identifies more vulnerabilities and increases the average accuracy of vulnerability detection by 7% over current methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (iii) IR- Fuzz generates an average of 350 test cases per second, in most cases orders-of-magnitude faster than conventional fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Our key contributions can be summarized as follows: We design and implement a novel framework IR-Fuzz for smart contract fuzzing, which consists of three key com- ponents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', function invocation sequence prolongation, branch-distance-driven seed optimization, and branch- importance-aware energy allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Within the framework, we present a sequence generation strategy to infer high-quality function invocation se- quences, steering fuzzing to explore unprecedented states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Further, we introduce a seed optimization paradigm that incorporates a branch distance-based measure to select and evolve test cases towards new branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, we develop a branch search algorithm to discover rare and vulnerable branches, and propose an energy allocation mechanism to concentrate on these critical branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We evaluate IR-Fuzz over large-scale real world smart contracts, and empirical results show that the proposed techniques are indeed useful in achieving high branch coverages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz surpasses state-of-the-art fuzzers and overall provides interesting insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As a side contribu- tion, we construct a large benchmark dataset for evaluat- ing smart contract fuzzing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Our implementa- tion and dataset are released, hoping to inspire others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Smart Contract Vulnerability Detection Since blockchain endows smart contracts with unalterable nature, there is no way to patch the vulnerabilities of a smart contract without forking the blockchain (almost an impossible mission), regardless of how much money the contract holds or how popular it is [2], [24]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, it is critical to conduct security vetting for smart contracts, especially before their deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Early works for smart contract vulnerability detection employ formal verification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, [14] introduces a framework to compile smart contracts to EVM bytecode and then put them into an existing verification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' [27] proposes a formal model and verifies smart contracts using the Isabelle/HOL tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Further, [28] and [29] define the formal semantics of contracts using the F* framework and the K framework, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Although these frameworks provide strong formal verification guarantees, they are still semi-automated and yield high false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Another stream of works rely on symbolic execution methods, such as Oyente [15], Slither [30], and Securify [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Oyente is one of the pioneering works to perform symbolic execution on smart contracts, which checks bugs based on expert- defined rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' [30] converts smart contracts into intermediate IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 3 representations and conducts static analysis to detect vulnera- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Whereas symbolic execution is a powerful technique for discovering bugs, it still suffers from the inherent problem of symbolic execution path explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Recent efforts resort to using fuzzing techniques for smart contract vulnerability detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' ContractFuzzer [20] is the first to apply fuzzing techniques to smart contracts and identi- fies vulnerabilities by monitoring runtime behaviors during fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' ReGuard [32] and Harvey [21] are dedicated to generating a number of test cases that cover as many paths as possible to trigger a vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' ILF [22] and sFuzz [16] aim to design a feedback-based seed mutation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Despite the practicality of fuzzing techniques, existing fuzzers still have difficulties in achieving high coverage and fuzzing efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Instead, our work alleviates the issues by carefully designing a sequence generation strategy, a seed optimization paradigm, and an energy allocation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Greybox Fuzzing Fuzzing techniques have been proven as an effective way to discover software vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' According to how much information is available about the program under test [33], fuzzing techniques can be cast into three categories: whitebox, blackbox, and greybox [34]–[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Put succinctly, blackbox test- ing conducts fuzzing without knowing any internal structure of the target program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In contrast, whitebox testing performs fuzzing while having full knowledge about the internal archi- tecture of the target program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Greybox fuzzing stands in the middle of blackbox fuzzing and whitebox fuzzing, where we have partial knowledge of the internal structure of the target program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, greybox fuzzing can be further divided into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' One spectrum of works [37], [38] aim at covering as many paths or branches as possible, expecting to reveal a bug in the program, namely coverage-guided greybox fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' AFL, one of the most well-known fuzzers, employs the lightweight instrumentation technique and genetic algorithm to improve coverage [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Some other researchers [16], [40] increase code coverage by smartly selecting and mutating test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Typically, these methods improve coverage by generating as many new test cases as possible to traverse previously uncovered program paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Another spectrum of works [35], [37], [41] are designed to direct greybox fuzzing towards a set of specific target locations, termed targeted greybox fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' There is a number of greybox fuzzers that focus on specific program locations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', low-frequency or uncovered branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, [19] utilizes a power schedule to collect feedback information and steer fuzzing towards target locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' AFLGo [35] calculates the distance between entry points and buggy code in the control flow graph, guiding seed mutation to cover the target locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Overall, targeted greybox fuzzing generates test cases to reach certain target locations, attempting to further trigger a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' MOTIVATING EXAMPLE As a motivating example, we present a real world smart contract GuessNum, which implements a gambling game on contract GuessNum { mapping(address => uint256) userBalance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' uint256 prizePool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' constructor () payable { prizePool = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } /* Initialize the prize pool */ function guess(uint256 num) payable external { uint256 luckyNum = uint256(keccak256(abi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='encodePacked( block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='difficulty, now)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' /* Generate a lucky number */ luckyNum = luckyNum % 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' prizePool = SafeAdd(prizePool, msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' /* Put funds into the prize pool */ if (num == luckyNum && msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value == 50 finney) { /* If a player guessesit, he obtains 40 times the participant fee */ userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender] = SafeAdd(userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender], msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value*40);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } function getReward() payable external { if (userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender] < prizePool && userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender] > 0) { msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value(userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender])();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' /* Reentrancy bug */ prizePool = SafeSub(prizePool, userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' userBalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } } 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1 A real-world smart contract written in Solidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Ethereum [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1 shows the simplified code2 of GuessNum, which is written in Solidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The contract realizes a guess number game that users play by submitting their guesses along with participation fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The fee is fixed to 50 finney each time and is poured into the prize pool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', prizePool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Function constructor() runs only once when the contract is created, and it puts the funds of the contract’s owner into the prize pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A player who wants to submit a guess can invoke function guess(), which compares the received guess number with the randomly generated lucky number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', luckyNum (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' If guess number exactly matches the luckyNum, the player will obtain 40 times the participation fee in return (line 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Players can get rewards by calling function getReward().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This GuessNum contract, unfortunately, suffers from a classical reentrancy vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' From line 18 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1, we observe that function getReward() invokes call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value to transfer money to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, due to the default settings of smart contracts, the transfer operation will automatically trigger the fallback function of the recipient contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, an attacker may set a malicious second- time invocation to getReward() in his fallback function for stealing extra money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Since getReward() waits for the first-time transfer to finish, the balance of the attacker is not reduced yet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', the user balance reduction operation at line 19 is behind call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value and is not executed yet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Function getReward() thus may wrongly believe that the attacker still has enough balance and transfers money to the attacker again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Limitation of Existing Fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Interestingly, this simple smart contract reveals three key challenges for most existing fuzzers to expose vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) The order of function invocations is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We observe that if the conditions at line 11 are not satisfied (namely the then-branch at line 13 is not reached), then the second condition at line 17 will never be met either (because userbalance[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender] is equal to 2Address: 0xd5e94f6350c7911015dd120b0b006420b6e85a58 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 4 Sequence Generation Source Code .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Json File ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Compile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Data Flow Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Seed Evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Sequence Prolongation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Seed Selection and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Mutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Bug Report ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Fuzzing Execution Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Branch Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Customized Energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Schedule ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Source Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Energy Allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Vulnerability Analysis & Report ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Function Invocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='New Test Cases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Branch Distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Measure Technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='High-Quality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Test Suite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Target Location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Rare Branches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Vulnerable Branches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Feedback Result ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Security Vulnerability Patterns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='TSInvocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='TSContaminate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='TSRandom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Initial Test Cases Order Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 2 A high-level overview of IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz has four main components, including (1) Sequence Generation, (2) Seed Optimization, (3) Energy Allocation, and (4) Vulnerability Analysis and Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As such, the fuzzer will be unable to reach the then- branch at lines 18–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) Generating a test case to satisfy the second condition at line 11 (msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value == 50 finney) is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' More specifically, the variable msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value has a size of 32 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Thus, when we generate a random value for msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value in fuzzing, we have only 1 2256 probability to obtain the value 50 to meet the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Indeed, existing fuzzers like AFL [39] are shown to have difficulties in electing a test case to enter the then-branch at line 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (3) Vulnerable branches that may contain vulnerabilities only take a small fraction of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, lines 18–20 are vulnerable code which only exists in one branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Current fuzzers [22], [23] tend to treat each branch equally, which may fail to discover the vulnerability due to insufficient fuzzing resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Fuzzing Policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We embrace three key designs in IR- Fuzz to tackle the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) IR-Fuzz leverages the variable read and write dependencies between functions to generate the ordered function invocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' It further extends the ordered sequence with another ordered sequence to explore more complex states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, this contract (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1) has two global variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', userBalance and prizePool, which both appear in functions guess() and getReward().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' By analyzing read and write dependencies of the global variable userBalance, IR-Fuzz recognizes that function getReward() depends on function guess(), awaring that guess() should be called before getReward().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Consequently, IR-Fuzz generates the function invocation sequence as: guess()→getReward().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) IR-Fuzz adopts a branch distance-based schema to select test cases according to how far a test case is from satisfying the condition of a just-missed branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, the distance of reaching the just-missed branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', then-branch at line 13) is calculated as |msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value - 50| since the branch condition at line 11 is msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value == 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Intuitively, the test case has a higher probability to enter the just-missed branch as the distance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' With the guidance of distance measure, IR- Fuzz iteratively evolves test cases to get increasingly closer to satisfying the branch condition at line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (3) IR-Fuzz engages a branch search algorithm to pick out vulnerable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', then- branch at lines 18–20) and rare branches, and then formulates an energy schedule to expend more fuzzing resources on these important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In our experiments, after only 26s, IR- Fuzz generates a test case to reach the then-branch at line 18 and exposes the reentrancy vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' METHOD Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The overall architecture of IR-Fuzz is outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Generally, IR-Fuzz consists of four key components: Sequence Generation: Given that a contract might contain multiple functions, to explore their possible function in- vocation sequences, IR-Fuzz first analyzes the data flow dependencies between functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, it defines a rule to compute the order priority of each function, and generates a function invocation sequence that successively calls the ordered functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Further, IR-Fuzz adopts a prolongation technique to extend the sequence, driving the fuzzer to dive into deeper states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed Evolution: To guide seed mutation so that the gen- erated cases could reach a target branch, IR-Fuzz utilizes a branch distance-based measure to select and evolve test cases iteratively according to how far a test case is from satisfying the condition of the target branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Energy Allocation: To further take care of the rare branches and branches that are likely to have vulnerabilities, IR-Fuzz introduces a branch search algorithm to analyze exercised branches and picks out those important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, IR- Fuzz formulates a customized energy schedule and utilizes two rules to flexibly guide fuzzing energy allocation towards these critical branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Vulnerability Analysis and Report: IR-Fuzz analyzes the generated logs and refers to vulnerability-specific patterns to discover vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Bug reports are generated for further manual inspections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In what follows, we will elaborate on the details of these components one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Sequence Generation Presented with the multiple functions of a smart contract, existing methods tend to generate a function invocation se- quence by randomly picking a function each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Scrutinizing 12K real-world smart contracts, we empirically observe that the state of a smart contract is often captured by the state of its global variables, and different functions do share and operate differently on the global variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Some functions 三IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 5 perform ‘read’ operations on the variables while some other functions may perform ‘write’ operations on the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Generating function invocation sequence randomly ignores such connections between functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The functions that per- form ‘write’ operations could change the state of the contract while the functions that perform only ‘read’ operations are unable to change the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, we assign higher order priority to functions that perform ‘write’ operations so that we may explore a broader contract state space and cover more branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' It is worth mentioning that initializing a variable will not affect the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Sequence Ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Motivated by this, we propose to model the global variable read and write dependencies be- tween functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, we first characterize the source code of a smart contract into an abstract syntax tree [42], from which we extract variable access operations such as assignments and comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, we leverage a data flow analyzer to capture the read and write dependencies of global variables between functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Afterwards, we calculate the order priority (OP) of each function and sort them according to their OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, we generate the ordered function invocation sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', the invocation sequence that calls the sorted functions successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Formally, we denote the set of functions in a smart contract as F = {F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', FN}, and global variables that appear in function Fi as V = {vi1, vi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', viM}, where M is the number of global variables appear in Fi and vik represents the k-th global variable in Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, each variable vik has a unique identifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', variable name), which is denoted as vID ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To further indicate the operation that Fi exerted on vik, we use vop ik = 1 to represent that Fi performs read operation on vik, and vop ik = 0 to denote that Fi conducts write operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Now, we define the order priority of functions as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Rule 1: Read & Write Dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' When a global variable appears in two different functions, the function that executes write operation on the variable should be called earlier than the function that executes read operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Put differently, given global variables vik in Fi and vjn in Fj (where vID ik = vID jn ), we suggest that Fi should have a higher order priority (OPFi > OPFj) when vop ik = 0 and vop jn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Guided by this, we may compute the order priority of each function by converting this rule into the following formula, which sums up the analysis on read & write dependencies (Rule 1) of all global variables in different functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' OPi = Mi � k=1 Mj � p=1 vop jp(1 − vop ik ) · cmp(vik, vjp) 1 ≤ i, j ≤ N && i ̸= j (1) where N is the number of functions in the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Mi and Mj denotes the number of the appearance of global variables in Fi and Fj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Notably, cmp(vik, vjp) compares the identifier of global variables in two different functions, which is given by: cmp(vik, vjp) = � 1, vID ik = vID jp 0, vID ik ̸= vID jp (2) where vik denotes k-th variable of Fi and vjp represents p-th variable of Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Mathematically, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1), a function accumulates one order priority score only if it writes on a global variable and the variable is read by another function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In this context, the functions that conduct write operations on more global variables get higher priority and are put in the front of the function invocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This drives the fuzzer to exercise more on the functions that could change the states and boost the fuzzing by encouraging it to encounter more states and reach more branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Interestingly, our experimental results show that branch coverage is significantly improved with such sequence ordering (see §V-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Here, we take the contract of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1 as an example to illustrate the order priority calculation of each function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1, we can observe that this contract has two global variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', userBalance and prizePool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Function guess() performs a read operation and a write operation on the two variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We represent the variables that appear in function guess() as Vguess = {vguess1, vguess2, vguess3, vguess4}, where vguess1 = vguess2 = prizePool, vop guess1 = 1, vop guess2 = 0, and vguess3 = vguess4 = userBalance, vop guess3 = 1, vop guess4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Meanwhile, cmp(userBalance, prizePool) = 0, while cmp(userBalance, userBalance) = cmp(prizePool, prizePool) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Similarly, the global variables appear in getReward() are denoted as VgetReward = {vgetReward1, vgetReward2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', vgetReward8}, where vop getReward1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' = vop getReward6 = 1, and vop getReward7 = vop getReward8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As such, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1), we calculate the order priority of function guess() as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' OPguess =vop getReward1(1 − vop guess4) + vop getReward2(1 − vop guess2) +vop getReward3(1 − vop guess4) + vop getReward4(1 − vop guess4) +vop getReward5(1 − vop guess2) + vop getReward6(1 − vop guess4) =6 (3) For function getReward(),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' we calculate its order priority as: OPgetReward =vop guess1(1 − vop getReward7) +vop guess3(1 − vop getReward8) =2 (4) According to the order priority calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' we can see that the order priority of calling function guess() is greater than that of function getReward().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, IR-Fuzz finally generates the function invocation sequence as: guess() → getReward().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Sequence Prolongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Another important insight is that a smart contract might go through many different states dur- ing its lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, current methods typically conduct fuzzing starting from the initial state of the contract, which expends too much energy revolving around the initial state and is usually incapable of unearthing bugs triggered by other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' These facts inspire us to explore richer starting states via sequence prolongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, we first exercise the ordered function invocation sequence S using various test cases, which result in different states of the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We then engage in the invocation sequence S again but execute S starting from these different states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', appending a new sequence S after S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For instance, presented with the crowdfunding contract in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 3, we observe that covering the if-branch at line 21 for traditional fuzzers is difficult, which requires at least twice IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 6 contract CrowdFunding { uint256 goal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' uint256 raised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' uint256 phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' address beneficiary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' constructor () { beneficiary = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' goal = 300 ether;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' raised = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' phase = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } /* 0: Active 1: Finished */ function donate(uint256 donations) payable public { /* Check if the crowdfunding goal is reached */ if (raised < goal) { raised += donations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } else { phase = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } function withdraw() public { /* The crowdfunding goal has been reached */ if (phase == 1) { beneficiary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='delegatecall(bytes4(keccak256(“transfer(uint256)”)), raised);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } } 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 3 An example contract for illustrating how the sequence prolongation technique is used in IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' invocations of function donate().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, this contract implements a simple crowdfunding project that allows users to donate money by calling donate().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The constructor() sets the goal of crowdfunding as 300 Ether (line 9), and the raised money is initialized to 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', raised = 0 at line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The status of the crowdfunding process is initialized to 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', phase = 0 at line 11), which represents unaccomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' According to Rule 1, we realize that the invocation to func- tion donate() should have a higher order priority than that of function withdraw().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As a result, the function invocation sequence is generated as: donate()→ withdraw().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, such a sequence fails to satisfy the branch condition at line 21 because call function donate() once cannot enter the else- branch at line 16 to set phase = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To reach the else-branch at line 16, function donate() needs to be called at least twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Namely, the required amount of money for the crowdfunding is accomplished at the first call (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', raised ≥ goal), and the second call enters the else-branch at line 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, to explore deeper states, we propose to prolong the function invocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In contrast to most fuzzing methods that conventionally call each function only once in the invocation sequence, we further extend the ordered function invocation sequence S by appending the same invocation sequence to S, namely S → S, such that the second sequence starts its execution from a different state rather than the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To explore different starting states for the second sequence, the first sequence is presented with different sets of parameters, which lead to different inner statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Technically, we first try to sufficiently exercise sequence S with various different input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A different set of input parameters to S translates to a variant of S, denoted as Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, we concatenate two sequences Si and Sj that have different input parameters and exercise the new concatenated sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The following rule formulates the Algorithm 1: SEED ITERATIVE OPTIMIZATION 1 currentTestSuite ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 2 currentTestCase ← initialTestCase();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 3 while ¬Terminated() do 4 Let BtestCase be covered branches by currentTestCase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5 Let Bmiss be just-missed branches in currentTestSuite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 6 for br ∈ BtestCase do 7 if br is new branch then 8 currentTestSuite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ADD(currentTestCase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 9 for br ∈ Bmiss do 10 for seed ∈ currentTestSuite && seed ̸= currentTestCase do 11 dist_1 ← distance(currentTestCase, br);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 12 dist_2 ← distance(seed, br);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 13 if dist_1 < dist_2 then 14 currentTestSuite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='REMOVE(seed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 15 currentTestSuite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ADD(currentTestCase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 16 energy ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 17 MutationEnergy ← AssignMutationEnergy();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 18 while energy < MutationEnergy do 19 testCase ← SelectInput(currentTestSuite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 20 newCase ← Mutation(testCase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 21 if ¬RepeatCheck(currentTestSuite, newCase) && ¬V alidityCheck(newCase) then 22 currentTestCase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ADD(newCase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 23 log ← FuzzInput(newCase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 24 energy ← UpdateEnergy(log, energy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 25 return currentTestSuite: A set of high-quality test cases sequence pair selection to generate a new prolonged sequence, which constrains that the input parameters of Si and Sj should be quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Rule 2: Sequence Pair Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) When the number of required function input parameters in the sequence is no less than 2, we select the sequence pairs iff at least one input paramter is different between the two sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, given three sequences S1: F1(x1) → F2(), S2: F1(x2) → F2(), and S3: F1(x2) → F2(), pairs P1(S1, S2) and P2(S1, S3) are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In contrast, pair P3(S2, S3) is not selected since S2 and S3 have the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) When the number of required function input parameters is larger than 2, we select the sequence pairs iff at least two input parameters are different between the two sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, given three sequences S1: F1(x1, y1) → F2(z1), S2: F1(x1, y2) → F2(z1), and S3: F1(x2, y2) → F2(z2), pairs P1(S1, S3) and P2(S2, S3) are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Upon each sequence pair selection, we obtain a prolonged function invocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For the crowdfunding contract demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 3, IR-Fuzz first generates two sequences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', S1: donate(300) → withdraw() and S2: donate(200) → withdraw().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' According to Rule 2, IR-Fuzz combines S1 and S2 as a new prolonged sequence S: donate(300) → withdraw()→ donate(200) → withdraw().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Since the goal of crowdfunding is 300, the first call to donate(300) in S satisfies the condition of the if-branch at line 15, and the second call to donate(200) reaches the else-branch at line 16 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', set phase = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The new sequence thus can reach the then-branch at line 22 on the IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 7 second call to withdraw() and expose the potential Ether frozen vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Promisingly, with the prolongation technique, IR-Fuzz greatly expands the scope of explored states and branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Note that the sequence prolongation technique yields little impact on the overhead of the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed Evolution To fuzz a function invocation sequence, the most intuitive and direct way is to generate test cases randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Despite its simplicity, this strategy is not favorable for reaching unex- plored conditional branches due to its random nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz, instead, incorporates a seed evolution paradigm to refine test cases iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The seed evolution framework is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' First, IR-Fuzz initializes an empty test suite and a set of test cases (lines 1-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, the first loop from lines 6 to 8 performs seed selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Whenever a test case covers a new branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', any branch not covered by test cases in the test suite), it is added to the suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Next, the loop from lines 9 to 15 evolves test cases iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, we propose a branch distance-based measure to select those test cases which are closer to satisfying the conditions of new branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Thereafter, the loop from lines 18 to 24 executes seed mutation, in which function Mutation() generates the mutated test cases based on the test cases selected from the suite (lines 19-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, we adopt seed verification strategies to guarantee the validity of mutated test cases (line 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This mutation process continues until a mutation energy upper- bound is reached (line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, a new test suite that contains a set of high-quality test cases is shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In what follows, we present the technical details of seed selection and seed mutation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In IR-Fuzz, we first try a classical seed selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' That is, IR-Fuzz monitors the execution of test cases and records the branches that each test case traverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A test case is added into the test suite as long as it covers a new branch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', a branch which is not covered by any test case in the suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Empirically, our experimental results show that this strategy could reveal a number of branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, it is still quite inefficient in reaching those complex branches with strict conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, the probability of satisfying the second condition (msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value == 50 finney) at line 11 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1 is 1 2256 , which is extremely low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To meet such strict branch conditions, we design a novel seed selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Inspired by [16], we adopt a distance function dist(T, br) to compute a branch distance indicating how far a test case is from covering a just-missed branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', uncovered then- branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' More specifically, let br be a just-missed branch, which is not covered by any test case T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We suppose that br is a branch of condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Note that C can be either x==k, x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' = k, x ≤ k, x < k, x ≥ k or x > k, where x and k are variables or constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The function dist(T, br) is given by: dist(T, br) = � � � � � � � � � |x − k|, if C is x == k 1, if C is x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' = k max(x − k, 0) if C is x ≤ k or x < k max(k − x, 0) if C is x ≥ k or x > k (5) where x and k are extracted from the stack information recorded by IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Intuitively, dist(T, br) is defined such Algorithm 2: BRANCH SEARCHING Input: Program P, Test case case, Vulnerable statements T Output: Brare and Bvulnerable 1 br ← FuzzRun(P, case);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 2 Brare ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' // Rare Branches 3 Bvulnerable ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' // Vulnerable Branches 4 R ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5 i ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 6 while i < |br| do 7 if IsConditionInstruction(i, Cb) then 8 R ← R + 1, bpre ← br[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='i + 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 9 c, state ← StateInference(bpre);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 10 if V ulnerableStatementReached(P, state, c, T ) then 11 Bvulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ADD(br);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 12 i ← i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 13 if R ⩾ 2 then 14 Brare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ADD(br);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' that the closer a test case T is from satisfying the condition of branch br, the smaller the distance is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, a test case with msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value = 100 is closer to satisfying the condition msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value == 50 than a test case with msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value = 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For each just-missed branch, IR-Fuzz selects a test case has the smallest dist(T, br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' With the feedback of the branch distance measurement, IR-Fuzz can quickly approach complex branches guarded by strict conditions, improving the overall branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Note that all selected test cases are added to the test suite and transferred to the seed mutation phase for generating new test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed mutation plays an important role in enriching the test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In IR-Fuzz, we refer to several mutation strategies from AFL and introduce new ones tailored for smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, we preferentially mutate those test cases with smaller branch distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Given a test case encoded in the form of a bit vector, sFuzz [16] engages a set of mutation operators to generate new test cases, such as bit flipping, interest value insertion, and key-value insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz additionally ensures the generated test cases are valid by advocating two principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) IR-Fuzz checks the validity and integrity of the mutated test cases by using a bit verification approach, which sets random bits of a given seed to random values while keeping other bits of the seed unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Thereafter, IR-Fuzz saves a new test case as a new seed based on whether the new test case detects a new branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, IR-Fuzz will discard invalid test cases which lead to fuzzing crashes or bring much overhead to the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) IR-Fuzz removes duplicate test cases by comparing the mutated cases with the test cases in the test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In practice, we also apply multiple heuristics to save the mutation energy of IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, any test case in the test suite that does not discover a new branch after a round of mutation will be assigned with low priority in the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz updates energy according to the generated logs during fuzzing (lines 23-24 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The process of seed mutation continues until the mutation energy upper-bound is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Each seed is assigned with a priority score which measures its ability to detect new branches after mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 8 contract Cheer { uint256 num = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' function random(uint256 x, uint256 y, uint256 z) returns (uint256) { if (x % 2 == 0) { num = 256;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' while (x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='= z) { num = SafeMul(x, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' if (x > z) { z = SafeAdd(z, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } else { x = SafeAdd(x, x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } if (y % 2 == 0) { num = uint256(keccak256(abi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='encodePacked( block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='number, now)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' /* Block number dependency */ } } else { num = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } return num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 4 An example contract for illustrating how IR-Fuzz allocates energy to the target branches flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Energy Allocation In this subsection, we introduce how IR-Fuzz performs fuzzing energy allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Recall that most existing fuzzers treat program branches equally, ignoring the fact that vulner- able code usually takes a tiny fraction of the entire code [18], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As a result, conventional fuzzers may waste massive resources in fuzzing normal branches instead of rare branches and branches that are more likely to possess bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To tackle this problem, we design an energy allocation mechanism, guiding IR-Fuzz to assign fuzzing resources towards these important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, this mechanism consists of two modules: branch analysis and energy schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Branch Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We remark that the first challenge is how to pick out the important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To address this, we introduce a branch search algorithm to analyze all branches discovered during fuzzing and focus on two types of branches: rare and vulnerable, which are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Definition 1 (Branch): Given a path p exercised by a test case in smart contract S, we say that br is a prefix subpath of p if br is a subpath of p and br begins with the same starting point as p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Further, br is a branch of p if br is a prefix subpath of p and br ends with the if-branch or then-branch of a conditional or recurrent statement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', if, require, for, while) in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Definition 2 (Rare Branch): We consider a branch br is a rare branch if br contains at least two nested conditional statements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g, two nested if).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Each rare branch is associated with a rarity factor R, which is set to the number of nested conditional statements (at the end of this branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Definition 3 (Vulnerable Branch): Given a branch br and a set of vulnerable statements T that may introduce bugs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='number and call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value), we say that br is a vulnerable branch when br contains a vulnerable statement t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' After empirically scrutinizing real-world smart contracts, we found that over 50% (65% and 86% in our experiments) of bugs are located in rare and vulnerable branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To pick out the two types of branches, we design a branch analysis algorithm shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Technically, IR-Fuzz employs the abstract interpreter A to pick out the important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' First, A analyzes all branches Br discovered during fuzzing (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, the loop from lines 6 to 12 checks whether there exists a branch br that reaches the vulnerable statement t ∈ T and computes the rarity factor R for each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Afterwards, A adds the branches with R ⩾ 2 into Brare (lines 13-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, A obtains a set of rare branches Brare and vulnerable branches Bvulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Energy Schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We remark that the second challenge is how to assign resources to these important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Towards this aim, we formulate a customized energy schedule Ω to manage the fuzzing energy allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This schedule adopts two rules for rare and vulnerable branches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Rule 3: Energy Allocation for Rare Branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Given branches br1 with R1 and br2 with R2 in a same path p, where Ri is rarity factor and R1 < R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To facilitate that rare branches with higher rarity factors are more sufficiently fuzzed, Ω assigns energy ∇1∗E to br1 and ∇2∗E to br2 where ∇1 < ∇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Rule 4: Energy Allocation for Vulnerable Branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Given a branch br1 in a path p1 and a branch br2 in a path p2, Ω assigns α∗E energy to br1 (α > 1) and E energy to br2 when R1 = R2 and br1 ∈ Bvulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Coefficient α controls the preference degree for vulnerable branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We use the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 4 to show how IR-Fuzz works with the energy allocation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, IR-Fuzz first analyzes all discovered branches and picks out br at line 12 as the vulnerable branch since it contains a vulner- able statement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='number) at line 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, IR-Fuzz calculates the rarity factor R of this branch, where R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As a result, IR-Fuzz assigns energy (α + ∇) ∗ E to branch br, according to Rule 3 and Rule 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In our experiments, an interesting observation is that state-of-the-art tools like sFuzz cannot reach the if-branch at line 12 since they waste too much energy in fuzzing the while branch at line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In contrast, IR- Fuzz successfully covers this branch and exposes the block number dependency vulnerability at line 13 in 8s on average (between 4s and 12s in 5 runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, IR-Fuzz also utilizes the feedback of energy allocation to guide seed mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, the test cases which cover the vulnerable branches will be selected and fuzzed first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Inspiringly, with the energy allocation mechanism, IR-Fuzz can flexibly assign fuzzing resources to the rare and vulnerable branches, increasing overall fuzzing efficiency and branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Vulnerability Analysis and Report In this stage, IR-Fuzz turns to vulnerability analysis and report generation, which reveals vulnerabilities in smart con- tracts and generates a detailed bug report for further manual confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In particular, we investigate previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' [44]) and define the specific patterns for eight types of vulnerabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' namely timestamp dependency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' block number dependency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' dangerous delegatecall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Ether frozen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 9 Execution & Analyze Execution & Analyze Smart Contract Generate compiled JSON file Seed selection with branch-distance-based feedback Branch energy allocation Seed mutation with energy-allocation-based feedback Bug reports Data flow dependency Function invocation sequence Seed Selection & Mutation Vulnerability analysis and report Seed evolution Generate function invocation sequence b1 b2 bn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' { 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' bytecode:{ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' }；' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' abi:{ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='} 1 1 4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' b1 b2 bn 2 3 2 k-1 k Rare Vulnerable b1 : e1 b2 : e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' bn : en bx: emax sx(seed of bx) 1 2 3 4 5 6 1 2 3 4 k k-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Energy schedule Branch Energy b1 e1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' bn en Abstract syntax tree Compile guess() getReward() Test Suite Log information Vulnerability-specific patterns Log CALL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Branch analysis Order Test Case Test Case Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5 The workflow of IR-Fuzz to reveal the reentrancy vulnerability in the real-world contract of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' unchecked external call, reentrancy, integer overflow, and dan- gerous Ether strict equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We have implemented a pattern analyzer to handle these patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz analyzes the fuzzing results and reveals bugs with the assistance of the pattern analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In the following, we show an example of how IR- Fuzz exposes a reentrancy vulnerability using patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Vulnerability Pattern Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Reentrancy vulnerability is considered as an invocation to call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value that can call back to itself through a chain of calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' That is, the in- vocation of call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value is successfully re-entered to perform unexpected repeat money transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, we design two patterns to expose the reentrancy vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The first pattern CALLValueInvocation checks if there exists an in- vocation to call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value in the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The second pattern RepeatedCallValue concerns whether a specific function with call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value invocation is called repeatedly during fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR- Fuzz reports that a function has a reentrancy vulnerability if it fulfills the combined pattern: CALLValueInvocation ∧ RepeatedCallValue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz Workflow Illustration with an Example In this subsection, we take the contract of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 1 as an example to show the workflow of IR-Fuzz on revealing a reentrancy vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The workflow consists of six steps, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Given the contract with source code as input, IR-Fuzz first compiles the source code to the JSON file, which consists of EVM bytecode and application binary interface (ABI) (\x82 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Second, IR-Fuzz extracts the abstract syntax tree and captures the data flow dependencies of global variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' By analyzing these dependencies, IR- Fuzz infers the function invocation sequence as: guess() → getReward() (\x83 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Thirdly, IR-Fuzz generates test cases for the two function calls and adds high-quality test cases into the test suite based on the branch distance-based measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As such, IR-Fuzz effectively generates test cases to cover new branches (\x84 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Furthermore, IR-Fuzz performs the branch analysis to pick out the important branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Then, it customizes an energy schedule to assign fuzzing resources ( in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz utilizes the feedback of energy allocation to guide seed selection and mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' After several rounds of fuzzing, IR-Fuzz reaches the then-branch at line 18 and triggers the execution of the transfer function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We develop an attack contract generator, which simulates an attack contract that calls the transfer function getReward() again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz records the instructions into a log file (\x86 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, IR-Fuzz analyzes the log and determines whether the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value was called multiple times, exposing the reentrancy vulnerability with the assistance of vulnerability- specific patterns (\x87 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Besides fuzzing the ordered invocation sequence, IR-Fuzz further prolongs the sequence to explore other complex states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' EXPERIMENTS In this section, we conduct extensive experiments to evaluate IR-Fuzz, seeking to address the following research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' RQ1: Can IR-Fuzz effectively detect contract vulnerabili- ties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' How is its performance against state-of-the-art tools?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' RQ2: Does IR-Fuzz achieve higher branch coverage than existing methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' RQ3: How efficient is IR-Fuzz in fuzzing smart contracts and generating test cases compared with other fuzzers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' RQ4: How much do different components of IR-Fuzz contribute to its performance in branch coverage and vul- nerability detection accuracy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We first introduce the experimental settings, then proceed to answer the above questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We also present a case study to allow for a better understanding of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Experimental Setup Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz in total contains 9K+ lines of C++ code, which is released for public use at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' com/Messi-Q/IR-Fuzz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We implemented IR-Fuzz on the basis of sFuzz [16] (a state-of-the-art smart contract fuzzer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In the experiments, we include seven open- source methods that either have a high number of citations IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 10 TABLE I Summary of vulnerability types supported by state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' TP is short for timestamp de- pendency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' BN represents block number dependency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' DG represents dangerous delegatecall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' EF represents Ether frozen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' UC represents unchecked external call;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' RE rep- resents reentrancy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' OF represents integer overflow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' SE represents dangerous Ether strict equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Methods Vulnerability Type #Citation or #GitHub Stars Publication TP BN DG EF UC RE OF SE Oyente [15] ✓ ✓ ✓ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='780 citations CCS’16 Osiris [45] ✓ ✓ ✓ 182 citations ACSAC’18 Securify [31] ✓ ✓ ✓ 604 citations CCS’18 ILF [22] ✓ ✓ ✓ ✓ 105 citations CCS’19 sFuzz [16] ✓ ✓ ✓ ✓ ✓ ✓ ✓ 91 citations ICSE’20 Mythril [46] ✓ ✓ ✓ ✓ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='900 GitHub stars White Paper ConFuzzius [23] ✓ ✓ ✓ ✓ ✓ ✓ 45 GitHub stars EuroS&P’21 IR-Fuzz ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ – – or receive many stars in GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The methods are summa- rized in Table I, where we illustrate the vulnerability types that they can detect, their numbers of citations or GitHub stars, and their publication information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For fuzzing tools, we select ConFuzzius [23], ILF [22], and sFuzz [16], which achieve state-of-the-art performance and support at least four vulnerability types on smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For static analysis tools, we select Mythril [46], Oyente [15], Osiris [45], and Securify [31], which are well-known vulnerability checkers for smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We compare IR-Fuzz with them in terms of branch coverage, effectiveness, and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' All experiments are conducted on a computer equipped with an Intel Core i9 CPU at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='3GHz, a GPU at 2080Ti, and 64GB Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Each experiment is repeated ten times, we report the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We obtain the dataset by crawling Etherscan [47] verified contracts, which are real-world smart contracts de- ployed on Ethereum Mainnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In practice, we removed 5,074 duplicate contracts by comparing the hash of the contract bi- nary code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Our final dataset contains a total 12,515 smart con- tacts that have source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' As listed in Table I, we focus on eight types of vulnerabilities in the dataset, namely timestamp dependency (TP), block number dependency (BN), dangerous delegatecall (DG), Ether frozen (EF), unchecked external call (UC), reentrancy (RE), integer overflow (OF), and dangerous Ether strict equality (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We deployed all smart contacts of the dataset to a local Ethereum test network for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For the ground truth labels of smart contracts, we define vulnerability-specific patterns for each kind of vulnerability to give a preliminary label and then manually check whether a smart contract in the dataset indeed has a certain vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In particular, using the defined vulnerability-specific patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', keyword matching), we could find smart contracts that may have vulnerabilities and save our time on labeling those contracts that are safe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', a contract with no ‘call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='value’ invocation will not have reentrancy vulnerabilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Effectiveness (RQ1) First, we benchmark IR-Fuzz against existing vulnerability detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We count the number of smart contracts that have vulnerabilities and are identified by each method, and present the accuracy, true positives, and false positives of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Comparing IR-Fuzz to State-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We first com- pare IR-Fuzz to other fuzzers and existing static analysis TABLE II Accuracy comparison (%) on different methods, including static analysis tools, fuzzers, and IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' ‘n/a’ denotes that a tool cannot detect the specific vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Methods Vulnerability Type (Accuracy) TP BN DG FE UC RE OF SE Mythril [46] n/a 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='97 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='95 n/a n/a 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='09 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='87 n/a Oyente [15] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='86 n/a n/a n/a n/a 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='61 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='76 n/a Osiris [45] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='56 n/a n/a n/a n/a 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='28 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='80 n/a Securify [31] n/a n/a n/a 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='42 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='24 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='52 n/a n/a ILF [22] n/a 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='53 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='99 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='65 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='71 n/a n/a n/a sFuzz [16] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='25 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='37 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='33 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='85 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='26 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='98 n/a ConFuzzius [23] n/a 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='70 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='47 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='91 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='68 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='33 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='35 n/a IR-Fuzz 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='25 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='18 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='33 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='05 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='77 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='79 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='73 TABLE III True and false positives of each method in identifying the eight types of smart contract vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Vulnerability Type (True / False Positives) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='TP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='DG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='FE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='UC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='RE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='OF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='SE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Mythril [46] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='4/63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20/20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='0/62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10/245 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Oyente [15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='12/6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='8/87 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='16/637 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Osiris [45] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='4/5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='12/139 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='12/632 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='Securify [31] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='0/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='7/208 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='4/194 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ILF [22] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='0/103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='8/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='0/3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='5/82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sFuzz [16] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='23/8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20/108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20/7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20/3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='7/100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10/68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='3/235 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='ConFuzzius [23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20/120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='4/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='0/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='8/86 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='6/131 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10/565 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='n/a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='IR-Fuzz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='92/5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='26/3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='58/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='65/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='83/36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='95/20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='21/10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='45/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='485 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Quantitative experimental results of each method are summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' From the table, we obtain the fol- lowing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) Compared with other methods, IR- Fuzz is able to identify more vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Inspiringly, IR- Fuzz has achieved a high accuracy (more than 90%) on all eight types of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) IR-Fuzz consistently outperforms state-of-the-art methods by a large margin in detecting each type of vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, for Ether frozen vulnerability (EF), IR-Fuzz gains 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='63% and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='14% accuracy improvements over Securify and ConFuzzius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' These strong empirical evidences suggest the great potential of IR- Fuzz to identify smart contract vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We attribute its superior performance to the key modules proposed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', sequence generation, seed optimization, and energy allocation, which boost the capability of IR-Fuzz in improving branch coverage and hunting vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (3) Promisingly, IR- Fuzz discovers a new kind of smart contract vulnerability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', dangerous Ether strict equality (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To the best of our knowledge, this vulnerability cannot yet be detected by current automatic tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We also present an illustrative case study on how our method detects this vulnerability in §V-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Analysis of True and False Positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To further eval- uate the effectiveness of IR-Fuzz, we examine the identified vulnerable contracts to see whether they are true positives or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Table III demonstrates the number of vulnerable contracts discovered by each method, as well as the numbers of true positives and false positives of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) From Ta- ble III, we observe that existing methods have not yet obtained a high true positive rate on the eight types of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, for unchecked external call vulnerability (UC), Securify and sFuzz generate 7 true positives, while ConFuzzius and ILF obtain 8 and 2 true positives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This is mainly due to the reason that conventional tools ignore handling exceptions for the return value of external calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) Moreover, we also find that existing methods have high false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For block number dependency vulnerability (BN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' fuzzing tools sFuzz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' ConFuzzius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' and ILF produce IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 11 0 10 20 30 40 Time (s) 0 20 40 60 80 100 Branch Coverage (%) IR-Fuzz sFuzz ILF ConFuzzius (a) Branch coverage of different meth- ods on small contracts 0 10 20 30 40 Time (s) 0 20 40 60 80 100 Branch Coverage (%) IR-Fuzz sFuzz ILF ConFuzzius (b) Branch coverage of different meth- ods on large contracts Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 6 Curves comparison: the tendency of branch coverage over time on different fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' over 100 false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For integer overflow vulnerability (OF), 632, 637, and 565 false positives are reported by Osiris, Oyente, and ConFuzzius, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The high false positives of these methods may stem from two facts: (i) Most methods tend to detect vulnerabilities using a few simple but imprecise patterns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', identifying block number vulnerability by crudely checking whether there is a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='number statement in the function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (ii) Many tools conservatively assume that all arithmetic operations returning a negative value are vulnerable, resulting in high false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz reports more true positives than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, for timestamp dependency vulnerability (TP), IR- Fuzz generates 92 true positives, 88, 80, and 69 more than Osiris, Oyente, and sFuzz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In total, IR-Fuzz finds vulnerabilities in 485 contracts, roughly 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='7 times more than sFuzz, which ranks the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For reentrancy vulnerability (RE), IR-Fuzz produces 95 true positives, which significantly outperforms the state-of-the-art tool Osiris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' More importantly, IR-Fuzz can precisely detect a new kind of vulnerability (SE) without reporting any false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We attribute the good performance of IR-Fuzz to the fact that it integrates the three presented new techniques, which are able to supplement each other for precise bug detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In summary, IR-Fuzz can effectively identify various vulnerabilities in smart contracts, surpassing existing static analysis tools and fuzzers by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Branch Coverage (RQ2) We now present evaluation results on branch coverage of IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We measure the number of distinct branches covered by the generated test cases in the test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, to examine the branch coverage on contracts with different sizes, we follow the settings of previous work [22] and split the dataset into 1,885 large contracts (≥3,600 instructions) and 10,630 small ones (<3,600 instructions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We compare with other fuzzers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', sFuzz, ILF, and Con- Fuzzius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, we visualize the comparison results on small contracts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 6(a) and on large contracts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 6(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We plot the tendency of branch coverage over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' It can be seen that IR-Fuzz consistently outperforms other fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Quantitatively, IR-Fuzz achieves 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10% cover- age on small contracts, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20%, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10%, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10% higher than sFuzz, ILF, and ConFuzzius, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' On large con- tracts, IR-Fuzz achieves 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='20%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='00%, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10% higher IR-Fuzz Mythril ConFuzziusOyente ILF Osirs sFuzz Securify Tool 0 50 100 150 200 250 300 Time (s) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='06 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='26 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='50 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='88 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='41 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='52 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='30 (a) Average execution time 20 40 60 80 100 120 Time (S) 0 5 10 15 20 25 30 35 40 Test Cases (K) sFuzz IR-Fuzz (b) Number of generated test cases Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 7 Visual comparison of efficiency on different tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' coverage, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, we also observe that IR-Fuzz reaches the highest coverage with less time required than other fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' On average, IR-Fuzz spent only 10s to achieve the highest coverage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10% on small contracts and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='10% on large contracts), while the other three fuzzers spent 18s, 16s, 13s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We conjecture that the advantages of IR-Fuzz in achieving high branch coverage come from three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' First, IR-Fuzz generates the high-quality function invocation sequence by adopting a dependency-aware sequence generation strategy, enforcing the fuzzer to tap into richer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Second, IR- Fuzz employs a branch distance-based measure to iteratively optimize the generated test cases, steering fuzzing towards covering new branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Thirdly, IR-Fuzz takes into account the significance of rare branches and branches that are likely to have vulnerabilities, and designs an energy allocation mech- anism to flexibly guide fuzzing energy allocation towards these critical branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, IR-Fuzz utilizes the feedback results generated by the energy allocation mechanism to guide seed mutation, which further increases branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Efficiency (RQ3) In this subsection, we systematically examine the efficiency of IR-Fuzz and compare it against other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' First, we conduct experiments to measure the overhead of IR-Fuzz by calculating the average execution time on each contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We run IR-Fuzz on the whole dataset, revealing that it spends 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='30s per contract on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 7(a) compares IR- Fuzz to other methods in terms of the average execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' From the figure, we observe that IR-Fuzz is significantly more efficient than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Particularly, its average execution time is 251s and 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='22s faster than Securify and sFuzz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We believe the reasons for the much faster speed of IR-Fuzz are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (1) IR-Fuzz can quickly infer the ordered function invocation sequence, accelerating fuzzing execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) IR-Fuzz adopts the branch distance-based measure to boost its efficiency in generating test cases, which requires much fewer mutations to reach a target branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (3) IR-Fuzz leverages the energy allocation mechanism to flexibly assign fuzzing resources, saving overall fuzzing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Next, we further measure the efficiency of IR-Fuzz by counting how many test cases are generated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specif- ically, each contract is run for 120 seconds in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We show the average statistics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 7(b), where the x- axis represents how long a contract is fuzzed, and the y-axis denotes the number of test cases generated during fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 12 TABLE IV Accuracy and coverage comparison (%) be- tween IR-Fuzz and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Method Vulnerability Type (Accuracy) Coverage TP BN DG FE UC RE OF SE IR-Fuzz-WSG 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='01 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='38 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='48 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='73 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='70 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='80 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='86 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='42 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='03 IR-Fuzz-WDM 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='94 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='56 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='31 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='15 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='26 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='51 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='02 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='12 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='89 IR-Fuzz-WEA 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='22 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='00 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='86 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='00 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='04 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='49 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='60 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='82 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='63 IR-Fuzz 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='05 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='79 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='06 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='48 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='03 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='73 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='73 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='73 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='65 From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 7(b), we can learn that (1) IR-Fuzz significantly generates more test cases than sFuzz within the same time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' On average, IR-Fuzz generates approximately 350 test cases per second, 290 more than sFuzz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' (2) The number of test cases generated by IR-Fuzz has increased rapidly over time while the process is slow in sFuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' These evidences reveal that IR-Fuzz can efficiently generate test cases for fuzzing smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Ablation Study (RQ4) By default, IR-Fuzz adopts the proposed sequence gener- ation strategy to generate the function invocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' It is interesting to see the effect of removing this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, IR-Fuzz utilizes a branch distance-based measure to select and evolve test cases iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We are curious about how much this method contributes to the performance of IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, IR-Fuzz introduces an energy allocation mechanism to flexibly guide fuzzing resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' It is useful to evaluate the contributions of this mechanism by removing it from IR-Fuzz as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In what follows, we conduct experiments to study the three components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Study on Sequence Generation Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We removed the sequence generation strategy (introduced in §IV-A) from IR-Fuzz and replaced it with a random sequence construction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This variant is denoted as IR-Fuzz-WSG, where WSG is short for without sequence generation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Quantitative results are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We can observe that the performance of IR-Fuzz is significantly better than IR-Fuzz- WSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' For example, on the reentrancy detection task, IR-Fuzz achieves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='97% and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='62% improvement in terms of accuracy and branch coverage, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Study on Branch Distance-based Seed Evolution Paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To evaluate the effect of the branch distance- based seed evolution paradigm, we analyze the performance of IR-Fuzz with and without it, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Towards this aim, we modify IR-Fuzz by removing this mechanism, utilizing random test case generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This variant is denoted as IR- Fuzz-WDM, where WDM is short for without the distance measure mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The empirical findings are demonstrated in Table IV, where we can observe that the accuracy and branch coverage of IR-Fuzz-WDM are lower than IR-Fuzz by an average of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='03% and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='76% on the eight types of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This reveals that incorporating the branch distance-based measure is necessary and critical to improve the performance of IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Study on Energy Allocation Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We further investigate the impact of the energy allocation mechanism in IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, we remove this mechanism while replacing it with assigning fuzzing energy equally to every contract Gamble { uint256 private number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' uint256 phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' address winner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' constructor (uint256 num) { require(num < 100);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' number = num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' phase = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } /* 0: guess 1: start a new game */ function guess(uint256 fee) payable external { require (phase == 0 && fee == 10 finney);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' if (address(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='balance == number * 10 finney) { /* Ether strict equality */ winner = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' phase = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } function newGame(uint256 num) external { require(phase == 1 && msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='sender == winner);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='transfer(address(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='balance);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' require(num < 100);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' number = num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' phase = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' } } } 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 8 An example contract where IR-Fuzz detects a new kind of vulnerability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', dangerous Ether strict equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This new variant is termed as IR-Fuzz-WEA, namely IR-Fuzz without an energy allocation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The com- parison results are presented in Table IV, where all eight types of vulnerabilities are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We can clearly see that the accuracy and branch coverage of IR-Fuzz-WEA are lower than IR-Fuzz by an average of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='87% and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='02%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This suggests that the energy allocation mechanism contributes to significant performance gains in IR-Fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Case Study We now present a case study on a new vulnerability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', dangerous Ether strict equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' To our knowledge, existing investigated methods cannot expose this vulnerability yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' 8 shows a simplified example that implements a gambling game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' A user can join the game by transferring participation fees with 10 finney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' If a user is the number-th participant, he will become the winner of the game (line 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The winner can obtain the whole balance of the contract by calling newGame() and starting a next round of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, if the contract owner had pre-stored some Ethers in the contract, the balance of the contract will never be equal to the sum of users’ participation fees (namely, the branch condition at line 14 will never be satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' This indicates that there will be no winner in the game, and the participation fees in the contract will be permanently frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We empirically checked this contract using existing tools and manually inspected their generated reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Unfortunately, the dangerous Ether strict equality vulnerability cannot yet be detected by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In contrast, IR-Fuzz successfully identifies this vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, T1: IR-Fuzz infers the function invocation sequence as: guess()→ newGame() and generates a test case to cover the requirement at line 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' T2: Record the instruction BALANCE when the fuzzing process IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 13 reaches line 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' T3: Check if BALANCE is followed by the jump or compare instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' T4: IR-Fuzz finds that line 14 is reachable and the vulnerability-specific patterns of dangerous Ether strict equality are triggered, outputting that the contract has such a vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' DISCUSSION In this section, we discuss the limitations of IR-Fuzz and potential future improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Sequence Generation Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz generates the ordered function invocation sequence with the guidance of the order priority computation rules mentioned in §IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' We calculate the order priority of function calls in the sequence by analyzing the data flow dependencies of global variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' In the case that several functions perform frequent write and read operations on global variables, the calculation of function order priority may bring a certain amount of computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Seed Mutation Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' IR-Fuzz refers to several seed mutation strategies adopted from AFL, usually using bit manipulation techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=', bit flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' However, such a method still bears the problem of generating repetitive and invalid test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Moreover, arbitrarily mutating bits of a test input may ignore certain critical parts of the input that should not be mutated, reducing the probability of hitting the branches guarded by strict conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Therefore, in the subsequent work, we may focus on enabling the fuzzer not to mutate these crucial parts of a test case, making the fuzzing trigger deep and complex states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' CONCLUSION In this paper, we present IR-Fuzz, a fully automatic fuzzing framework equipped with invocation ordering and crucial branch revisiting, to detect vulnerabilities in smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Specifically, we propose a sequence generation strategy con- sisting of invocation ordering and prolongation to generate the high-quality function invocation sequence, enforcing the fuzzer to trigger complex and deep states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Furthermore, we design a seed optimization paradigm that engages a branch distance-based measure to evolve test cases iteratively to- wards a target branch, alleviating the randomness of test case generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Finally, we develop an energy allocation mechanism to flexibly guide fuzzing resource allocation to- wards rare and vulnerable branches, improving the overall fuzzing efficiency and branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Experimental results demonstrate that IR-Fuzz significantly surpasses state-of-the- art fuzzing approaches by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' Our implementation and dataset are released to facilitate future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' The presented techniques in IR-Fuzz might also be transferable to fuzz other software programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content=' REFERENCES [1] Z.' 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https://etherscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} +page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfhAd2/content/2301.03943v1.pdf'} diff --git a/xtE5T4oBgHgl3EQfNA53/content/tmp_files/2301.05486v1.pdf.txt b/xtE5T4oBgHgl3EQfNA53/content/tmp_files/2301.05486v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c7e174c1494f4fbfe11c1626513f5bf9e6b6490 --- /dev/null +++ b/xtE5T4oBgHgl3EQfNA53/content/tmp_files/2301.05486v1.pdf.txt @@ -0,0 +1,992 @@ +Significant Unconventional Anomalous Hall Effect in Heavy Metal/Antiferromagnetic +Insulator Heterostructures +Yuhan Liang,1, ∗ Liang Wu,2, † Minyi Dai,3 Yujun Zhang,4 Qinghua Zhang,5 Jie Wang,2 +Nian Zhang,6, 7 Wei Xu,4 Le Zhao,8 Hetian Chen,1 Ji Ma,2 Jialu Wu,1 Yanwei Cao,9, 10 +Di Yi,1 Jing Ma,1 Wanjun Jiang,8 Jia-Mian Hu,3 Ce-Wen Nan,1, ‡ and Yuan-Hua Lin1, § +1School of Materials Science and Engineering, Tsinghua University, Beijing, 100084, China +2Faculty of Materials Science and Engineering, Kunming University +of Science and Technology, Kunming, 650093, Yunnan, China +3Department of Materials Science and Engineering, +University of Wisconsin-Madison, Madison, WI, USA +4Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China +5Institute of Physics, Chinese Academy of Sciences, Beijing 100049, China +6State Key Laboratory of Functional Materials for Informatics, +Shanghai Institute of Microsystem and Information Technology, +Chinese Academy of Sciences, Shanghai 200050, China +7CAS Center for Excellence in Superconducting Electronics(CENSE), +Chinese Academy of Sciences, Shanghai 200050, China +8Department of Physics, Tsinghua University, Beijing 100084, China +9Ningbo Institute of Materials Technology and Engineering, +Chinese Academy of Sciences, Ningbo 315201, China +10Center of Materials Science and Optoelectronics Engineering, +University of Chinese Academy of Sciences, Beijing 100049, China +(Dated: January 16, 2023) +The anomalous Hall effect (AHE) is a quantum coherent transport phenomenon that convention- +ally vanishes at elevated temperatures because of thermal dephasing. Therefore, it is puzzling that +the AHE can survive in heavy metal (HM)/antiferromagnetic (AFM) insulator (AFMI) heterostruc- +tures at high temperatures yet disappears at low temperatures. In this paper, we report that an +unconventional high-temperature AHE in HM/AFMI is observed only around the N´eel temperature +of AFM, with large anomalous Hall resistivity up to 40 nΩ cm. This mechanism is attributed to the +emergence of a noncollinear AFM spin texture with a non-zero net topological charge. Atomistic +spin dynamics simulation shows that such a unique spin texture can be stabilized by the subtle in- +terplay among the collinear AFM exchange coupling, interfacial Dyzaloshinski-Moriya interaction, +thermal fluctuation, and bias magnetic field. +I. +INTRODUCTION +Heavy metal (HM)/antiferromagnetic (AFM) insula- +tor (AFMI) heterostructures are an emerging essential +system for investigating the interaction between spin cur- +rent and antiferromagnetic (AFM) order, which have the +potential for applications in energy-efficient, ultrafast, +and robust spintronics devices.[1–10] In particular, the +reflected spin current after interactions with the magnetic +order at the interface can provide valuable information +for determining the magnetic order, which has already +been well established in the counterpart HM/ferromagnet +(FM) heterostructures, such as the spin Hall magnetore- +sistance (SMR) and its derivative spin Hall-anomalous +Hall effect (SH–AHE).[11, 12] Most recently, consider- +able attention has been paid to SMR in HM/AFMI het- +erostructures, which is considered a probe for current- +induced switching of the AFM order.[2, 13–16] Despite +∗ These authors contributed equally +† These authors contributed equally; liangwu@kust.edu.cn +‡ cwnan@tsinghua.edu.cn +§ linyh@tsinghua.edu.cn +the extensive investigations of SMR in HM/AFMI het- +erostructures, the anomalous Hall effect (AHE) in such +systems has not yet been subjected to the same exam- +ination, therefore, its origin, particularly whether it is +caused by SMR as in HM/FM heterostructures, remains +unclear. +Recently, a high-temperature AHE was observed in +HM/AFMI heterostructures, which are based on Ta, Pt, +and W grown on Cr2O3 (AFMI) with an anomalous Hall +resistivity (ρAHE) of approximately 1 nΩ cm.[3–5] De- +spite the controversy regarding the underlying mecha- +nism, a ubiquitous phenomenon insensitive to the selec- +tion of the HM and crystalline orientation of AFMI was +observed. That is, a superparamagnetism-like AHE sig- +nal with a zero-coercive field (Hc) exists at temperatures +significantly higher than the bulk Cr2O3 N´eel tempera- +ture TN (307 K) but disappears in the low-temperature +region (< 200 K).[3–5] However, for heterostructures con- +sisting of AFMI with significantly higher bulk TN, for ex- +ample, the α-Fe2O3 (TN ∼ 950 K) and NiO (TN ∼ 523 +K), an AHE at approximately room temperature has not +been reported. Thus, it is desirable to extend the AHE +to other AFMI-based heterostructures, and understand +arXiv:2301.05486v1 [cond-mat.mes-hall] 13 Jan 2023 + +2 +400 +300 +200 +100 +0 +Temperature (K) +6 +5 +4 +3 +2 +1 +NiO Thickness (uc) +MgO (001) +NiO (3 uc) +Pt (3 nm) +(a) +(c) +Pt(3 nm)/NiO(3 uc)/MgO(001) +Pt (3 nm)/NiO(x uc) +Pt (x nm)/NiO(3 uc) +2 nm +-30 +-20 +-10 +0 +10 +20 +30 +ρAHE (nΩ·cm) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +µ0H (T) + 250 K + 300 K + 350 K + 400 K + 5 K + 60 K + 100 K + 150 K + 200 K +7 +6 +5 +4 +3 +2 +Pt Thickness (nm) +40 +30 +20 +10 +0 +-ρAHE (nΩ·cm) +(d) +(b) +Pt (x nm)/NiO(3 uc) +FIG. 1. Observation of AHE in Pt/NiO/MgO heterostructures. (a) The STEM of Pt(3 nm)/NiO(3 uc)/MgO(001) (the inset +is the SAED pattern). (b) ρAHE of the same sample at varying temperatures. (c-d) Evolution of the ρAHE with temperature +and thickness of NiO and Pt. +its physical mechanism. +In this study, we demonstrate that the AHE can +be induced in HM/NiO heterostructures at elevated +temperatures (up to 400 K) by controlling the AFM– +paramagnetic (PM) (AFM–PM) transition of an ultra- +thin epitaxial NiO film. The value of ρAHE was approx- +imately 40 nΩ cm, which is higher than the recorded +ρAHE in HM/magnetic insulator heterostructures. The +origin of the AHE in HM/AFMI heterostructures is at- +tributed to the emergence of AFM spin textures with +uncompensated topological charges during the AFM–PM +phase transition. Moreover, the line shape of the AHE +signal depends on the defect at the HM/AFMI interface, +which can induce topological-Hall-effect-like (THE-like) +signals based on the two-channel AHE scenario. +II. +RESULTS AND DISCUSSIONS +We first employed scanning transmission electron mi- +croscopy (STEM) to characterize the structural prop- +erties of the HM/NiO heterostructures with ultrathin +NiO. The STEM image and selected area electron diffrac- +tion (SAED) pattern of a representative Pt(3 nm)/NiO(3 +uc)/MgO(001) heterostructure are shown in Figure 1a, +where uc denotes unit cells (see details in Experimen- +tal Section). +The NiO was fully epitaxial on the +MgO(001) substrate with a rock-salt crystalline struc- +ture, whereas the Pt overlayer was polycrystalline. The +field-dependent magnetization measurements showed no +observable macroscopic ferromagnetism (see Figure S1, +Supporting Information). To investigate both the Pt and +NiO thickness dependence of the ρAHE, we performed +Hall measurements at 5–400 K, with subtraction of the +linear ordinary Hall signal (Complete AHE data are pre- +sented in Figure S2, Supporting Information). +A low +driving current density of ∼102 A cm−2 was used to sup- +press the Joule heating. The absence of an AHE in the +reference sample without the NiO layer is shown in Fig- +ure S3 (Supporting Information). +A temperature-dependent AHE was observed in the +Pt(3 nm)/NiO(3 uc)/MgO(001) heterostructure (Figure +1b). The AHE was only observed at high temperatures +(250–400 K). This trend is consistent with that of Cr2O3- + +3 +Pt(3 nm)/NiO(3 uc)/MgO(111) +W(3 nm)/NiO(3 uc)/MgO(001) +-30 +-20 +-10 +0 +10 +20 +30 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +µ0H (T) + 250 K + 300 K + 350 K + 400 K + 5 K + 60 K + 100 K + 150 K + 200 K +(b) +(a) +-30 +-20 +-10 +0 +10 +20 +30 +ρAHE (nΩ·cm) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +µ0H (T) + 250 K + 300 K + 350 K + 400 K + 5 K + 60 K + 100 K + 150 K + 200 K +FIG. 2. Insensitivity of AHE to AFMI orientation and HM type. (a-b) ρAHE of Pt(3 nm)/NiO(3 uc)/MgO(111) and W(3 +nm)/NiO(3 uc)/MgO(001) heterostructures at various temperatures. +based HM/AFMI heterostructures.[3–5] Comprehensive +contour plots of ρAHE in Pt(3 nm)/NiO(x uc)/MgO(001) +and Pt(x nm)/NiO(3 uc)/MgO(001) are shown in Fig- +ure 1c and Figure 1d. The AHE mainly existed above +150 K, and its magnitude significantly depended on the +NiO and Pt thicknesses. As the NiO thickness increased, +the minimum temperature required for the appearance +of ρAHE increased. For NiO thicker than 6 uc, we ob- +served no apparent ρAHE signals within the accessible +temperature (5–400 K) and field (±8 T) ranges. Figure +1d shows the decay of ρAHE as Pt becomes thicker, indi- +cating the interfacial nature of the AHE. It is noted that +a local maximum of ρAHE appeared at approximately +5 K, which could be induced by the skew scattering of +electrons by localized paramagnetic centers (Giovannini– +Kondo model).[17, 18] +As a G-type collinear AFM, the superexchange in- +teraction between Ni ions results in antiferromagneti- +cally stacking of ferromagnetic {111} planes.[10] To in- +vestigate how the crystallographic orientation of AFMI +and the type of HM influence the AHE, we further +measured ρAHE in the Pt(3 nm)/NiO(3 uc)/MgO(111) +and W(3 nm)/NiO(3 uc)/MgO(001) heterostructures. A +similar temperature-dependent AHE behavior was ob- +served (Figure 2), indicating that the AHE was insen- +sitive to the crystal orientation of AFMI and the type +of HM. In particular, W showed the opposite sign of +the spin Hall angle to Pt[19], however, the sign of AHE +remained the same.[4] In addition, the line shape of +ρAHE for W(3 nm)/NiO(3 uc)/MgO(001) was slightly +different from that of Pt(3 nm)/NiO(3 uc)/MgO(001), +which will be discussed subsequently. +Additional con- +trol experiments were performed to clarify the role of +the HM and concomitant spin Hall effect (SHE) and +the strong spin-orbit coupling on the AHE. First, no +AHE signal was observed for Ti(3 nm)/Cu(3 nm)/NiO(3 +uc)/MgO(001) as expected (Figure S4, Supporting In- +formation). Second, a weaker AHE signal was observed +in Pt(3 nm)/Cu(1.2 nm)/NiO(3 uc)/MgO(001) (Figure +3a), ruling out the proximity-induced-ferromagnetism +in Pt. +Finally, Pt/W/NiO/MgO(001) heterostructures +were fabricated, and the competing of spin current[19] +initially decreased the magnitude of the AHE, and even- +tually reversed the polarity (sign) of the AHE with thick- +ened Pt (Figure S5, Supporting Information). The ex- +periments demonstrate that the SHE in HM plays a +critical role, and the scattering of the spin current by +AFMI (in a reflective manner) induces the AHE.The +experimental results, along with the ρAHE values re- +ported for HM/magnetic insulator heterostructures, are +presented in Figure 3c. +We achieved ρAHE values up +to 40 nΩ cm in the Pt/NiO heterostructures, which is +higher than the recorded value in HM/magnetic insula- +tor heterostructures.[4, 20–23] Additionally, the scaling +relation of the HM/AFMI heterostructure is shown in +Figure S6 (Supporting Information), which is similar to +that of the gating-induced ferromagnetic Pt.[24, 25] +An additional THE-like Hall effect (the bump and dip +feature added to the AHE signals) was observed in W(3 +nm)/NiO (3 uc)/MgO(001) and Pt/Cu(1.2 nm)/NiO (3 +uc)/MgO(001) compared to Pt/NiO (3 uc)/MgO(001) +(Figure 2b and 3b). From the Ellingham diagram, it can +be observed that the Cu/Cu2O, Ni/NiO and W/WO3 +lines are relatively close, whereas the Pt/PtO2 line is +significantly higher than all the above lines.[26] This im- +plies that the oxygen can migrate from NiO to Cu and W +to induce additional oxygen vacancies in NiO, resulting +in weak ferromagnetism.[27–30] To confirm this defect- +induced THE-like signal, we deposited a NiO layer with +a significantly lower oxygen pressure to enrich the oxy- +gen vacancies. +A THE-like signal was detected again +(Figure 3b), indicating that the THE-like signal was re- +lated to oxygen vacancy-induced dilute ferromagnetism. +Thus, the THE-like signal can be regarded as an al- + +4 +Pt(3 nm)/NiO(3 uc, 10-4 Pa)/MgO(001) +(b) +Pt(3 nm)/Cu(1.2 nm)/NiO(3 uc)/MgO(001) +-10 +-5 +0 +5 +10 +ρAHE (nΩ·cm) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +µ0H (T) + 250 K + 300 K + 350 K + 400 K + 10 K + 50 K + 100 K + 150 K + 200 K +(a) +(c) +40 +30 +20 +10 +0 +40 +30 +20 +10 +0 +|ρAHE| (nΩ·cm) +400 +300 +200 +100 +0 +Temperature (K) +Pt(3 nm)/NiO(4 uc)/MgO(001) +Pt(3 nm)/NiO(3 uc)/MgO(001) +Pt(3 nm)/NiO(3 uc)/MgO(111) +Pt(3 nm)/NiO(3 uc, 10 +-4 Pa)/MgO(001) +Pt(3 nm)/Cu(1.2 nm)/NiO(3 uc)/MgO(001) +Pt/Cr2O3 Ref. [4] +Pt/TmIG Ref. [20] +Pt/TmIG Ref. [21] +Pt/YIG Ref. [22] +Pt/Cr2Ge2Te6 Ref. [23] +-10 +-5 +0 +5 +10 +ρAHE (nΩ·cm) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +µ0H (T) + 340 K + 360 K + 400 K + 200 K + 250 K + 300 K +FIG. 3. Control experimental results for AHE in HM/AFMI heterostructures. (a-b) ρAHE of Pt(3 nm)/NiO(3 uc)/MgO(111) +and W(3 nm)/NiO(3 uc)/MgO(001) at various temperatures. (c) Temperature-dependent saturated ρAHE from our data and +reported HM/magnetic insulators.[4, 20–23] The four-pointed stars represent our data, and the circles represent data from +references. +ternative two-channel AHE scenario engineered by the +defects.[31, 32] One of the channels results from the AFM +order, whereas the other is related to the dilute ferromag- +netism, similar to the HM/FM heterostructures.[12, 33] +Next, we discuss the relationship between the TN of +the AFMI layer and the temperature-dependent AHE +behavior. The TN of the ultrathin NiO is highly depen- +dent on its thickness and boundary conditions. It has +been reported that an ultrathin NiO film grown on MgO +can have a significantly reduced TN from its bulk value, +for example, the TN of NiO(3 uc)/MgO(001) was lower +than 40 K, and its TN could be considerably enhanced up +to 390 K simply by Ag capping owing to image charge +screening.[34] Therefore, the TN in the same 3 uc NiO +film in our Pt/NiO/MgO(001) should also be enhanced +by the Pt overlayer to induce a high-temperature AHE. +Note that, the short-range magnetic order above TN[35– +38] could also support the high-temperature AHE. +To determine the TN of NiO in our Pt/NiO/MgO sam- +ple, we performed temperature-dependent X-ray absorp- +tion spectroscopy (XAS) measurements at the Ni L2 +edge. The decrease in the L2 ratio with increasing tem- +perature indicates an AFM–PM transition.[34, 39] The +continuous decrease in the Ni L2 ratio of Pt(3 nm)/NiO(3 +uc)/MgO(001) at 298–523 K indicates that this temper- +ature range is within the continuous second-order AFM– +PM transition range of NiO (left panel of Figure 4a), +and the TN should be as high as 523 K and comparable +to that of the bulk NiO (Here, the TN is defined as the +temperature of the absence of the long-range AFM or- +der via thermal fluctuation). This indicates that Pt can +impose a more pronounced effect than Ag on stabilizing +the AFM order in NiO. However, for Pt(3 nm)/NiO(6 +uc)/MgO(001), the Ni L2 ratio started to decline above +400 K, showing that the AFM–PM transition in this sam- + +5 +Temperature +AFM +PM +AFM⎯PM +AHE +AFM order strength +Thermally driven topological AFM spin texture +(a) +(b) +868 +870 +872 +874 +868 +870 +872 +874 +Pt(3 nm)/NiO(3 uc)/MgO(001) +XAS (a. u.) +300 +400 +500 +1.3 +1.4 +L2 ratio +L2 ratio +T (K) +298 K +523 K +Pt(3 nm)/NiO(6 uc)/MgO(001) +Photon energy (eV) +300 +400 +500 +1.2 +1.3 +T (K) +298 K +523 K +~𝑇! +FIG. 4. +Correlation between AHE and AFM–PM transition. +a) Temperature-dependent XAS for Pt(3 nm)/NiO(3 +uc)/MgO(001) and Pt(3 nm)/NiO(6 uc)/MgO(001). b) Curves are extracted from AHE (red symbols) and XAS (blue symbols) +data from above two samples (circles for the sample with 3 uc NiO, and squares for that with 6 uc NiO), normalized by the +putative AFM–PM transition temperature ranges. The AFM order strength is represented by the L2 ratio of XAS data. The +insets show the calculated spin structures in the ultrathin NiO, represented by the distribution of N’eel vector n, and the vector +length is not proportional to the magnitude of the n at various temperatures. +ple was initiated at 400 K. W exhibited a similar effect +on 3 uc NiO, that is, the AFM–PM transition covered +the temperature range of 298–523K (Figure S7, Sup- +porting Information). A comparison between the AHE +and XAS data showed that the AHE coincided with that +of the AFM–PM transition in the common temperature +range of 300–400 K. Notably, the absence of AHE in +Pt(3 nm)/NiO(6 uc)/MgO(001) below 400 K can be ex- +plained by the AFM–PM transition starting at tempera- +tures higher than 400 K. +Based on the consistent temperature range between the +AFM–PM transition and the AHE, the thermally soft- +ened AFM order during the AFM–PM transition plays a +crucial role in the AHE of HM/AFMI heterostructures. +In Cr2O3-based HM/AFMI heterostructures, the AFM– +PM transition generates the temperature-dependent un- +compensated magnetic moment on the HM/AFMI inter- +face based on SH–AHE scenario, which is however ques- +tioned by X-ray magnetic dichroism measurement.[3–5] +In our case, the magnetically compensated (100)-facet +and uncompensated (111)-facet in NiO exhibited simi- +lar temperature-dependent AHE. Thus, the conventional +SH–AHE scenario based on uncompensated magnetic +moments should not account for the appearance of the +AHE. +Three mechanisms cause the conventional AHE: intrin- +sic contribution by the Berry phase and extrinsic contri- +butions by skew scattering and side jump.[40, 41] In this +study, such an AHE in the heterostructure system oc- +curs owing to the interaction between the spin current +(SHE in HM) and the magnetic structure (magnetic in- +sulator, in this case, AFMI). Recently, it has been re- +ported that an AFM topological spin texture emerges +around the phase-transition temperature of Pt/α-Fe2O3 +heterostructures.[42] A significant Fert–Levy-type inter- +facial Dzyaloshinskii–Moriya interaction (DMI) exists +and favors noncollinear spin textures, considering the +space-inversion symmetry breaking on the interface.[43] + +6 +Thus, the AFM topological spin textures could exist in +HM/AFMI heterostructures at room temperature, re- +sponsible for the AHE by scattering spin-polarized elec- +trons with an effective field ⟨bz⟩ = +√ +3φ0/(2λ2) for per +unit topological charge, where ⟨bz⟩ is the out-of-plane ef- +fective field, φ0 is the unit flux and λ is the size of spin +texture.[44, 45] +We performed atomistic spin dynamics simulations to +investigate the formation of AFM topological spin tex- +tures. We found that a noncollinear AFM spin texture +could emerge during the AFM–PM transition (Figure 4b) +owing to the interaction among the collinear AFM ex- +change coupling, DMI, thermal field, and external mag- +netic field (see the details in Experimental Section). In +the intermediate-temperature region, the thermal en- +ergy kBT is sufficiently high to destabilize the otherwise +collinear AFM order (for example, the spin structure in +Figure 4b with kBT = 0), such that interfacial DMI can +induce the noncollinear AFM spin texture. At high tem- +peratures, the thermal perturbation dominates, yielding +a PM-like spin state. The curves in Figure 4b are guides +for the eye to show the coincident temperature range +between the AFM–PM transition and the AHE occur- +rence (the putative AFM–PM transition ranges for sam- +ple Pt(3 nm)/NiO(3 uc)/MgO(001) and Pt(3 nm)/NiO(6 +uc)/MgO(001) are approximately 200–523 K and 400– +523 K, respectively.). +The topological charges associated with the two spin +sublattices are denoted as QA and QB, respectively, and +can be calculated from the spatial distribution of the N´eel +vector (see Experimental Section and Figure S8, Sup- +porting Information). For a canonical AFM skyrmion, +the local effective fields induced by QA and QB have the +same magnitude but opposite signs because QA+QB = 0 +and |QA| = |QB| = 1 leading to the generation of a zero +net effective field.[45] Therefore, an AFM skyrmion can- +not cause an AHE because the time-reverse symmetry is +preserved. However, applying a large out-of-plane mag- +netic field can break the time-reverse symmetry as the +two spin sublattices are canted (Figure S9, Supporting +Information), yielding a non-zero net topological charge +(QA + QB ̸= 0) and hence, an AHE. +III. +CONCLUSION +In this paper, we reported an unconventional high- +temperature AHE insensitive to the selection of the HM +and AFMI in HM/AFMI heterostructures, which was +suppressed and even eliminated in a low-temperature re- +gion. This is explained by an extended SMR model that +includes the AFM spin texture. The emergence of the +AFM spin texture (for example, skyrmion or meron) is +considered to be stabilized by the DMI at the HM/AFMI +interface, and can only occur at an intermediate temper- +ature during the AFM–PM phase transition, as demon- +strated via micromagnetic simulations. +The THE-like +Hall effect can be triggered by controlling the fabrication +conditions of the samples, which can be regarded as an +additional AHE channel induced by defect-induced weak +FM. The room-temperature AHE in AFMI heterostruc- +tures demonstrates the strong interaction between the +AFM order and spin current, which aids in developing +AFM-based spintronics devices. +IV. +EXPERIMENTAL SECTION +Sample fabrication +The epitaxial NiO thin films were fabricated using pulsed +laser deposition (PLD). The growth conditions were as +follows: a growth temperature of 650 ◦C (measured using +a pyrometer), oxygen background pressure of 50 mTorr, +an excimer laser with a wavelength of 248 nm, a rep- +etition rate of 3 Hz, an energy density of 1.4 J cm−2, +and target-substrate distance of 5.5 cm. After deposition +and cooling down to room temperature, the NiO films +were immediately transferred to a magnetron sputtering +chamber (AJA International, Inc.) with a background +pressure better than 2 × 10−8 torr, and heated to 150 ◦C +for 15 minutes, to prevent possible surface gas absorp- +tion. Then, DC sputtering was then employed to deposit +the Pt layer at room temperature with a power of 30 W, +background Ar of 3 mTorr, resulting in a deposition rate +of 2.6 nm/min. The W and Cu were deposited under the +same conditions at deposition rates of 1.3 nm/min and +3.6 nm/min, respectively. +Structural, +electrical and magnetic properties +characterization +The structural properties were characterized using a +high-resolution X-ray diffractometer (XRD, Malvern +Panalytical). +The magnetic hysteresis loop was mea- +sured using a superconducting quantum interference de- +vice (MPMS, Quantum Design). +The transport prop- +erties were measured using the van der Pauw method +in a Physical Property Measurement System (PPMS, +Quantum Design DynaCool system). The measured Hall +signals were firstly treated using an asymmetry proce- +dure, and the ordinary Hall signals were then subtracted. +Cross-sectional high-resolution transmission electron mi- +croscopy samples were prepared using a focused ion beam +(FIB, Zeiss Auriga) with Ga+ ions. +X-ray absorption spectroscopy +Element-specific X-ray absorption spectroscopy (XAS) +was measured using total electron yield (TEY) at Beam- +line BL02B02 of the Shanghai Synchrotron Radiation Fa- +cility (SSRF) and Beamline 4B9B of the Beijing Syn- +chrotron Radiation Facility (BSRF). To evaluate the +magnetic properties of these NiO thin films, we utilized +the magnetic linear dichroism (MLD) effect in the Ni +L2 XAS. The normal incidence of X-rays resulted in the +electric field component of X-rays being parallel to the +sample plane. The sample temperature was controlled +by laser heating the sample holder. A linear background + +7 +connecting raw data points at 870.8 eV and 871.8 eV was +first subtracted from the raw data, and the spectra were +normalized from 0 to 1. The L2 ratio was defined as the +normalized peak intensity at 871.8 eV. +Atomistic spin dynamics simulations +Micromagnetic simulations were performed to simulate +the spin structure in the AFM NiO with interfacial DMI. +The total Hamiltonian of the AFM thin-film H included +the Hamiltonians of the exchange interaction Hexch, uni- +axial anisotropy Hanis, external fields Hext, and the +Hamiltonian of the interfacial Dyzaloshinskii–Moriya in- +teraction (DMI) HDMI, expressed as follows: +H =Hexch + Hanis + Hext + HDMI += − +� + +JSk · Sl − +� +k +Ka (Sk · ne)2 ++ +� + +Dkl · (Sk × Sl) , +(1) +where Sk and Sl are the orientation vectors of the lo- +cal spin and neighboring spins, respectively; J is the ex- +change constant, Ka is the uniaxial anisotropy constant, +ne is the uniaxial easy axis, and Dkl is the DMI constant. +In the simulation, a two-dimensional system of 64 × +64 grids was considered in which each grid with a size +of 1 nm × 1 nm, contained two neighboring spins. By +linearly combining Sm and Sn inside the same gird, the +net magnetization vector m = (Sm +Sn)/2 and the N´eel +vector n = (Sm − Sn)/2 were defined. The expression +of the free energy was derived by substituting Sm and +Sn with m and n, respectively, in the total Hamiltonian. +Under the continuum approximation[46], the total free +energy is expressed as follows: +Etot = +� +dV +� +A∗ � +(∂xn)2 + (∂yn)2� +− K (n · ne)2 ++ D [nz(∇ · n) − (n · ∇)nz] − Hext · n +� +, +(2) +where A∗ = −2.42 pJ m−1, is the continuum-scale ex- +change constant, which was converted from the AFM ex- +change interaction J = −19.01 meV[47]. We assumed the +existence of a perpendicular magnetic anisotropy (i.e., +ne//z) in the ultrathin (3 u.c.) NiO owing to the inter- +facial symmetry breaking, with a continuum anisotropy +constant K = 1.5× 105 J m−3(which is a typical number +utilized in existing micromagnetic simulation investiga- +tions of AFM skyrmions)[48], D∗ = −3.5×10−3 J m−2 is +the continuum DMI constant, which is a value calculated +from the first-principles in a similar NiO/Au system.[49] +The free energy expression of the N´eel vector n in +Eq. (2) is the same as that of the magnetization in the +ferromagnetic materials. +Therefore, we performed mi- +cromagnetic simulations using the open-source software +MuMax3[50] to calculate the equilibrium configuration of +the N´eel vector. The evolution of n was determined by +solving the Landau–Lifshitz–Gilbert (LLG) equation, +∂n +∂t = − +γ0 +1 + α2 (n × Heff + αn × n × Heff) , +(3) +where α = 0.01 is the Gilbert damping coefficient[49] and +γ0 is the gyromagnetic ratio; Heff = − +1 +µ0Ms +δEtot +δn +is the +effective field, where µ0 is the vacuum permeability, amd +Ms = 6.2 × 105 A m−1 is the saturation magnetization +of NiO[49]. +The influence of thermal fluctuations was modeled by +adding a thermal fluctuation field Htherm to Heff, ex- +pressed by Htherm = η +� +2αkBT +µ0Msγ0∆V ∆t, where kBT is the +Boltzmann constant, T is the temperature, ∆V is the vol- +ume of each simulation cell, and ∆t is the time interval +in a real unit. η = η(r, t) = (ηx, ηy, ηz) is a white-noise +distributed stochastic vector uncorrelated in space and +time. +The temporal mean value of ηi(i = x, y, z) was +zero. The LLG equation was solved using fourth-order +Runge–Kutta methods, with a discretized time interval +of 20 fs. A uniform distribution of n in the z-direction +was used as the initial state for the low-temperature re- +gion (< 8 K), and an artificially generated single AFM +skyrmion was used for the higher-temperature region (> +8 K). The equilibrium distribution of n was assumed to +be reached when the total free energy density Etot no +longer changes significantly with time. +The topological charge of the two sublattices of n in the +equilibrium state, QA and QB were calculated as follows +to quantify the topological state of the AFM thin film[51]: +QA = 1 +8π +� +site A +qijk ; +QB = 1 +8π +� +site B +qijk , +(4) +where qijk is calculated for each unique triangle with +grids i, j, and k as vertices using Equation (5) +tan(qijk +2 ) = +ni · (nj × nk) +1 + ni · nj + ni · nk + nj · nj +. +(5) +The equilibrium state of n was determined based on the +values of QA and QB. If QA + QB = 0, the distribution +of n is identified as a collinear distribution, otherwise, it +is a topological AFM spin texture. If QA ̸= QB, there is +no evident ordered spin texture[46]. +Acknowledgements +Y. L. and L. W. contributed equally to this study. We +acknowledge the insightful discussions with Shuai Dong +and Jiahao Han. We thank the staff at Beamline 4B9B of +the BSRF and Beamline 02B02 of the SSRF for fruitful +discussions and experimental assistance. L. W. acknowl- +edges the support from the Natural Science Foundation +of China (Grant No. +52102131) and Yunnan Funda- +mental Research Projects (Grant Nos. 202101BE070001- +012 and 202201AT070171). C.-W. N and Y.-H. 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‡ and Yuan-Hua Lin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' § 1School of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 2Faculty of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Kunming University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Kunming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 650093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Yunnan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 3Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' University of Wisconsin-Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' WI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' USA 4Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 5Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 6State Key Laboratory of Functional Materials for Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Shanghai Institute of Microsystem and Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Shanghai 200050,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 7CAS Center for Excellence in Superconducting Electronics(CENSE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Shanghai 200050,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 8Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 9Ningbo Institute of Materials Technology and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Ningbo 315201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China 10Center of Materials Science and Optoelectronics Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' China (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 2023) The anomalous Hall effect (AHE) is a quantum coherent transport phenomenon that convention- ally vanishes at elevated temperatures because of thermal dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Therefore, it is puzzling that the AHE can survive in heavy metal (HM)/antiferromagnetic (AFM) insulator (AFMI) heterostruc- tures at high temperatures yet disappears at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' In this paper, we report that an unconventional high-temperature AHE in HM/AFMI is observed only around the N´eel temperature of AFM, with large anomalous Hall resistivity up to 40 nΩ cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' This mechanism is attributed to the emergence of a noncollinear AFM spin texture with a non-zero net topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Atomistic spin dynamics simulation shows that such a unique spin texture can be stabilized by the subtle in- terplay among the collinear AFM exchange coupling, interfacial Dyzaloshinski-Moriya interaction, thermal fluctuation, and bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' INTRODUCTION Heavy metal (HM)/antiferromagnetic (AFM) insula- tor (AFMI) heterostructures are an emerging essential system for investigating the interaction between spin cur- rent and antiferromagnetic (AFM) order, which have the potential for applications in energy-efficient, ultrafast, and robust spintronics devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [1–10] In particular, the reflected spin current after interactions with the magnetic order at the interface can provide valuable information for determining the magnetic order, which has already been well established in the counterpart HM/ferromagnet (FM) heterostructures, such as the spin Hall magnetore- sistance (SMR) and its derivative spin Hall-anomalous Hall effect (SH–AHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [11, 12] Most recently, consider- able attention has been paid to SMR in HM/AFMI het- erostructures, which is considered a probe for current- induced switching of the AFM order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [2, 13–16] Despite ∗ These authors contributed equally † These authors contributed equally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' liangwu@kust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='cn ‡ cwnan@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='cn § linyh@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='cn the extensive investigations of SMR in HM/AFMI het- erostructures, the anomalous Hall effect (AHE) in such systems has not yet been subjected to the same exam- ination, therefore, its origin, particularly whether it is caused by SMR as in HM/FM heterostructures, remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Recently, a high-temperature AHE was observed in HM/AFMI heterostructures, which are based on Ta, Pt, and W grown on Cr2O3 (AFMI) with an anomalous Hall resistivity (ρAHE) of approximately 1 nΩ cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [3–5] De- spite the controversy regarding the underlying mecha- nism, a ubiquitous phenomenon insensitive to the selec- tion of the HM and crystalline orientation of AFMI was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' That is, a superparamagnetism-like AHE sig- nal with a zero-coercive field (Hc) exists at temperatures significantly higher than the bulk Cr2O3 N´eel tempera- ture TN (307 K) but disappears in the low-temperature region (< 200 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [3–5] However, for heterostructures con- sisting of AFMI with significantly higher bulk TN, for ex- ample, the α-Fe2O3 (TN ∼ 950 K) and NiO (TN ∼ 523 K), an AHE at approximately room temperature has not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Thus, it is desirable to extend the AHE to other AFMI-based heterostructures, and understand arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='05486v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='mes-hall] 13 Jan 2023 2 400 300 200 100 0 Temperature (K) 6 5 4 3 2 1 NiO Thickness (uc) MgO (001) NiO (3 uc) Pt (3 nm) (a) (c) Pt(3 nm)/NiO(3 uc)/MgO(001) Pt (3 nm)/NiO(x uc) Pt (x nm)/NiO(3 uc) 2 nm 30 20 10 0 10 20 30 ρAHE (nΩ·cm) 8 6 4 2 0 2 4 6 8 µ0H (T) 250 K 300 K 350 K 400 K 5 K 60 K 100 K 150 K 200 K 7 6 5 4 3 2 Pt Thickness (nm) 40 30 20 10 0 ρAHE (nΩ·cm) (d) (b) Pt (x nm)/NiO(3 uc) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Observation of AHE in Pt/NiO/MgO heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (a) The STEM of Pt(3 nm)/NiO(3 uc)/MgO(001) (the inset is the SAED pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (b) ρAHE of the same sample at varying temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (c-d) Evolution of the ρAHE with temperature and thickness of NiO and Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' its physical mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' In this study, we demonstrate that the AHE can be induced in HM/NiO heterostructures at elevated temperatures (up to 400 K) by controlling the AFM– paramagnetic (PM) (AFM–PM) transition of an ultra- thin epitaxial NiO film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The value of ρAHE was approx- imately 40 nΩ cm, which is higher than the recorded ρAHE in HM/magnetic insulator heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The origin of the AHE in HM/AFMI heterostructures is at- tributed to the emergence of AFM spin textures with uncompensated topological charges during the AFM–PM phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Moreover, the line shape of the AHE signal depends on the defect at the HM/AFMI interface, which can induce topological-Hall-effect-like (THE-like) signals based on the two-channel AHE scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS We first employed scanning transmission electron mi- croscopy (STEM) to characterize the structural prop- erties of the HM/NiO heterostructures with ultrathin NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The STEM image and selected area electron diffrac- tion (SAED) pattern of a representative Pt(3 nm)/NiO(3 uc)/MgO(001) heterostructure are shown in Figure 1a, where uc denotes unit cells (see details in Experimen- tal Section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The NiO was fully epitaxial on the MgO(001) substrate with a rock-salt crystalline struc- ture, whereas the Pt overlayer was polycrystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The field-dependent magnetization measurements showed no observable macroscopic ferromagnetism (see Figure S1, Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' To investigate both the Pt and NiO thickness dependence of the ρAHE, we performed Hall measurements at 5–400 K, with subtraction of the linear ordinary Hall signal (Complete AHE data are pre- sented in Figure S2, Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A low driving current density of ∼102 A cm−2 was used to sup- press the Joule heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The absence of an AHE in the reference sample without the NiO layer is shown in Fig- ure S3 (Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A temperature-dependent AHE was observed in the Pt(3 nm)/NiO(3 uc)/MgO(001) heterostructure (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The AHE was only observed at high temperatures (250–400 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' This trend is consistent with that of Cr2O3- 3 Pt(3 nm)/NiO(3 uc)/MgO(111) W(3 nm)/NiO(3 uc)/MgO(001) 30 20 10 0 10 20 30 8 6 4 2 0 2 4 6 8 µ0H (T) 250 K 300 K 350 K 400 K 5 K 60 K 100 K 150 K 200 K (b) (a) 30 20 10 0 10 20 30 ρAHE (nΩ·cm) 8 6 4 2 0 2 4 6 8 µ0H (T) 250 K 300 K 350 K 400 K 5 K 60 K 100 K 150 K 200 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Insensitivity of AHE to AFMI orientation and HM type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (a-b) ρAHE of Pt(3 nm)/NiO(3 uc)/MgO(111) and W(3 nm)/NiO(3 uc)/MgO(001) heterostructures at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' based HM/AFMI heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [3–5] Comprehensive contour plots of ρAHE in Pt(3 nm)/NiO(x uc)/MgO(001) and Pt(x nm)/NiO(3 uc)/MgO(001) are shown in Fig- ure 1c and Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The AHE mainly existed above 150 K, and its magnitude significantly depended on the NiO and Pt thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' As the NiO thickness increased, the minimum temperature required for the appearance of ρAHE increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' For NiO thicker than 6 uc, we ob- served no apparent ρAHE signals within the accessible temperature (5–400 K) and field (±8 T) ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Figure 1d shows the decay of ρAHE as Pt becomes thicker, indi- cating the interfacial nature of the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' It is noted that a local maximum of ρAHE appeared at approximately 5 K, which could be induced by the skew scattering of electrons by localized paramagnetic centers (Giovannini– Kondo model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [17, 18] As a G-type collinear AFM, the superexchange in- teraction between Ni ions results in antiferromagneti- cally stacking of ferromagnetic {111} planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [10] To in- vestigate how the crystallographic orientation of AFMI and the type of HM influence the AHE, we further measured ρAHE in the Pt(3 nm)/NiO(3 uc)/MgO(111) and W(3 nm)/NiO(3 uc)/MgO(001) heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A similar temperature-dependent AHE behavior was ob- served (Figure 2), indicating that the AHE was insen- sitive to the crystal orientation of AFMI and the type of HM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' In particular, W showed the opposite sign of the spin Hall angle to Pt[19], however, the sign of AHE remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [4] In addition, the line shape of ρAHE for W(3 nm)/NiO(3 uc)/MgO(001) was slightly different from that of Pt(3 nm)/NiO(3 uc)/MgO(001), which will be discussed subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Additional con- trol experiments were performed to clarify the role of the HM and concomitant spin Hall effect (SHE) and the strong spin-orbit coupling on the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' First, no AHE signal was observed for Ti(3 nm)/Cu(3 nm)/NiO(3 uc)/MgO(001) as expected (Figure S4, Supporting In- formation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Second, a weaker AHE signal was observed in Pt(3 nm)/Cu(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='2 nm)/NiO(3 uc)/MgO(001) (Figure 3a), ruling out the proximity-induced-ferromagnetism in Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Finally, Pt/W/NiO/MgO(001) heterostructures were fabricated, and the competing of spin current[19] initially decreased the magnitude of the AHE, and even- tually reversed the polarity (sign) of the AHE with thick- ened Pt (Figure S5, Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The ex- periments demonstrate that the SHE in HM plays a critical role, and the scattering of the spin current by AFMI (in a reflective manner) induces the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='The experimental results, along with the ρAHE values re- ported for HM/magnetic insulator heterostructures, are presented in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' We achieved ρAHE values up to 40 nΩ cm in the Pt/NiO heterostructures, which is higher than the recorded value in HM/magnetic insula- tor heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [4, 20–23] Additionally, the scaling relation of the HM/AFMI heterostructure is shown in Figure S6 (Supporting Information), which is similar to that of the gating-induced ferromagnetic Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [24, 25] An additional THE-like Hall effect (the bump and dip feature added to the AHE signals) was observed in W(3 nm)/NiO (3 uc)/MgO(001) and Pt/Cu(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='2 nm)/NiO (3 uc)/MgO(001) compared to Pt/NiO (3 uc)/MgO(001) (Figure 2b and 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' From the Ellingham diagram, it can be observed that the Cu/Cu2O, Ni/NiO and W/WO3 lines are relatively close, whereas the Pt/PtO2 line is significantly higher than all the above lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [26] This im- plies that the oxygen can migrate from NiO to Cu and W to induce additional oxygen vacancies in NiO, resulting in weak ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [27–30] To confirm this defect- induced THE-like signal, we deposited a NiO layer with a significantly lower oxygen pressure to enrich the oxy- gen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A THE-like signal was detected again (Figure 3b), indicating that the THE-like signal was re- lated to oxygen vacancy-induced dilute ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Thus, the THE-like signal can be regarded as an al- 4 Pt(3 nm)/NiO(3 uc, 10-4 Pa)/MgO(001) (b) Pt(3 nm)/Cu(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='2 nm)/NiO(3 uc)/MgO(001) 10 5 0 5 10 ρAHE (nΩ·cm) 8 6 4 2 0 2 4 6 8 µ0H (T) 250 K 300 K 350 K 400 K 10 K 50 K 100 K 150 K 200 K (a) (c) 40 30 20 10 0 40 30 20 10 0 |ρAHE| (nΩ·cm) 400 300 200 100 0 Temperature (K) Pt(3 nm)/NiO(4 uc)/MgO(001) Pt(3 nm)/NiO(3 uc)/MgO(001) Pt(3 nm)/NiO(3 uc)/MgO(111) Pt(3 nm)/NiO(3 uc, 10 4 Pa)/MgO(001) Pt(3 nm)/Cu(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='2 nm)/NiO(3 uc)/MgO(001) Pt/Cr2O3 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [4] Pt/TmIG Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [20] Pt/TmIG Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [21] Pt/YIG Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [22] Pt/Cr2Ge2Te6 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [23] 10 5 0 5 10 ρAHE (nΩ·cm) 8 6 4 2 0 2 4 6 8 µ0H (T) 340 K 360 K 400 K 200 K 250 K 300 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Control experimental results for AHE in HM/AFMI heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (a-b) ρAHE of Pt(3 nm)/NiO(3 uc)/MgO(111) and W(3 nm)/NiO(3 uc)/MgO(001) at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (c) Temperature-dependent saturated ρAHE from our data and reported HM/magnetic insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [4, 20–23] The four-pointed stars represent our data, and the circles represent data from references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' ternative two-channel AHE scenario engineered by the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [31, 32] One of the channels results from the AFM order, whereas the other is related to the dilute ferromag- netism, similar to the HM/FM heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [12, 33] Next, we discuss the relationship between the TN of the AFMI layer and the temperature-dependent AHE behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The TN of the ultrathin NiO is highly depen- dent on its thickness and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' It has been reported that an ultrathin NiO film grown on MgO can have a significantly reduced TN from its bulk value, for example, the TN of NiO(3 uc)/MgO(001) was lower than 40 K, and its TN could be considerably enhanced up to 390 K simply by Ag capping owing to image charge screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [34] Therefore, the TN in the same 3 uc NiO film in our Pt/NiO/MgO(001) should also be enhanced by the Pt overlayer to induce a high-temperature AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Note that, the short-range magnetic order above TN[35– 38] could also support the high-temperature AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' To determine the TN of NiO in our Pt/NiO/MgO sam- ple, we performed temperature-dependent X-ray absorp- tion spectroscopy (XAS) measurements at the Ni L2 edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The decrease in the L2 ratio with increasing tem- perature indicates an AFM–PM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [34, 39] The continuous decrease in the Ni L2 ratio of Pt(3 nm)/NiO(3 uc)/MgO(001) at 298–523 K indicates that this temper- ature range is within the continuous second-order AFM– PM transition range of NiO (left panel of Figure 4a), and the TN should be as high as 523 K and comparable to that of the bulk NiO (Here, the TN is defined as the temperature of the absence of the long-range AFM or- der via thermal fluctuation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' This indicates that Pt can impose a more pronounced effect than Ag on stabilizing the AFM order in NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' However, for Pt(3 nm)/NiO(6 uc)/MgO(001), the Ni L2 ratio started to decline above 400 K, showing that the AFM–PM transition in this sam- 5 Temperature AFM PM AFM⎯PM AHE AFM order strength Thermally driven topological AFM spin texture (a) (b) 868 870 872 874 868 870 872 874 Pt(3 nm)/NiO(3 uc)/MgO(001) XAS (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=') 300 400 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='4 L2 ratio L2 ratio T (K) 298 K 523 K Pt(3 nm)/NiO(6 uc)/MgO(001) Photon energy (eV) 300 400 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='3 T (K) 298 K 523 K ~𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Correlation between AHE and AFM–PM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' a) Temperature-dependent XAS for Pt(3 nm)/NiO(3 uc)/MgO(001) and Pt(3 nm)/NiO(6 uc)/MgO(001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' b) Curves are extracted from AHE (red symbols) and XAS (blue symbols) data from above two samples (circles for the sample with 3 uc NiO, and squares for that with 6 uc NiO), normalized by the putative AFM–PM transition temperature ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The AFM order strength is represented by the L2 ratio of XAS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The insets show the calculated spin structures in the ultrathin NiO, represented by the distribution of N’eel vector n, and the vector length is not proportional to the magnitude of the n at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' ple was initiated at 400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' W exhibited a similar effect on 3 uc NiO, that is, the AFM–PM transition covered the temperature range of 298–523K (Figure S7, Sup- porting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A comparison between the AHE and XAS data showed that the AHE coincided with that of the AFM–PM transition in the common temperature range of 300–400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Notably, the absence of AHE in Pt(3 nm)/NiO(6 uc)/MgO(001) below 400 K can be ex- plained by the AFM–PM transition starting at tempera- tures higher than 400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Based on the consistent temperature range between the AFM–PM transition and the AHE, the thermally soft- ened AFM order during the AFM–PM transition plays a crucial role in the AHE of HM/AFMI heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' In Cr2O3-based HM/AFMI heterostructures, the AFM– PM transition generates the temperature-dependent un- compensated magnetic moment on the HM/AFMI inter- face based on SH–AHE scenario, which is however ques- tioned by X-ray magnetic dichroism measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [3–5] In our case, the magnetically compensated (100)-facet and uncompensated (111)-facet in NiO exhibited simi- lar temperature-dependent AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Thus, the conventional SH–AHE scenario based on uncompensated magnetic moments should not account for the appearance of the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Three mechanisms cause the conventional AHE: intrin- sic contribution by the Berry phase and extrinsic contri- butions by skew scattering and side jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [40, 41] In this study, such an AHE in the heterostructure system oc- curs owing to the interaction between the spin current (SHE in HM) and the magnetic structure (magnetic in- sulator, in this case, AFMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Recently, it has been re- ported that an AFM topological spin texture emerges around the phase-transition temperature of Pt/α-Fe2O3 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [42] A significant Fert–Levy-type inter- facial Dzyaloshinskii–Moriya interaction (DMI) exists and favors noncollinear spin textures, considering the space-inversion symmetry breaking on the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [43] 6 Thus, the AFM topological spin textures could exist in HM/AFMI heterostructures at room temperature, re- sponsible for the AHE by scattering spin-polarized elec- trons with an effective field ⟨bz⟩ = √ 3φ0/(2λ2) for per unit topological charge, where ⟨bz⟩ is the out-of-plane ef- fective field, φ0 is the unit flux and λ is the size of spin texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [44, 45] We performed atomistic spin dynamics simulations to investigate the formation of AFM topological spin tex- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' We found that a noncollinear AFM spin texture could emerge during the AFM–PM transition (Figure 4b) owing to the interaction among the collinear AFM ex- change coupling, DMI, thermal field, and external mag- netic field (see the details in Experimental Section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' In the intermediate-temperature region, the thermal en- ergy kBT is sufficiently high to destabilize the otherwise collinear AFM order (for example, the spin structure in Figure 4b with kBT = 0), such that interfacial DMI can induce the noncollinear AFM spin texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' At high tem- peratures, the thermal perturbation dominates, yielding a PM-like spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The curves in Figure 4b are guides for the eye to show the coincident temperature range between the AFM–PM transition and the AHE occur- rence (the putative AFM–PM transition ranges for sam- ple Pt(3 nm)/NiO(3 uc)/MgO(001) and Pt(3 nm)/NiO(6 uc)/MgO(001) are approximately 200–523 K and 400– 523 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The topological charges associated with the two spin sublattices are denoted as QA and QB, respectively, and can be calculated from the spatial distribution of the N´eel vector (see Experimental Section and Figure S8, Sup- porting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' For a canonical AFM skyrmion, the local effective fields induced by QA and QB have the same magnitude but opposite signs because QA+QB = 0 and |QA| = |QB| = 1 leading to the generation of a zero net effective field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [45] Therefore, an AFM skyrmion can- not cause an AHE because the time-reverse symmetry is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' However, applying a large out-of-plane mag- netic field can break the time-reverse symmetry as the two spin sublattices are canted (Figure S9, Supporting Information), yielding a non-zero net topological charge (QA + QB ̸= 0) and hence, an AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' CONCLUSION In this paper, we reported an unconventional high- temperature AHE insensitive to the selection of the HM and AFMI in HM/AFMI heterostructures, which was suppressed and even eliminated in a low-temperature re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' This is explained by an extended SMR model that includes the AFM spin texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The emergence of the AFM spin texture (for example, skyrmion or meron) is considered to be stabilized by the DMI at the HM/AFMI interface, and can only occur at an intermediate temper- ature during the AFM–PM phase transition, as demon- strated via micromagnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The THE-like Hall effect can be triggered by controlling the fabrication conditions of the samples, which can be regarded as an additional AHE channel induced by defect-induced weak FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The room-temperature AHE in AFMI heterostruc- tures demonstrates the strong interaction between the AFM order and spin current, which aids in developing AFM-based spintronics devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' EXPERIMENTAL SECTION Sample fabrication The epitaxial NiO thin films were fabricated using pulsed laser deposition (PLD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The growth conditions were as follows: a growth temperature of 650 ◦C (measured using a pyrometer), oxygen background pressure of 50 mTorr, an excimer laser with a wavelength of 248 nm, a rep- etition rate of 3 Hz, an energy density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='4 J cm−2, and target-substrate distance of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' After deposition and cooling down to room temperature, the NiO films were immediately transferred to a magnetron sputtering chamber (AJA International, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=') with a background pressure better than 2 × 10−8 torr, and heated to 150 ◦C for 15 minutes, to prevent possible surface gas absorp- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Then, DC sputtering was then employed to deposit the Pt layer at room temperature with a power of 30 W, background Ar of 3 mTorr, resulting in a deposition rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='6 nm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The W and Cu were deposited under the same conditions at deposition rates of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='3 nm/min and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='6 nm/min, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Structural, electrical and magnetic properties characterization The structural properties were characterized using a high-resolution X-ray diffractometer (XRD, Malvern Panalytical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The magnetic hysteresis loop was mea- sured using a superconducting quantum interference de- vice (MPMS, Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The transport prop- erties were measured using the van der Pauw method in a Physical Property Measurement System (PPMS, Quantum Design DynaCool system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The measured Hall signals were firstly treated using an asymmetry proce- dure, and the ordinary Hall signals were then subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Cross-sectional high-resolution transmission electron mi- croscopy samples were prepared using a focused ion beam (FIB, Zeiss Auriga) with Ga+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' X-ray absorption spectroscopy Element-specific X-ray absorption spectroscopy (XAS) was measured using total electron yield (TEY) at Beam- line BL02B02 of the Shanghai Synchrotron Radiation Fa- cility (SSRF) and Beamline 4B9B of the Beijing Syn- chrotron Radiation Facility (BSRF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' To evaluate the magnetic properties of these NiO thin films, we utilized the magnetic linear dichroism (MLD) effect in the Ni L2 XAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The normal incidence of X-rays resulted in the electric field component of X-rays being parallel to the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The sample temperature was controlled by laser heating the sample holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A linear background 7 connecting raw data points at 870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='8 eV and 871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='8 eV was first subtracted from the raw data, and the spectra were normalized from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The L2 ratio was defined as the normalized peak intensity at 871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Atomistic spin dynamics simulations Micromagnetic simulations were performed to simulate the spin structure in the AFM NiO with interfacial DMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The total Hamiltonian of the AFM thin-film H included the Hamiltonians of the exchange interaction Hexch, uni- axial anisotropy Hanis, external fields Hext, and the Hamiltonian of the interfacial Dyzaloshinskii–Moriya in- teraction (DMI) HDMI, expressed as follows: H =Hexch + Hanis + Hext + HDMI = − � JSk · Sl − � k Ka (Sk · ne)2 + � Dkl · (Sk × Sl) , (1) where Sk and Sl are the orientation vectors of the lo- cal spin and neighboring spins, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' J is the ex- change constant, Ka is the uniaxial anisotropy constant, ne is the uniaxial easy axis, and Dkl is the DMI constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' In the simulation, a two-dimensional system of 64 × 64 grids was considered in which each grid with a size of 1 nm × 1 nm, contained two neighboring spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' By linearly combining Sm and Sn inside the same gird, the net magnetization vector m = (Sm +Sn)/2 and the N´eel vector n = (Sm − Sn)/2 were defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The expression of the free energy was derived by substituting Sm and Sn with m and n, respectively, in the total Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Under the continuum approximation[46], the total free energy is expressed as follows: Etot = � dV � A∗ � (∂xn)2 + (∂yn)2� − K (n · ne)2 + D [nz(∇ · n) − (n · ∇)nz] − Hext · n � , (2) where A∗ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='42 pJ m−1, is the continuum-scale ex- change constant, which was converted from the AFM ex- change interaction J = −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='01 meV[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' We assumed the existence of a perpendicular magnetic anisotropy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=', ne//z) in the ultrathin (3 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=') NiO owing to the inter- facial symmetry breaking, with a continuum anisotropy constant K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='5× 105 J m−3(which is a typical number utilized in existing micromagnetic simulation investiga- tions of AFM skyrmions)[48], D∗ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='5×10−3 J m−2 is the continuum DMI constant, which is a value calculated from the first-principles in a similar NiO/Au system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [49] The free energy expression of the N´eel vector n in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (2) is the same as that of the magnetization in the ferromagnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Therefore, we performed mi- cromagnetic simulations using the open-source software MuMax3[50] to calculate the equilibrium configuration of the N´eel vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The evolution of n was determined by solving the Landau–Lifshitz–Gilbert (LLG) equation, ∂n ∂t = − γ0 1 + α2 (n × Heff + αn × n × Heff) , (3) where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='01 is the Gilbert damping coefficient[49] and γ0 is the gyromagnetic ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Heff = − 1 µ0Ms δEtot δn is the effective field, where µ0 is the vacuum permeability, amd Ms = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='2 × 105 A m−1 is the saturation magnetization of NiO[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The influence of thermal fluctuations was modeled by adding a thermal fluctuation field Htherm to Heff, ex- pressed by Htherm = η � 2αkBT µ0Msγ0∆V ∆t, where kBT is the Boltzmann constant, T is the temperature, ∆V is the vol- ume of each simulation cell, and ∆t is the time interval in a real unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' η = η(r, t) = (ηx, ηy, ηz) is a white-noise distributed stochastic vector uncorrelated in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The temporal mean value of ηi(i = x, y, z) was zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The LLG equation was solved using fourth-order Runge–Kutta methods, with a discretized time interval of 20 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' A uniform distribution of n in the z-direction was used as the initial state for the low-temperature re- gion (< 8 K), and an artificially generated single AFM skyrmion was used for the higher-temperature region (> 8 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The equilibrium distribution of n was assumed to be reached when the total free energy density Etot no longer changes significantly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The topological charge of the two sublattices of n in the equilibrium state, QA and QB were calculated as follows to quantify the topological state of the AFM thin film[51]: QA = 1 8π � site A qijk ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' QB = 1 8π � site B qijk , (4) where qijk is calculated for each unique triangle with grids i, j, and k as vertices using Equation (5) tan(qijk 2 ) = ni · (nj × nk) 1 + ni · nj + ni · nk + nj · nj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' (5) The equilibrium state of n was determined based on the values of QA and QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' If QA + QB = 0, the distribution of n is identified as a collinear distribution, otherwise, it is a topological AFM spin texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' If QA ̸= QB, there is no evident ordered spin texture[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Acknowledgements Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' contributed equally to this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' We acknowledge the insightful discussions with Shuai Dong and Jiahao Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' We thank the staff at Beamline 4B9B of the BSRF and Beamline 02B02 of the SSRF for fruitful discussions and experimental assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' acknowl- edges the support from the Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 52102131) and Yunnan Funda- mental Research Projects (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 202101BE070001- 012 and 202201AT070171).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' N and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' L were supported by the Basic Science Center Project of the 8 National Natural Science Foundation of China (NSFC) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 51788104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The work conducted at the University of Wisconsin-Madison was supported by the National Science Foundation (NSF) under the Award CBET-2006028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' The atomistic spin dynamics simula- tions were performed using Bridges at the Pittsburgh Su- percomputing Center under Allocation TGDMR180076, which is a part of the Extreme Science and Engineering Discovery Environment (XSEDE) and supported by NSF Grant ACI-154856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' would like to thank the support from the Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='52002370).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [1] R.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Saitoh, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 110, 206601 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Takahashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Nakayama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Althammer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Goennenwein, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 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+page_content=' Manchon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Status Solidi RRL 11, 1600409 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Gepr¨ags, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Opel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Fischer, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Gomonay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Schwenke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Althammer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Huebl, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Gross, J Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 127, 243902 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Giovannini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 11, 489 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE5T4oBgHgl3EQfNA53/content/2301.05486v1.pdf'} +page_content=' 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a/zdE2T4oBgHgl3EQfMgbP/content/tmp_files/2301.03726v1.pdf.txt b/zdE2T4oBgHgl3EQfMgbP/content/tmp_files/2301.03726v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..377188251ed30cfc3df3e316605303cb5c85af3d --- /dev/null +++ b/zdE2T4oBgHgl3EQfMgbP/content/tmp_files/2301.03726v1.pdf.txt @@ -0,0 +1,1867 @@ +Neighborhood-Regularized Self-Training for Learning with Few Labels +Ran Xu1, Yue Yu2, Hejie Cui1, Xuan Kan1, Yanqiao Zhu3, Joyce Ho1, Chao Zhang2, Carl Yang1* +1 Emory University, 2 Georgia Institute of Technology, 3 University of California at Los Angeles +{ran.xu,hejie.cui,xuan.kan,joyce.c.ho,j.carlyang}@emory.edu, {yueyu,chaozhang}@gatech.edu, yzhu@cs.ucla.edu +Abstract +Training deep neural networks (DNNs) with limited supervi- +sion has been a popular research topic as it can significantly +alleviate the annotation burden. Self-training has been suc- +cessfully applied in semi-supervised learning tasks, but one +drawback of self-training is that it is vulnerable to the la- +bel noise from incorrect pseudo labels. Inspired by the fact +that samples with similar labels tend to share similar rep- +resentations, we develop a neighborhood-based sample se- +lection approach to tackle the issue of noisy pseudo labels. +We further stabilize self-training via aggregating the predic- +tions from different rounds during sample selection. Exper- +iments on eight tasks show that our proposed method out- +performs the strongest self-training baseline with 1.83% and +2.51% performance gain for text and graph datasets on av- +erage. Our further analysis demonstrates that our proposed +data selection strategy reduces the noise of pseudo labels by +36.8% and saves 57.3% of the time when compared with the +best baseline. Our code and appendices will be uploaded to +https://github.com/ritaranx/NeST. +1 +Introduction +In the era of deep learning, neural network models have +achieved promising performance in most supervised learn- +ing settings, especially when combined with self-supervised +learning techniques (Chen et al. 2020; Devlin et al. 2019; +Hu et al. 2020; Zhu et al. 2022). However, they still require +a sufficient amount of labels to achieve satisfactory perfor- +mances on many downstream tasks. For example, in the text +domain, curating NLP datasets often require domain experts +to read thousands of documents and carefully label them +with domain knowledge. Similarly, in the graph domain, +molecules are examples naturally represented as graphs, and +characterizing their properties relies on density functional +theory (DFT) (Cohen, Mori-S´anchez, and Yang 2012) which +often takes several hours. Such a dependency on labeled data +is one of the barriers to deploy deep neural networks (DNNs) +in real-world applications. +To better adapt the DNNs to target tasks with limited la- +bels, one of the most popular approaches is semi-supervised +learning (SSL), which jointly leverages unlabeled data and +*Corresponding author. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +labeled data to improve the model’s generalization power on +the target task (Yang et al. 2021; Wang et al. 2022). Although +generative models (Gururangan et al. 2019) and consistency- +based regularization (Tarvainen and Valpola 2017; Miy- +ato et al. 2018; Xie et al. 2020a) methods have been pro- +posed for semi-supervised learning, they either suffer from +the issue of limited representation power (Tsai, Lin, and +Fu 2022) or require additional resources to generate high- +quality augmented samples (e.g., for text classification, Xie +et al. (2020a) generate augmented text via back-translation, +which rely on a Machine Translation model trained with +massive labeled sentence pairs). Consequently, they cannot +be readily applied to low-resource scenarios. +Self-training is a proper tool to deal with the deficiency +of labeled data via gradually enlarging the training set with +pseudo-labeled data (Rosenberg, Hebert, and Schneiderman +2005). Specifically, it can be interpreted as a teacher-student +framework: the teacher model generates pseudo labels for +the unlabeled data, and the student model updates its pa- +rameters by minimizing the discrepancy between its predic- +tions and the pseudo labels (Xie et al. 2020b; Mukherjee and +Awadallah 2020). Though conceptually simple, self-training +has achieved superior performance for various tasks with +limited labels, such as image classification (Sohn et al. 2020; +Rizve et al. 2021), natural language understanding (Du et al. +2020), sequence labeling (Liang et al. 2020), and graph +learning (Hao et al. 2020). Self-training has also been suc- +cessfully extended to other settings including weak super- +vision (Zhang et al. 2021b) and zero-shot learning (Li, +Savarese, and Hoi 2022). +However, one major challenge of self-training is that it +suffers from confirmation bias (Arazo et al. 2020) — when +the teacher model memorizes some biases and generates +incorrect pseudo labels, the student model will be rein- +forced to train with these wrong biases. As a result, the +biases may amplify over iterations and deteriorates the fi- +nal performance. To suppress the noisy pseudo labels in +self-training, Xu et al. (2021); Zhang et al. (2021a); Sohn +et al. (2020); Kim et al. (2022b) leverage model predic- +tive confidence with a thresholding function, +Mukherjee +and Awadallah (2020); Tsai, Lin, and Fu (2022) propose to +leverage model uncertainty to select samples with low un- +certainty, and Wang et al. (2021) use meta-learning to con- +duct instance reweighting for sequence labeling. Although +arXiv:2301.03726v1 [cs.LG] 10 Jan 2023 + +these approaches attempt to reduce the label noise, they se- +lect the data for self-training based on the model prediction +only. However, the predictions of the deep neural network +can be over-confident and biased (Guo et al. 2017; Kong +et al. 2020; Kan, Cui, and Yang 2021), and directly using +such predictions without any intervention to filter pseudo la- +bels cannot effectively resolve the label noise issue. Another +problem from self-training is training instability, as it se- +lects pseudo-labeled data only based on the prediction of the +current round. Due to the stochasticity involved in training +neural networks (e.g., random initialization, training order), +the prediction can be less stable (Yu et al. 2022b), especially +for the noisy pseudo-labeled data (Xia et al. 2022). Conse- +quently, the noise in the previous rounds may propagate to +later rounds, which deteriorate the final performance. +Motivated by the above, we propose NeST, a simple yet +powerful approach guided by the data representations, to +boost the performance of self-training for few-shot learn- +ing. Inspired by recent works indicating that the represen- +tations from deep neural networks can be discriminative and +less affected by noisy labels (Li et al. 2021), we harness the +features learned from the neural models to select the most +reliable samples in self-training. In addition, several works +have indicated that samples within the same category tend to +share similar representations, such as category-guided text +mining (Meng et al. 2020) and motif-driven graph learn- +ing (Zhang et al. 2020). Similarly, we hypothesize that a +sample’s pseudo label is more likely to be correct only if its +prediction is similar to the neighbor labeled instances in the +embedding space. To fulfill the denoising purpose, NeST +creates the neighborhood for each unlabeled data by find- +ing the top-k nearest labeled samples, then calculates the +divergence between its current prediction and the label of its +neighbors to rank the unlabeled data. As a result, only the in- +stances with the lowest divergence will be selected for self- +training, which mitigates the issue of label noise. Moreover, +to robustly select the training samples for self-training, we +aggregate the predictions on different iterations to promote +samples that have lower uncertainty over multiple rounds for +self-training. +We remark that NeST is an efficient substitution for exist- +ing self-training approaches and can be combined with vari- +ous neural architectures. The contributions of this paper are: +• We propose NeST to improve the robustness of self- +training for learning with few labels only. +• We +design +two +additional +techniques, +namely +neighborhood-regularized +sample +selection +to +re- +duce label noise, and prediction aggregation to alleviate +the training instability issue. +• Experiments on 4 text datasets and 4 graph datasets with +different volumes of labeled data verify that NeST im- +proves the performance by 1.83% and 2.51% respec- +tively and saves the running time by 57.3%. +2 +Related Work +Self-training is one of the earliest approaches to semi- +supervised learning (Rosenberg, Hebert, and Schneiderman +2005). The method uses a teacher model to generate new la- +bels on which a student model is fitted. The major drawback +of self-training is that it is vulnerable to label noise (Arazo +et al. 2020). There are several popular approaches to stabi- +lize the self-training process, such as using sample selec- +tion (Mukherjee and Awadallah 2020; Sohn et al. 2020) and +reweighting strategies (Zhou, Kantarcioglu, and Thuraising- +ham 2012; Wang et al. 2021) to filter noisy labels or design- +ing noise-aware loss functions (Liang et al. 2020; Yu et al. +2022a; Tsai, Lin, and Fu 2022) to improve the model’s ro- +bustness against incorrectly labeled data. In addition, data +augmentation methods (Kim et al. 2022a; Chen, Yang, and +Yang 2020; Zhang, Yu, and Zhang 2020) are also com- +bined with self-training to improve the model’s generaliza- +tion ability. +Leveraging representation information has also been ex- +plored in semi-supervised learning. For example, Li, Xiong, +and Hoi (2021); Zhao et al. (2022); Yu et al. (2021) improve +the representation via contrastive learning to assist semi- +supervised learning. Moreover, ACPL (Liu et al. 2022) and +SimMatch (Zheng et al. 2022) aggregates the labels from +their neighbors in the feature space. While these approaches +also attempt to harness sample representations, they do not +directly denoise the pseudo labeled data for boosting the per- +formance of self-training, which is the focus of our work. +One concurrent work (Lang, Vijayaraghavan, and Sontag +2022) combines data representations with the cut statistic +to select high quality training data. In particular, it aims to +select reliable subsets directly from the weakly-labeled data. +Instead, our work focuses on using clean labeled data to bet- +ter denoise instances in self-training. +3 +Preliminaries +In this section, we first present the setup of semi-supervised +learning and self-training, and then point out issues of the +existing sample selection algorithms for self-training. +3.1 +Task Definition +In this paper, we study the semi-supervised learning prob- +lem, which is defined as follows. Given a few labeled Xl = +{(xi, yi)}L +i=1 and unlabeled data Xu = {xj}U +j=1 (L ≪ U), +we seek to learn a predictor f(x; θ) : X +→ Y. Here +X = Xl ∪ Xu denotes all the input data and Y is the label +set, which can either be discrete (for classification) or con- +tinuous (for regression). f(x; θ) is either a C-dimensional +probability simplex for classification where C is the number +of classes or a continuous value for regression. +3.2 +Introduction to Self-training +Self-training can be interpreted as a teacher-student frame- +work (Mukherjee and Awadallah 2020; Xie et al. 2020b), +with θt and θs denoting the teacher and student model, re- +spectively. The process of the self-training is in Alg. 1. We +discuss the key components in self-training as belows. +Initialization of Models. +The labeled data Xl are used to +initialize the models as θ(0) +s += θ(0) +t += θinit, where +θinit = min +θ +Lsup(θ) = E(xi,yi)∈Xlℓsup (f(xi; θ), yi) . +(1) + +Algorithm 1: Procedures of Self-training. +Input: Labeled and unlabeled samples Xl, Xu; +Neural prediction model f(·; θ); Unlabeled +set � +Xu; Number of self-training iterations T; +Number of steps in each iteration T1. +// Train the model on labeled data Xl as initialization. +Update θs, θt by Eq. 1 using Adam. +// Self-training. +for t = 1, 2, · · · , T do +Select � +X tu (|� +X tu| = b) with θt by Eq. 4. +Adding � +X tu for self-training � +Xu = � +Xu ∪ � +X tu. +Update pseudo labels �y by Eq. 2 or 3 for � +X tu. +for k = 1, 2, · · · , T1 do +Sample a minibatch B from � +Xu. +Update θs with loss L in Eq. 5 using Adam. +Update teacher model θt ← θs. +Output: Final model f(·; θs). +ℓsup(·; ·) represents the supervised loss, which is the cross- +entropy loss for classification and the mean squared error +loss for regression. +Pseudo Label Generation with Teacher Model θt. +We +use the teacher model’s prediction f(x; θt) to generate +pseudo labels for Xu. For classification problems, the +pseudo labels can be written as +�yhard,j = +�1, +if j = argmax +k∈Y +[f(x; θt)]k ; +0, +else. +(2) +For the regression task, since the output is a continuous +value, the teacher model’s output is directly used as the +pseudo label +�y = f(x; θt). +(3) +Sample selection. +Directly using all the pseudo-labeled +data for self-training often yields sub-optimal results, as the +erroneous pseudo labels hurt the model performance. To +mitigate this issue, recent works attempt to select only a +subset of the unlabeled data for self-training. We denote the +sample policies as ψ(·), which can be generally written as +� +Xu = ψ (Xu, f(x; θt)) . +(4) +We omit the superscript for simplicity. The common choice +for ψ(·) including using predictive confidence (Sohn et al. +2020; Zhang et al. 2021a) or model uncertainty (Mukherjee +and Awadallah 2020; Tsai, Lin, and Fu 2022). +Model Training and Update. +With the generated pseudo +labels, we then train a student model θs to minimize the loss +for both labeled and unlabeled data by solving +min +θs +λLsup(θs) + (1 − λ)Exj∈ � +Xuℓst (f(xj; θs), �yj) , +(5) +where Lsup is defined in Eq. 1, � +Xu is obtained via Eq. 4, and +ℓst = 1{[f(xj; θs)]�yj > γ} · ℓsup is the loss function for +0 +10 +20 +30 +Error Rate of Selected +Pseudo Labels (in %) +Elec +0 +10 +20 +30 +Chemprot +All the Unlabeled Data +Selected w/ Confidence (ST) +Selected w/ Uncertainty (UST) +1 +2 +3 +4 +5 +Iterations +50 +60 +70 +80 +90 +Performance (in %) +Elec upper bound +Chemprot upper bound +Elec - Confidence (ST) +Elec - Uncertainty (UST) +Chemprot - Confidence (ST) +Chemprot - Uncertainty (UST) +1 +2 +3 +4 +5 +85 +90 +1 +2 +3 +4 +5 +Iterations +50 +55 +Performance (in %) +Figure 1: Left: The average error rate of all pseudo labels +and the selected pseudo labels with different selection strate- +gies. Middle: The performance on different self-training it- +erations. The upper bounds are the accuracy with full clean +labels. Right: Zoom in of the performance in the middle. +unlabeled data with the thresholding function (Sohn et al. +2020; Xie et al. 2020a). We iterate the process by treating the +trained student model as the teacher to generate new pseudo +labels and train a new student model based on the new gen- +erated labels until the model converges. +3.3 +Challenges of Self-training +To illustrate that the existing sample selection approaches +are flawed and cannot resolve the label noise issue, we first +demonstrate the performance of two widely-used selection +criteria: predictive confidence (ST (Rosenberg, Hebert, and +Schneiderman 2005; Du et al. 2020; Liang et al. 2020)) and +model uncertainty (UST (Mukherjee and Awadallah 2020)) +for self-training. The details of these two approaches are dis- +cussed in Appendix B. Note that, for these two approaches, +we follow the original implementation to select the unla- +beled set Xu in each iteration. We use a binary sentiment +classification dataset Elec and a chemical relation extraction +dataset Chemprot with ten classes as an example for easier +and harder task, respectively. For both datasets, we first train +the model with 30 clean labels per class. +Figure 1 shows the error rate of pseudo labels selected fol- +lowing these two criteria. We observe that these two meth- +ods are effective on easier tasks, where the selected data +has a relatively low error rate. It achieves comparable per- +formance with the fully-supervised method (95%) with less +than 1% of the clean labeled data. However, for more chal- +lenging tasks with a larger number of classes, the perfor- +mance of the initial model may not be satisfactory. The er- +ror rate of pseudo labels increases up to 16% on Chemprot +compared with Elec. Consequently, the gap between semi- +supervised learning and fully-supervised learning is even +larger — more than 25% in terms of F1 score. This phe- +nomenon suggests that the label noise issue is still the major +challenge that hampers the self-training performance. More- +over, using model uncertainty (Gal and Ghahramani 2016) +for sample selection does not fully address this challenge; +the gain can be marginal on harder datasets. +Apart from the label noise, we also observe performance +fluctuations over different self-training rounds. We name +this as the training instability issue, which occurs when the +teacher model only looks at the previous round and memo- +rizes the label noise specifically in that round. Then in the +next iteration, the student model can easily overfit the noise. + +Student model +Teacher model +Momentum +update +Step 3: Continue training the student model +Step 2: Select samples for self-training +Step 1: Initialize +with labeled data +Selected unlabeled data +with pseudo labels +Low +divergence +High +divergence +Unlabeled +data +Labeled +data +ACnXicbVHbhMxEHW1kuTe +GRB1ZESIWHaLcK0MdKgMQDlyI1TaV4Fc06k8aq17Zs20ia7+EV/ +go/gZvGiGSMpKl4zMz9pk5RaWkozT93Ylu3b5z97O/fjBw0ePd +7t7T06dqa3AoTDK2LMCHCqpcUiSFJ5VFqEsFI6Ki/dtfnSJ1kmjT +2hZYV7CuZYzKYACNenucpojwYQTLsi7ZtLtpf10FclNkK1Bj63je +LXueRTI+oSNQkFzo2ztKLcgyUpFDYxrx1WIC7gHMcBaijR5X6l +vEleBmazIwNR1OyYv/t8FA6tyLUFkCzd12riX/lxvXNDvMvdRV +TajF9UezWiVknYNyVRaFKSWAYCwMmhNxBwsCArL2vilKDdnOMly +34ptn90oVLAMLTemvkvnXuNV7RYSY75Bwzbsvgl3L5VaIGMfe2 +5MDo403juBCgszMJnzTjrv8k93+dV0KcUKv6qaeI4OJVt+3ITnB7 +0s7f9wfdB72iw9myHPWMv2D7L2Dt2xD6xYzZkgtXsB/vJfkXPo4/ +R5+jrdWnUWfc8ZRsRjf4AVj3RFg=\s +ACnXicbVHbhMxEHW1kuTe +GRB1ZESIWHaLcK0MdKgMQDlyI1TaV4Fc06k8aq17Zs20ia7+EV/ +go/gZvGiGSMpKl4zMz9pk5RaWkozT93Ylu3b5z97O/fjBw0ePd +7t7T06dqa3AoTDK2LMCHCqpcUiSFJ5VFqEsFI6Ki/dtfnSJ1kmjT +2hZYV7CuZYzKYACNenucpojwYQTLshTM+n20n6iuQmyNagx9ZxP +NnrXPKpEXWJmoQC58ZWlHuwZIUCpuY1w4rEBdwjuMANZTocr9S +3iQvAzNZsaGoylZsf92eCidW5ZFqCyB5m4715L/y41rmh3mXuq +JtTi+qNZrRIySbuGZCotClLAEBYGbQmYg4WBIVlbfxSlJsznGS5 +b8W2z24UKlgGFpvzfyXzr3GK1qsJMf8A4ZtWfwSbt8qtEDGvZ +cGB2caTx3AhQWZuGzZpz13+Se7/Mq6FMKFX/VNHEcnMq2fbkJTg/ +62dv+4PugdzRYe7bDnrEXbJ9l7B07Yp/YMRsywWr2g/1kv6Ln0cf +oc/T1ujTqrHueso2IRn8AWGfRFw=\t +! "! += !" "!, % ++ '!#(%) +Aggregate over +iterations +Figure 2: The framework of NeST. Red and blue points +stand for labeled data with different labels. White points rep- +resent unlabeled data. Light red and blue points stand for +predictions of unlabeled data. +4 +Method +We present NeST to improve the stability of self-training +by tackling the challenges mentioned in the previous sec- +tion. The overview of NeST is in Figure 2. Notably, we fo- +cus on the sample selection step (Eq. 4), and we propose +two key components, namely neighborhood-regularized se- +lection strategy and prediction aggregation to promote the +performance. The details of the two designs will be dis- +cussed in Section 4.1 and Section 4.2 respectively. +4.1 +Neighborhood-regularized Sample Selection +Prior works have demonstrated that leveraging embeddings +from the deep neural networks can identify the noisy labeled +data (Zhu, Dong, and Liu 2022). Motivated by this, we pro- +pose to harness the similarities in the embedding space to +mitigate the issue of erroneous pseudo labels in self-training. +Concretely, for each unlabeled sample xj with representa- +tion vj, we adopt the k-nearest neighbors (KNN) algorithm +to find the most similar labeled samples in the feature space: +Nj = {xi | xi ∈ Xl ∩ KNN(vj, Xl, k)}, +(6) +where KNN(vj, Xl, k) denotes k labeled examples in Xl that +are nearest to vj. +Divergence-based Sample Selection. +We then calculate +the scores for unlabeled samples xj ∈ Xu based on the +weighted divergence +D(xj) = Du(xj, N) + βDl(N), +(7) +where unlabeled divergence Du and labeled divergence Dl +are defined below, and β is a hyperparameter. This score +D(xj) will be further used for sample selection. +Unlabeled Divergence Du. +For each sample xj in the un- +labeled set with the neighbor set N, we calculate the diver- +gence between the prediction of xj and labeled data in N as +Du(xj, N) = +� +(xi,yi)∈N +d(f(xj; θt), yi), +(8) +where d is the Kullback–Leibler (KL) divergence for classi- +fication and L2 distance for regression (same as follows). To +interpret Eq. 8, we note that samples having the prediction +close to the nearest labeled instances will have lower Du. +Labeled Divergence Dl. +It measures the divergence +among the labels within the neighbor set N. We first cal- +culate the average label y = � +(xi,yi)∈N +yi +|N |, and then mea- +sure the labeled divergence as +Dl(N) = +� +(xi,yi)∈N +d(y, yi). +(9) +For each group N, samples with similar labels will have +smaller divergence Dl(N). +To summarize, a low divergence score D(xj) indicates +that the prediction of the unlabeled data point xj is close to +its neighbors, and the labels of its neighbors are consistent. +Thus, we hypothesize that such samples are more likely to +be correct and use them for self-training. +4.2 +Robust Aggregation of Predictions from +Different Iterations +The results in Figure 1c clearly demonstrate that only using +the prediction on the current iteration for sample selection +cause training instabilities. To effectively mitigate the bias in +the current iteration and stabilize the self-training, we pro- +pose to exploit the training prediction at different training +iterations more robustly (Xia et al. 2022). To achieve this, +we aggregate the value D(t)(xj) in the t-th round as +µ(t)(xj) = (1−m)×µ(t−1)(xj)+m× +� +D(t)(xj) +� +, (10) +where m is a hyperparameter bounded between 0 and 1 that +controls the weight for previous rounds. +To interpret Eq. 10, we argue that µ(xj) will be small only +when the model outputs consistently low scores for a sam- +ple xj in different iterations of self-training, as the model +is more certain about these samples. On the contrary, if the +model gives inconsistent predictions in different iterations, +then the model is potentially uncertain about the prediction, +thus adding its pseudo label in the next iteration may hurt the +self-training process. Motivated by this idea, we remove the +sample with inconsistent predictions over different iterations +to further suppress noisy labels. +To put the above two strategies together, our policy for +sample selection in the t-th round ψ(·) is mainly based on +the value of µ(t)(xj) in Eq. 10. Specifically, in t-th iteration, +we sample instances xj ∈ Xu without replacement using the +probability +p(xj) ∝ +W − µ(t)(xj) +� +xu∈Xu (W − µ(t)(xu)), +(11) +where W = maxx∈Xu(µ(t)(x)) is the normalizing factor. +Remark. +Our method introduces little computation over- +head. For each unlabeled data, the neighborhood regularized +sampling requires one extra kNN operation, which can be +efficiently supported via FAISS (Johnson, Douze, and J´egou +2021). The µ(t)(xj) from previous iterations can be cached +on disk and merged when selecting the training data for the +new iteration. Other than the sample selection method ψ(·), +NeST keeps other components intact and can be plugged- +in with any noise-robust learning techniques (Menon et al. +2021) and neural architectures. + +Dataset +Domain +Task +# Train / Test +# Class +Metric +Elec +Reviews +Sentiment Analysis +25K / 25K +2 +Acc. +AG News +News +Topic Classification +120K / 7.6K +4 +Acc. +NYT +News +Topic Classification +30K / 3.0K +9 +Acc. +Chemprot +Chemical +Relation Classification +12K / 1.6K +10 +F1 +BBBP +Physiology +Classification +1.6k / 204 +2 +ROC-AUC +BACE +Biophysics +Classification +1.2k / 151 +2 +ROC-AUC +Esol +Physical Chemistry +Regression +902 / 112 +— +RMSE +Lipophilicity +Physical Chemistry +Regression +3.3k / 420 +— +RMSE +Table 1: Statistics of text and graph datasets. +5 +Experiments +5.1 +Experiment Setup +We conduct experiments for semi-supervised learning on +eight datasets to demonstrate the efficacy of NeST. Four of +them are text-related tasks, including text classification and +relation extraction. We employ the pre-trained BERT from +the HuggingFace (Wolf et al. 2019) codebase for the imple- +mentation. The other four are graph-based tasks, where we +choose molecular property prediction as the main task and +use pre-trained Grover-base 1 (Rong et al. 2020) as the +backbone. The same backbone is used for both NeST and +baselines to ensure a fair comparison. +Semi-supervised Learning Settings. +For each dataset, we +train our method and baselines with different numbers of +labeled data from {30, 50, 100} per class. The remaining +in the training set is considered as unlabeled data. As sug- +gested by Bragg et al. (2021), we keep the size of the val- +idation set to be the same as the number of labeled data to +simulate the realistic setting. For each dataset, we apply 3 +runs on 3 splits and report the mean and standard deviations. +Parameter Settings. +We use Adam (Kingma and Ba +2014) as the optimizer and tune the learning rate in {1e- +5, 2e-5, 5e-5}. The batch size is selected from {8, 16, 32}. +Other hyperparameters in NeST include T, T1, γ for self- +training, β, b, k for sample selection in Eq. 7, and λ in +Eq. 5. We set β = 0.1, γ = 0.9, λ = 0.5, m = 0.6, +T = 5, T1 = 1000 for all datasets, and tune b = c|Xl| +with c ∈ {3, 5, 10, 20} for text datasets and c ∈ {1, 3, 5} for +graph datasets. We study the effect of k and c in Section 5.4. +Details for each dataset are in Appendix C.2. +Baselines. +We compare NeST with the following base- +lines. We use † to represent baselines designed for text-based +tasks and ‡ to represent baselines for graph-based tasks. +• BERT† (Devlin et al. 2019; Lee et al. 2020) is the super- +vised baseline for text-based tasks. +• Grover‡ (Rong et al. 2020) is the supervised baseline for +molecular property prediction. +• Mean-Teacher (MT) (Tarvainen and Valpola 2017) up- +dates the teacher model as a moving average of the stu- +1Since we do not focus on developing better self-supervised +learning methods, we do not compare with other GNN pretraining +models (You et al. 2021; Zhu et al. 2021) +dent model’s weight and adds a consistency regulariza- +tion between the student and teacher model. +• Virtual Adversarial Training (VAT) (Miyato et al. +2018) adds a regularization term between the sample +with the adversarial noise and its prediction. +• Self-training (ST) (Rosenberg, Hebert, and Schneider- +man 2005) is a conventional self-training method that +adds most confident pseudo labeled data to labeled set. +• UST (Mukherjee and Awadallah 2020) selects data with +lowest uncertainty using MC-dropout for self-training. +• UDA† (Xie et al. 2020a) adopts back translation and TF- +IDF word replacement as the data augmentation and adds +consistency loss on predictions on the augmented data. +• MixText† (Chen, Yang, and Yang 2020) interpolates +training data in the hidden space via Mixup as the data +augmentation to improve model performance. +• CEST† (Tsai, Lin, and Fu 2022) improves the UST +method by designing the contrastive loss over sample +pairs and noise-aware loss function. +• InfoGraph‡ (Sun et al. 2020) is a semi-supervised graph +classification method via maximizing mutual informa- +tion between graph and substructures. +• ASGN‡ (Hao et al. 2020) is a semi-supervised molecular +property prediction that jointly exploits information from +molecular structure and overall distribution. +5.2 +Semi-supervised Learning on Text +Datasets. +We conduct experiments on four widely used +datasets in NLP. We adopt Elec (McAuley and Leskovec +2013) for sentiment classification, AGNews (Zhang, Zhao, +and LeCun 2015) and NYT (Meng et al. 2020) for topic clas- +sification, and Chemprot (Taboureau et al. 2010) for chemi- +cal relation extraction in this set of experiments. The statis- +tics and evaluation metrics for each dataset are shown in Ta- +ble 1. We use BioBERT (Lee et al. 2020) as the backbone +for Chemprot as it is a domain specific dataset (Cui et al. +2022) and use RoBERTa-base for other datasets. +Results. +Table 2 summarizes the experimental results on +text datasets. We observe that NeST outperforms all base- +lines across all the four datasets under different volumes of +labeled data, and the performance gain compared to the best +baseline is around 1.83% on average. Note that UDA and +MixText require additional data augmentation, which can be + +Method +AG News (Accuracy, ↑) +Elec (Accuracy, ↑) +NYT (Accuracy, ↑) +Chemprot (F1, ↑) +30 +50 +100 +30 +50 +100 +30 +50 +100 +30 +50 +100 +BERT (2019) +80.6±1.4 +83.1±1.6 +86.0±1.1 +85.0±1.9 +87.2±1.0 +90.2±1.2 +79.4±1.6 +83.0±1.1 +85.7±0.5 +49.1±2.3 +51.2±1.7 +54.9±1.4 +Mean-Teacher (2017) +81.8±1.2 +83.9±1.4 +86.9±1.1 +87.6±0.9 +88.5±1.0 +91.7±0.7 +80.2±1.1 +83.5±1.3 +86.1±1.1 +50.0±0.7 +54.1±0.8 +56.8±0.4 +VAT (2018) +82.1±1.2 +85.0±0.8 +87.5±0.9 +87.9±0.8 +89.8±0.5 +91.5±0.4 +80.7±0.7 +84.4±0.9 +86.5±0.6 +50.7±0.7 +53.8±0.4 +57.0±0.5 +UDA (2020a) +86.5±0.9 +87.1±1.2 +87.8±1.2 +89.6±1.1 +91.2±0.6 +92.3±1.0 +— +— +— +— +— +— +MixText† (2020) +87.0±1.2 +87.7±0.9 +88.2±1.0 +91.0±0.9 +91.8±0.4 +92.4±0.5 +— +— +— +— +— +— +ST (2005; 2020) +86.0±1.4 +86.9±1.0 +87.8±0.6 +89.6±1.2 +91.4±0.4 +92.1±0.5 +85.4±0.9 +86.9±0.5 +87.5±0.5 +54.1±1.1 +55.3±0.7 +59.3±0.5 +UST (2020) +86.9∗ +87.4∗ +87.9∗ +90.0∗ +91.6∗ +91.9∗ +85.0±0.6 +86.7±0.4 +87.1±0.3 +53.5±1.3 +55.7±0.4 +59.5±0.7 +CEST‡ (2022) +86.5∗ +87.0∗ +88.4∗ +91.5∗ +92.1∗ +92.5∗ +— +— +— +— +— +— +NeST +87.8±0.8 +88.4±0.7 +89.5±0.3 +92.0±0.3 +92.4±0.2 +93.0±0.2 +86.5±0.7 +88.2±0.7 +88.6±0.6 +56.5±0.7 +57.2±0.4 +62.0±0.5 +Supervised +93.0∗ +95.3∗ +93.6±0.5 +82.5±0.4 +Table 2: Performance on four datasets with various amounts of labeled data. The higher value always indicates better perfor- +mance. Bold and underline indicate the best and second best results for each dataset, respectively (Same as below). ∗: The +number is reported from the original paper. The implementation of CEST is not publicly available. †: The result is lower than +the reported result in the original paper since they use a much larger development set. ‡: We remove the noise-aware loss as +well as graph-based regularization for a fair comparison. The effect of these two terms is presented in table 3. +Method +AG News (Accuracy, ↑) +Elec (Accuracy, ↑) +30 +50 +100 +30 +50 +100 +CEST +86.5 +87.5 +88.4 +91.5 +92.1 +92.5 +CEST w/ NRL +87.1 +88.0 +88.9 +92.2 +92.4 +92.8 +NeST +87.8±0.8 +88.4±0.7 +89.5±0.3 +92.0±0.3 +92.4±0.2 +93.0±0.2 +NeST w/ NRL +88.3±0.5 +88.9±0.6 +89.8±0.2 +92.3±0.2 +92.7±0.2 +93.1±0.3 +Table 3: Performance comparison of CEST (Tsai, Lin, and +Fu 2022) and NeST with noise-robust loss functions (NRL). +computationally expensive. Instead, NeST does not lever- +age any external resources but achieves better performance. +For other self-training baselines, we observe that they +cannot outperform our proposed method. As we keep other +components unchanged, the gain is mainly due to the +pseudo-labeled data denosing benefit of NeST. We will il- +lustrate this in Section 5.5. We also notice that the per- +formance gain is more prominent on NYT and Chemprot +datasets which have more classes, indicating NeST can be +better adapted to tasks with fine-grained classes. +Incorporating Noise-robust Loss Functions. +To demon- +strate NeST can be combined with other loss functions, Ta- +ble 3 further compares NeST with CEST (Tsai, Lin, and +Fu 2022) which adds additional noise-aware loss (Menon +et al. 2021) and graph-based regularization. The results show +that these components can further improve the performance +of NeST. Under both settings, NeST outperforms CEST, +which justifies the efficacy of our proposed strategies. +5.3 +Semi-supervised Learning on Graphs +Datasets. +We choose molecular property prediction as +the target task for graph classification. We conduct ex- +periments on four widely used datasets from the Molecu- +leNet (Wu et al. 2018), including BBBP (Martins et al. +2012), BACE (Subramanian et al. 2016), Esol (Delaney +2004) and Lipophilicity (Gaulton et al. 2012). The statistics +and evaluation metrics for each dataset are shown in Table 1. +Experiment results. +From the results in Table 4, we can +see that NeST outperforms all the baselines on all the +datasets. In particular, the performance gain compared to the +best baseline is around 2.5% on average. Compared to the +Grover model using labeled data only, the gain is around +8.5% on average. Notice that the traditional self-training +method (ST) sometimes performs even worse than Grover +fine-tuned on labeled data only, because confidence-based +selection introduces large label noise, which leads to many +wrong predictions. With proper control of noisy pseudo la- +bels, UST generally outperforms other baselines. However, +since they do not consider neighbor information, their per- +formance is not as good as NeST. +Adapting NeST to different backbones. +We use two +datasets as an example to demonstrate that NeST can be +adapted to different backbones. Table 5 shows the results of +training an AttentiveFP (Xiong et al. 2019), another popu- +lar GNN backbone based on graph attention networks for +molecular property prediction. Unlike Grover, AttentiveFP +is not pre-trained on massive molecules, but trained from +scratch in our experiments. We can see that the perfor- +mance of AttentiveFP is worse than Grover in most cases. +A key reason is that pre-trained Grover has considerably +more parameters than AttentiveFP, and incorporates rich do- +main knowledge with self-supervised learning on unlabeled +molecules. Nevertheless, NeST still outperforms all the +baselines by 1.5% on two datasets. This indicates that NeST +does not rely on any specific architecture, and it serves as an +effective plug-in module for different GNN models. +5.4 +Parameter and Ablation Studies +We study the effect of different parameters of NeST on NYT +and Chemprot with 30 labels per class, shown in Figure 3. +Results on other datasets are in Appendix D. The perfor- +mance first improves as k increases, because larger k allows +more labeled data in each neighborhood, introducing less +randomness and regularizing divergence calculation. When +k > 7, the performance drops as the neighborhood includes +labeled data far away, no longer serving as an appropriate +regularizer. Similarly, as c gets larger, the performance first +increases and then decreases. This indicates that when c is + +Method +BBBP (ROC-AUC, ↑) +BACE (ROC-AUC, ↑) +Esol (RMSE, ↓) +Lipo (RMSE, ↓) +30 +50 +100 +30 +50 +100 +30 +50 +100 +30 +50 +100 +Grover (2020) +69.0±1.9 +79.3±2.2 +86.4±1.1 +68.6±1.1 +75.2±3.1 +78.1±1.5 +1.573±0.061 +1.340±0.028 +1.147±0.022 +1.209±0.025 +1.133±0.036 +1.088±0.020 +Mean-Teacher (2017) +71.8±1.2 +80.9±1.4 +87.2±0.6 +69.5±0.8 +76.6±0.7 +80.2±0.2 +1.500±0.018 +1.298±0.020 +1.104±0.021 +1.154±0.027 +1.091±0.018 +1.067±0.035 +VAT (2018) +72.2±1.0 +80.0±1.6 +88.5±0.5 +69.8±0.5 +76.5±0.4 +78.6±0.8 +1.492±0.039 +1.264±0.031 +1.056±0.014 +1.199±0.016 +1.089±0.021 +1.050±0.027 +ASGN (2020) +72.1±1.3 +80.5±1.3 +87.3±0.9 +69.6±1.0 +77.2±1.0 +79.3±0.4 +1.449±0.030 +1.249±0.018 +1.096±0.004 +1.123±0.033 +1.084±0.039 +1.023±0.032 +InfoGraph (2020) +72.5±0.9 +81.2±0.3 +88.5±0.5 +70.2±0.6 +77.8±0.8 +80.4±0.5 +1.414±0.041 +1.222±0.017 +1.082±0.013 +1.125±0.024 +1.082±0.027 +1.039±0.020 +ST (2005) +71.0±2.0 +80.4±1.4 +87.8±1.4 +67.9±1.3 +75.8±2.0 +78.9±1.0 +1.463±0.043 +1.225±0.030 +1.105±0.025 +1.135±0.047 +1.090±0.043 +1.030±0.013 +UST (2020) +71.7±1.1 +81.8±0.7 +88.8±0.4 +69.9±0.3 +78.0±0.4 +80.4±0.5 +1.408±0.026 +1.174±0.034 +1.023±0.010 +1.115±0.020 +1.069±0.014 +1.010±0.009 +NeST +75.4±1.0 +83.5±0.8 +90.0±0.4 +70.5±0.2 +79.3±0.3 +81.6±0.3 +1.325±0.024 +1.130±0.026 +1.001±0.002 +1.088±0.011 +1.039±0.021 +0.992±0.013 +Supervised⋄ (2020) +93.6 +87.8 +0.888 +0.563 +Table 4: Performance on four datasets with various amounts of labeled data. For classification datasets (BBBP, BACE), the +higher value indicates better performance, while for regression datasets (Esol, Lipo), the lower value stands for better perfor- +mance. Bold and underline indicate the best and second best results, respectively (Same as below). +Method +BBBP (ROC-AUC, ↑) +BACE (ROC-AUC, ↑) +30 +50 +100 +30 +50 +100 +AttentiveFP (2019) +66.4±1.3 +75.6±1.1 +82.9±0.5 +68.0±1.4 +72.2±1.3 +74.0±0.9 +Mean-Teacher (2017) +67.9±0.7 +76.3±0.3 +83.4±0.2 +70.1±0.3 +73.8±0.3 +75.5±0.6 +VAT (2018) +68.3±0.4 +76.7±0.5 +84.0±0.5 +70.5±0.3 +73.4±0.5 +76.0±0.4 +ASGN (2020) +70.0±0.4 +77.1±0.2 +84.1±0.3 +70.9±0.5 +74.9±0.7 +77.9±0.6 +InfoGraph (2020) +68.5±0.4 +79.3±0.8 +83.8±0.4 +70.7±0.3 +75.3±0.2 +78.8±0.3 +ST (2005) +69.8±0.2 +76.0±1.4 +84.0±1.1 +67.5±0.5 +71.4±3.4 +76.0±0.8 +UST (2020) +70.7±0.3 +79.2±0.5 +84.5±0.7 +70.3±0.4 +75.5±0.3 +78.7±0.6 +NeST +71.2±0.4 +80.7±0.5 +85.2±0.5 +72.2±0.4 +76.6±0.5 +80.7±0.5 +Table 5: Performance on two datasets with various amounts +of labeled data using AttentiveFP as the backbone. +1 +3 +5 +7 +9 +54 +55 +56 +57 +58 +83 +84 +85 +86 +87 +Chemprot +NYT +(a) k +3 +5 +10 +20 +54 +55 +56 +57 +58 +80 +82 +84 +86 +88 +Chemprot +NYT +(b) c +1 +2 +3 +4 +5 +48 +50 +52 +54 +56 +Performance (in %) +w/ Confidence (ST) +w/ Uncertainty (UST) +NeST w/o Agg. +NeST +(c) Acc. over iters. +Figure 3: Effect of different components of NeST. +too small, the data are insufficient to train an effective stu- +dent model, and when c is too large, unlabeled data tend to +be noisier and can hurt the performance. +We also inspect components of NeST, shown in Fig- +ure 3c. It is observed that both two strategies help to im- +prove performance. The aggregation module stabilizes the +self-training as the fluctuation issue has been mitigated. +5.5 +Analysis +We take a closer look at the performance of NeST and other +self-training algorithms using four text datasets. For each +dataset, we study the setting with 30 labels per class. +Error of Pseudo Labels. +To demonstrate how NeST re- +duces the noise of pseudo labels, we compare the error rate +of pseudo labels selected by ST, UST and NeST. From Fig- +ure 4 we can notice that ST and UST tend to have high er- +ror rates due to their sole reliance on model prediction, and +UST cannot stably improve the denoising ability of ST. In +contrast, NeST significantly reduces the pseudo labels error +rate by 36.8% on average compared to the best baseline. As +a result, cleaner pseudo labels lead to performance gain. +2 +3 +4 +5 +Error Rate of Selected +Pseudo Labels (in %) +Amazon +1 +3 +5 +7 +NYT +17 +19 +21 +23 +Chemprot +2 +4 +6 +8 +AGNews +Selected w/ Confidence (ST) +Selected w/ Uncertainty (UST) +NeST w/o Agg. +NeST +Figure 4: Error rates of pseudo labels selected by different +methods. Agg. is the aggregation technique in Section 4.2. +0 +500 +1000 +Running Time (in sec) +Amazon +0 +250 +500 +750 +NYT +0 +100 +200 +Chemprot +0 +500 +1000 +1500 +AGNews +UST +CEST +NeST +Figure 5: Running time of different methods. +Running Time. +We compare the running time for one +self-training iteration of NeST with UST and CEST, which +are the two strongest baselines using model uncertainty. As +shown in Figure 5, the running time is reduced by 57.3% on +average. The gain is even more significant on larger datasets +(e.g., AGNews) where inference multiple times becomes the +efficiency bottleneck. Instead, the KNN operation takes less +than 2 seconds with FAISS. To sum up, NeST is more effi- +cient and can be readily combined with self-training. +6 +Conclusion +In this paper, we propose NeST to improve sample selec- +tion in self-training for robust label efficient learning. We +design a neighborhood-regularized approach to select more +reliable samples based on representations for self-training. +Moreover, we propose to aggregate the predictions on differ- +ent iterations to stabilize self-training. Experiments on four +text datasets and four graph datasets show that NeST outper- +forms the baselines by 1.83% and 2.51% on average. NeST +also significantly reduce the noise in pseudo labels by 36.8% +and reduce the running time by 57.3% when compared with +the strongest baseline. For future works, we plan to extend +NeST to other application domains and modalities. + +Acknowledgement +We thank the anonymous reviewers for the valuable feed- +backs. This research was partially supported by the in- +ternal funds and GPU servers provided by the Computer +Science Department of Emory University. In addition, YY +and CZ were partly supported by NSF IIS-2008334, IIS- +2106961, and CAREER IIS-2144338. JH was supported by +NSF grants IIS-1838200 and IIS-2145411. +References +Arazo, E.; Ortego, D.; Albert, P.; O’Connor, N. E.; and +McGuinness, K. 2020. +Pseudo-labeling and confirmation +bias in deep semi-supervised learning. In IJCNN. +Bragg, J.; Cohan, A.; Lo, K.; and Beltagy, I. 2021. Flex: +Unifying evaluation for few-shot nlp. NeurIPS. +Chen, J.; Yang, Z.; and Yang, D. 2020. +Mixtext: +Linguistically-informed interpolation of hidden space for +semi-supervised text classification. In ACL. +Chen, T.; Kornblith, S.; Norouzi, M.; and Hinton, G. 2020. +A simple framework for contrastive learning of visual repre- +sentations. In ICML. +Cohen, A. J.; Mori-S´anchez, P.; and Yang, W. 2012. Chal- +lenges for density functional theory. 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Detecting corrupted +labels without training a model to predict. In ICML. +A +Dataset Information +• AGNews (Zhang, Zhao, and LeCun 2015): This dataset +is a collection of more than one million news articles. It is +constructed by Zhang, Zhao, and LeCun (2015) choosing +the 4 largest topic classes from the original corpus. The +total number of training samples is 120K and both vali- +dation and testing are 7.6K. +• Elec (McAuley and Leskovec 2013): This dataset is a +subset of Amazon’s product reviews for binary senti- +ment classification. It is constructed by McAuley and +Leskovec (2013), including 25K training samples, and +25K testing samples. +• NYT (Meng et al. 2020): This dataset is collected using +the New York Times API. It is constructed by Meng et al. +(2020), including 30K training samples, and 3K testing +samples. +• ChemProt (Taboureau et al. 2010). This is a 10-class re- +lation extraction dataset constructed by Taboureau et al. +(2010), containing 12K training samples and 1.6K testing +samples. +• BBBP (Martins et al. 2012): This is a binary classifica- +tion task for predicting whether a compound carries the +permeability property of penetrating the blood-brain bar- +rier. It contains 2039 molecules in total. +• BACE (Subramanian et al. 2016): This is a binary clas- +sification task for predicting compounds which could act +as the inhibitors of human β-secretase 1 in the past few +years. It contains 1513 molecules in total. +• Esol (Delaney 2004) is a regression task for predicting +the solubility of compounds. It contains 1128 molecules +in total. +• Lipophilicity (Gaulton et al. 2012) is a regression task +for predicting the property that affects the molecular +membrane permeability and solubility. It contains 4200 +molecules in total. +B +Introduction of Confidence-based and +Uncertainty-based Selection Criterion +B.1 +Confidence-based Method +For confidence-based strategy, we grant higher selection +probability yi = p(xi) for sample xi with higher predictive +confidence +p(xi) ∝ +max (yi) +� +xu∈Xu max (yu) +(12) +B.2 +Uncertainty-based Method +For uncertainty-based method, the overall goal is to select +samples that model is less uncertain about. For each data +1 +3 +5 +7 +9 +68 +70 +72 +74 +76 +AUC +69 +70 +71 +72 +73 +BBBP +BACE +(a) k +1 +3 +5 +73 +74 +75 +76 +AUC +69 +70 +71 +72 +BBBP +BACE +(b) c +Figure 6: Effect of different hyperparameters of NeST for +two graph learning datasets. +sample xi ∈ Xu , the information gain B with respect to its +expected label yi as +B (yi, θ | xi, DU) = H (y | x, Xu) − Ep(θ|DU)[H(y | x, θ)] +(13) +As the above equation is computationally intractable, +BALD (Gal and Ghahramani 2016) is used to approximate +the model uncertainty as +�B (yi, θ | xi, Xu) = − +C +� +c=1 +� +1 +M +M +� +m=1 +ym +c +� +log +� +1 +M +M +� +m=1 +ym +c +� ++ 1 +M +C +� +c=1 +M +� +m=1 +ym +c log(ym +c ) +(14) +Then, the sample selection probability is calculated as +p(xi) ∝ +1 − �B (yi | xi, Xu) +� +xu∈Xu 1 − �B (yu | xu, Xu) +. +(15) +Note that, in the above equations, M denotes the number +of inferences. Usually M is set to a large value (M = 10 +is used in this work). This causes the inefficiency issue for +deep neural networks, especially for large datasets. +C +Implementations +C.1 +Computing Infrastructure +System: Ubuntu 18.04.3 LTS; Python 3.7; Pytorch 1.8. +CPU: Intel(R) Core(TM) i7-5930K CPU @ 3.50GHz. +GPU: NVIDIA A5000. +C.2 +Hyperparameters +Details of hyperparameters in text datasets is shown in Ta- +ble 6. Details of hyperparameters in graph datasets is shown +in Table 7. +D +Hyperparameter Study on Graph Datasets +We use BBBP and BACE as an example to study the ef- +fect of different hyperparameters on graph datasets, shown +in Figure 6. We find that for these two datasets, k = 5 and +c = 3 lead to the best performance. + +Hyper-parameter +AGNews +Elec +NYT +Chemprot +Dropout Ratio +0.1 +Maximum Tokens +128 +256 +160 +64 +Batch Size for Labeled Data +16 +8 +16 +32 +Batch Size for Unlabeled Data +32 +Learning Rate +2e-5 +2e-5 +1e-5 +5e-5 +Initialization Epochs +12 +15 +15 +15 +k +5 +7 +9 +7 +c +10 +20 +10 +10 +Table 6: Hyper-parameter configurations for semi-supervised text classification. +Hyper-parameter +BBBP +BACE +Esol +Lipo +Dropout Ratio +0.1 +Batch Size for Labeled Data +16 +16 +8 +8 +Batch Size for Unlabeled Data +16 +Weight Decay +1e-4 +Learning Rate +5e-5 +Initialization Epochs +10 +10 +12 +12 +k +5 +5 +3 +5 +c +3 +3 +1 +3 +Table 7: Hyper-parameter configurations for semi-supervised graph learning. +10 +12 +14 +16 +Error Rate of Selected +Pseudo Labels (in %) +BBBP +10.0 +12.5 +15.0 +17.5 +BACE +0.3 +0.4 +0.5 +0.6 +Esol +0.3 +0.4 +0.5 +Lipo +Selected w/ Confidence (ST) +Selected w/ Uncertainty (UST) +NeST w/o Agg. +NeST +Figure 7: Error rates of pseudo labels selected by different +methods. Agg. is the aggregation technique in Section 4.2. +E +Error of Pseudo Labels and Running Time +on Graph Datasets +Figure 7 shows the error of pseudo labels for four graph +learning datasets. Note that for BBBP and BACE, we show +the error rate of pseudo labels, same as the text datasets. For +Esol and Lipo, since they are regression datasets, we show +the average RMSE of the pseudo labels. Note that the lower +value indicates higher quality of pseudo labels. The results +indicate that our proposed two strategies also improve the +quality of pseudo labels. +Figure 8 shows the running time of the best baseline +(UST) and NeST. Note that CEST is designed for text data, +and cannot be directly used for molecular data. From the re- +sults, we find that NeST saves 54.1%–65.1% of the running +time. We point out that the improvement of running time is +rather smaller when compared with the text datasets. This is +mainly because the size of datasets are much smaller, thus +the additional time for multiple inferences in UST is not ex- +cessive. That being said, these results can well illustrate that +NeST is more efficient and will serve as a complement to +0 +50 +100 +150 +Running Time (in sec) +BBBP +0 +50 +100 +150 +BACE +0 +50 +100 +Esol +0 +100 +200 +300 +Lipo +UST +NeST +Figure 8: Running time of NeST and the best baseline +(UST) on four graph datasets. +existing self-training approach. + diff --git a/zdE2T4oBgHgl3EQfMgbP/content/tmp_files/load_file.txt b/zdE2T4oBgHgl3EQfMgbP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c61e57a9a942f209cf8c27f9591c048c0c9ead7 --- /dev/null +++ b/zdE2T4oBgHgl3EQfMgbP/content/tmp_files/load_file.txt @@ -0,0 +1,2084 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf,len=2083 +page_content='Neighborhood-Regularized Self-Training for Learning with Few Labels Ran Xu1, Yue Yu2, Hejie Cui1, Xuan Kan1, Yanqiao Zhu3, Joyce Ho1, Chao Zhang2, Carl Yang1* 1 Emory University, 2 Georgia Institute of Technology, 3 University of California at Los Angeles {ran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='xu,hejie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='cui,xuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='kan,joyce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='ho,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='carlyang}@emory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='edu, {yueyu,chaozhang}@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='edu, yzhu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='edu Abstract Training deep neural networks (DNNs) with limited supervi- sion has been a popular research topic as it can significantly alleviate the annotation burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Self-training has been suc- cessfully applied in semi-supervised learning tasks, but one drawback of self-training is that it is vulnerable to the la- bel noise from incorrect pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Inspired by the fact that samples with similar labels tend to share similar rep- resentations, we develop a neighborhood-based sample se- lection approach to tackle the issue of noisy pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We further stabilize self-training via aggregating the predic- tions from different rounds during sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Exper- iments on eight tasks show that our proposed method out- performs the strongest self-training baseline with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='83% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='51% performance gain for text and graph datasets on av- erage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Our further analysis demonstrates that our proposed data selection strategy reduces the noise of pseudo labels by 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8% and saves 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3% of the time when compared with the best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Our code and appendices will be uploaded to https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='com/ritaranx/NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 1 Introduction In the era of deep learning, neural network models have achieved promising performance in most supervised learn- ing settings, especially when combined with self-supervised learning techniques (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' However, they still require a sufficient amount of labels to achieve satisfactory perfor- mances on many downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For example, in the text domain, curating NLP datasets often require domain experts to read thousands of documents and carefully label them with domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Similarly, in the graph domain, molecules are examples naturally represented as graphs, and characterizing their properties relies on density functional theory (DFT) (Cohen, Mori-S´anchez, and Yang 2012) which often takes several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Such a dependency on labeled data is one of the barriers to deploy deep neural networks (DNNs) in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To better adapt the DNNs to target tasks with limited la- bels, one of the most popular approaches is semi-supervised learning (SSL), which jointly leverages unlabeled data and Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' labeled data to improve the model’s generalization power on the target task (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Although generative models (Gururangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2019) and consistency- based regularization (Tarvainen and Valpola 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Miy- ato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020a) methods have been pro- posed for semi-supervised learning, they either suffer from the issue of limited representation power (Tsai, Lin, and Fu 2022) or require additional resources to generate high- quality augmented samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=', for text classification, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2020a) generate augmented text via back-translation, which rely on a Machine Translation model trained with massive labeled sentence pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Consequently, they cannot be readily applied to low-resource scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Self-training is a proper tool to deal with the deficiency of labeled data via gradually enlarging the training set with pseudo-labeled data (Rosenberg, Hebert, and Schneiderman 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Specifically, it can be interpreted as a teacher-student framework: the teacher model generates pseudo labels for the unlabeled data, and the student model updates its pa- rameters by minimizing the discrepancy between its predic- tions and the pseudo labels (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Mukherjee and Awadallah 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Though conceptually simple, self-training has achieved superior performance for various tasks with limited labels, such as image classification (Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Rizve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021), natural language understanding (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020), sequence labeling (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020), and graph learning (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Self-training has also been suc- cessfully extended to other settings including weak super- vision (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021b) and zero-shot learning (Li, Savarese, and Hoi 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' However, one major challenge of self-training is that it suffers from confirmation bias (Arazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) — when the teacher model memorizes some biases and generates incorrect pseudo labels, the student model will be rein- forced to train with these wrong biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' As a result, the biases may amplify over iterations and deteriorates the fi- nal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To suppress the noisy pseudo labels in self-training, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2021a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2022b) leverage model predic- tive confidence with a thresholding function, Mukherjee and Awadallah (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Tsai, Lin, and Fu (2022) propose to leverage model uncertainty to select samples with low un- certainty, and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2021) use meta-learning to con- duct instance reweighting for sequence labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Although arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='03726v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='LG] 10 Jan 2023 these approaches attempt to reduce the label noise, they se- lect the data for self-training based on the model prediction only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' However, the predictions of the deep neural network can be over-confident and biased (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Kan, Cui, and Yang 2021), and directly using such predictions without any intervention to filter pseudo la- bels cannot effectively resolve the label noise issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Another problem from self-training is training instability, as it se- lects pseudo-labeled data only based on the prediction of the current round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Due to the stochasticity involved in training neural networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=', random initialization, training order), the prediction can be less stable (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022b), especially for the noisy pseudo-labeled data (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Conse- quently, the noise in the previous rounds may propagate to later rounds, which deteriorate the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Motivated by the above, we propose NeST, a simple yet powerful approach guided by the data representations, to boost the performance of self-training for few-shot learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Inspired by recent works indicating that the represen- tations from deep neural networks can be discriminative and less affected by noisy labels (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021), we harness the features learned from the neural models to select the most reliable samples in self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In addition, several works have indicated that samples within the same category tend to share similar representations, such as category-guided text mining (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) and motif-driven graph learn- ing (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Similarly, we hypothesize that a sample’s pseudo label is more likely to be correct only if its prediction is similar to the neighbor labeled instances in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To fulfill the denoising purpose, NeST creates the neighborhood for each unlabeled data by find- ing the top-k nearest labeled samples, then calculates the divergence between its current prediction and the label of its neighbors to rank the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' As a result, only the in- stances with the lowest divergence will be selected for self- training, which mitigates the issue of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Moreover, to robustly select the training samples for self-training, we aggregate the predictions on different iterations to promote samples that have lower uncertainty over multiple rounds for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We remark that NeST is an efficient substitution for exist- ing self-training approaches and can be combined with vari- ous neural architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The contributions of this paper are: We propose NeST to improve the robustness of self- training for learning with few labels only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We design two additional techniques, namely neighborhood-regularized sample selection to re- duce label noise, and prediction aggregation to alleviate the training instability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Experiments on 4 text datasets and 4 graph datasets with different volumes of labeled data verify that NeST im- proves the performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='83% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='51% respec- tively and saves the running time by 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2 Related Work Self-training is one of the earliest approaches to semi- supervised learning (Rosenberg, Hebert, and Schneiderman 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The method uses a teacher model to generate new la- bels on which a student model is fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The major drawback of self-training is that it is vulnerable to label noise (Arazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' There are several popular approaches to stabi- lize the self-training process, such as using sample selec- tion (Mukherjee and Awadallah 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) and reweighting strategies (Zhou, Kantarcioglu, and Thuraising- ham 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021) to filter noisy labels or design- ing noise-aware loss functions (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Tsai, Lin, and Fu 2022) to improve the model’s ro- bustness against incorrectly labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In addition, data augmentation methods (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Chen, Yang, and Yang 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhang, Yu, and Zhang 2020) are also com- bined with self-training to improve the model’s generaliza- tion ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Leveraging representation information has also been ex- plored in semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For example, Li, Xiong, and Hoi (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2021) improve the representation via contrastive learning to assist semi- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Moreover, ACPL (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022) and SimMatch (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022) aggregates the labels from their neighbors in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' While these approaches also attempt to harness sample representations, they do not directly denoise the pseudo labeled data for boosting the per- formance of self-training, which is the focus of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' One concurrent work (Lang, Vijayaraghavan, and Sontag 2022) combines data representations with the cut statistic to select high quality training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In particular, it aims to select reliable subsets directly from the weakly-labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Instead, our work focuses on using clean labeled data to bet- ter denoise instances in self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 3 Preliminaries In this section, we first present the setup of semi-supervised learning and self-training, and then point out issues of the existing sample selection algorithms for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Task Definition In this paper, we study the semi-supervised learning prob- lem, which is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Given a few labeled Xl = {(xi, yi)}L i=1 and unlabeled data Xu = {xj}U j=1 (L ≪ U), we seek to learn a predictor f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ) : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Here X = Xl ∪ Xu denotes all the input data and Y is the label set, which can either be discrete (for classification) or con- tinuous (for regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ) is either a C-dimensional probability simplex for classification where C is the number of classes or a continuous value for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 Introduction to Self-training Self-training can be interpreted as a teacher-student frame- work (Mukherjee and Awadallah 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020b), with θt and θs denoting the teacher and student model, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The process of the self-training is in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We discuss the key components in self-training as belows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Initialization of Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The labeled data Xl are used to initialize the models as θ(0) s = θ(0) t = θinit, where θinit = min θ Lsup(θ) = E(xi,yi)∈Xlℓsup (f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ), yi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (1) Algorithm 1: Procedures of Self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Input: Labeled and unlabeled samples Xl, Xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Neural prediction model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Unlabeled set � Xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Number of self-training iterations T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Number of steps in each iteration T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' // Train the model on labeled data Xl as initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Update θs, θt by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 1 using Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' // Self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' for t = 1, 2, · · · , T do Select � X tu (|� X tu| = b) with θt by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Adding � X tu for self-training � Xu = � Xu ∪ � X tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Update pseudo labels �y by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2 or 3 for � X tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' for k = 1, 2, · · · , T1 do Sample a minibatch B from � Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Update θs with loss L in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5 using Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Update teacher model θt ← θs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Output: Final model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ℓsup(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ·) represents the supervised loss, which is the cross- entropy loss for classification and the mean squared error loss for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Pseudo Label Generation with Teacher Model θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We use the teacher model’s prediction f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θt) to generate pseudo labels for Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For classification problems, the pseudo labels can be written as �yhard,j = �1, if j = argmax k∈Y [f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θt)]k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 0, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2) For the regression task, since the output is a continuous value, the teacher model’s output is directly used as the pseudo label �y = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (3) Sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Directly using all the pseudo-labeled data for self-training often yields sub-optimal results, as the erroneous pseudo labels hurt the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To mitigate this issue, recent works attempt to select only a subset of the unlabeled data for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We denote the sample policies as ψ(·), which can be generally written as � Xu = ψ (Xu, f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θt)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (4) We omit the superscript for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The common choice for ψ(·) including using predictive confidence (Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021a) or model uncertainty (Mukherjee and Awadallah 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Tsai, Lin, and Fu 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Model Training and Update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' With the generated pseudo labels, we then train a student model θs to minimize the loss for both labeled and unlabeled data by solving min θs λLsup(θs) + (1 − λ)Exj∈ � Xuℓst (f(xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θs), �yj) , (5) where Lsup is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 1, � Xu is obtained via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 4, and ℓst = 1{[f(xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θs)]�yj > γ} · ℓsup is the loss function for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Error Rate of Selected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Pseudo Labels (in %) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Elec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Chemprot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='All the Unlabeled Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Selected w/ Confidence (ST) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Selected w/ Uncertainty (UST) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Performance (in %) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Elec upper bound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Chemprot upper bound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Elec - Confidence (ST) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Elec - Uncertainty (UST) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Chemprot - Confidence (ST) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Chemprot - Uncertainty (UST) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Performance (in %) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Figure 1: Left: The average error rate of all pseudo labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='and the selected pseudo labels with different selection strate- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Middle: The performance on different self-training it- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The upper bounds are the accuracy with full clean labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Right: Zoom in of the performance in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' unlabeled data with the thresholding function (Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We iterate the process by treating the trained student model as the teacher to generate new pseudo labels and train a new student model based on the new gen- erated labels until the model converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 Challenges of Self-training To illustrate that the existing sample selection approaches are flawed and cannot resolve the label noise issue, we first demonstrate the performance of two widely-used selection criteria: predictive confidence (ST (Rosenberg, Hebert, and Schneiderman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020)) and model uncertainty (UST (Mukherjee and Awadallah 2020)) for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The details of these two approaches are dis- cussed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Note that, for these two approaches, we follow the original implementation to select the unla- beled set Xu in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We use a binary sentiment classification dataset Elec and a chemical relation extraction dataset Chemprot with ten classes as an example for easier and harder task, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For both datasets, we first train the model with 30 clean labels per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Figure 1 shows the error rate of pseudo labels selected fol- lowing these two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We observe that these two meth- ods are effective on easier tasks, where the selected data has a relatively low error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It achieves comparable per- formance with the fully-supervised method (95%) with less than 1% of the clean labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' However, for more chal- lenging tasks with a larger number of classes, the perfor- mance of the initial model may not be satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The er- ror rate of pseudo labels increases up to 16% on Chemprot compared with Elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Consequently, the gap between semi- supervised learning and fully-supervised learning is even larger — more than 25% in terms of F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This phe- nomenon suggests that the label noise issue is still the major challenge that hampers the self-training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' More- over, using model uncertainty (Gal and Ghahramani 2016) for sample selection does not fully address this challenge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' the gain can be marginal on harder datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Apart from the label noise, we also observe performance fluctuations over different self-training rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We name this as the training instability issue, which occurs when the teacher model only looks at the previous round and memo- rizes the label noise specifically in that round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Then in the next iteration, the student model can easily overfit the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Student model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Teacher model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Momentum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Step 3: Continue training the student model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Step 2: Select samples for self-training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Step 1: Initialize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='with labeled data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Selected unlabeled data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='with pseudo labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='divergence ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=", % + '!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='#(%) Aggregate over iterations Figure 2: The framework of NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Red and blue points stand for labeled data with different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' White points rep- resent unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Light red and blue points stand for predictions of unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 4 Method We present NeST to improve the stability of self-training by tackling the challenges mentioned in the previous sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The overview of NeST is in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Notably, we fo- cus on the sample selection step (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 4), and we propose two key components, namely neighborhood-regularized se- lection strategy and prediction aggregation to promote the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The details of the two designs will be dis- cussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Neighborhood-regularized Sample Selection Prior works have demonstrated that leveraging embeddings from the deep neural networks can identify the noisy labeled data (Zhu, Dong, and Liu 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Motivated by this, we pro- pose to harness the similarities in the embedding space to mitigate the issue of erroneous pseudo labels in self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Concretely, for each unlabeled sample xj with representa- tion vj, we adopt the k-nearest neighbors (KNN) algorithm to find the most similar labeled samples in the feature space: Nj = {xi | xi ∈ Xl ∩ KNN(vj, Xl, k)}, (6) where KNN(vj, Xl, k) denotes k labeled examples in Xl that are nearest to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Divergence-based Sample Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We then calculate the scores for unlabeled samples xj ∈ Xu based on the weighted divergence D(xj) = Du(xj, N) + βDl(N), (7) where unlabeled divergence Du and labeled divergence Dl are defined below, and β is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This score D(xj) will be further used for sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Unlabeled Divergence Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For each sample xj in the un- labeled set with the neighbor set N, we calculate the diver- gence between the prediction of xj and labeled data in N as Du(xj, N) = � (xi,yi)∈N d(f(xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θt), yi), (8) where d is the Kullback–Leibler (KL) divergence for classi- fication and L2 distance for regression (same as follows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To interpret Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 8, we note that samples having the prediction close to the nearest labeled instances will have lower Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Labeled Divergence Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It measures the divergence among the labels within the neighbor set N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We first cal- culate the average label y = � (xi,yi)∈N yi |N |, and then mea- sure the labeled divergence as Dl(N) = � (xi,yi)∈N d(y, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (9) For each group N, samples with similar labels will have smaller divergence Dl(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To summarize, a low divergence score D(xj) indicates that the prediction of the unlabeled data point xj is close to its neighbors, and the labels of its neighbors are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Thus, we hypothesize that such samples are more likely to be correct and use them for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 Robust Aggregation of Predictions from Different Iterations The results in Figure 1c clearly demonstrate that only using the prediction on the current iteration for sample selection cause training instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To effectively mitigate the bias in the current iteration and stabilize the self-training, we pro- pose to exploit the training prediction at different training iterations more robustly (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To achieve this, we aggregate the value D(t)(xj) in the t-th round as µ(t)(xj) = (1−m)×µ(t−1)(xj)+m× � D(t)(xj) � , (10) where m is a hyperparameter bounded between 0 and 1 that controls the weight for previous rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To interpret Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 10, we argue that µ(xj) will be small only when the model outputs consistently low scores for a sam- ple xj in different iterations of self-training, as the model is more certain about these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' On the contrary, if the model gives inconsistent predictions in different iterations, then the model is potentially uncertain about the prediction, thus adding its pseudo label in the next iteration may hurt the self-training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Motivated by this idea, we remove the sample with inconsistent predictions over different iterations to further suppress noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To put the above two strategies together, our policy for sample selection in the t-th round ψ(·) is mainly based on the value of µ(t)(xj) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Specifically, in t-th iteration, we sample instances xj ∈ Xu without replacement using the probability p(xj) ∝ W − µ(t)(xj) � xu∈Xu (W − µ(t)(xu)), (11) where W = maxx∈Xu(µ(t)(x)) is the normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Our method introduces little computation over- head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For each unlabeled data, the neighborhood regularized sampling requires one extra kNN operation, which can be efficiently supported via FAISS (Johnson, Douze, and J´egou 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The µ(t)(xj) from previous iterations can be cached on disk and merged when selecting the training data for the new iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Other than the sample selection method ψ(·), NeST keeps other components intact and can be plugged- in with any noise-robust learning techniques (Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021) and neural architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Dataset Domain Task # Train / Test # Class Metric Elec Reviews Sentiment Analysis 25K / 25K 2 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' AG News News Topic Classification 120K / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6K 4 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' NYT News Topic Classification 30K / 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0K 9 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Chemprot Chemical Relation Classification 12K / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6K 10 F1 BBBP Physiology Classification 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6k / 204 2 ROC-AUC BACE Biophysics Classification 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2k / 151 2 ROC-AUC Esol Physical Chemistry Regression 902 / 112 — RMSE Lipophilicity Physical Chemistry Regression 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3k / 420 — RMSE Table 1: Statistics of text and graph datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5 Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Experiment Setup We conduct experiments for semi-supervised learning on eight datasets to demonstrate the efficacy of NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Four of them are text-related tasks, including text classification and relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We employ the pre-trained BERT from the HuggingFace (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2019) codebase for the imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The other four are graph-based tasks, where we choose molecular property prediction as the main task and use pre-trained Grover-base 1 (Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The same backbone is used for both NeST and baselines to ensure a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Semi-supervised Learning Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For each dataset, we train our method and baselines with different numbers of labeled data from {30, 50, 100} per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The remaining in the training set is considered as unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' As sug- gested by Bragg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2021), we keep the size of the val- idation set to be the same as the number of labeled data to simulate the realistic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For each dataset, we apply 3 runs on 3 splits and report the mean and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Parameter Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We use Adam (Kingma and Ba 2014) as the optimizer and tune the learning rate in {1e- 5, 2e-5, 5e-5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The batch size is selected from {8, 16, 32}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Other hyperparameters in NeST include T, T1, γ for self- training, β, b, k for sample selection in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 7, and λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We set β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='9, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6, T = 5, T1 = 1000 for all datasets, and tune b = c|Xl| with c ∈ {3, 5, 10, 20} for text datasets and c ∈ {1, 3, 5} for graph datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We study the effect of k and c in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Details for each dataset are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We compare NeST with the following base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We use † to represent baselines designed for text-based tasks and ‡ to represent baselines for graph-based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' BERT† (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) is the super- vised baseline for text-based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Grover‡ (Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) is the supervised baseline for molecular property prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Mean-Teacher (MT) (Tarvainen and Valpola 2017) up- dates the teacher model as a moving average of the stu- 1Since we do not focus on developing better self-supervised learning methods, we do not compare with other GNN pretraining models (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021) dent model’s weight and adds a consistency regulariza- tion between the student and teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Virtual Adversarial Training (VAT) (Miyato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2018) adds a regularization term between the sample with the adversarial noise and its prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Self-training (ST) (Rosenberg, Hebert, and Schneider- man 2005) is a conventional self-training method that adds most confident pseudo labeled data to labeled set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' UST (Mukherjee and Awadallah 2020) selects data with lowest uncertainty using MC-dropout for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' UDA† (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020a) adopts back translation and TF- IDF word replacement as the data augmentation and adds consistency loss on predictions on the augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' MixText† (Chen, Yang, and Yang 2020) interpolates training data in the hidden space via Mixup as the data augmentation to improve model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' CEST† (Tsai, Lin, and Fu 2022) improves the UST method by designing the contrastive loss over sample pairs and noise-aware loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' InfoGraph‡ (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) is a semi-supervised graph classification method via maximizing mutual informa- tion between graph and substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ASGN‡ (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) is a semi-supervised molecular property prediction that jointly exploits information from molecular structure and overall distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 Semi-supervised Learning on Text Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We conduct experiments on four widely used datasets in NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We adopt Elec (McAuley and Leskovec 2013) for sentiment classification, AGNews (Zhang, Zhao, and LeCun 2015) and NYT (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) for topic clas- sification, and Chemprot (Taboureau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2010) for chemi- cal relation extraction in this set of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The statis- tics and evaluation metrics for each dataset are shown in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We use BioBERT (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020) as the backbone for Chemprot as it is a domain specific dataset (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022) and use RoBERTa-base for other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Table 2 summarizes the experimental results on text datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We observe that NeST outperforms all base- lines across all the four datasets under different volumes of labeled data, and the performance gain compared to the best baseline is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='83% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Note that UDA and MixText require additional data augmentation, which can be Method AG News (Accuracy, ↑) Elec (Accuracy, ↑) NYT (Accuracy, ↑) Chemprot (F1, ↑) 30 50 100 30 50 100 30 50 100 30 50 100 BERT (2019) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 54.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 Supervised 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0∗ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3∗ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 Table 2: Performance on four datasets with various amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The higher value always indicates better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Bold and underline indicate the best and second best results for each dataset, respectively (Same as below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ∗: The number is reported from the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The implementation of CEST is not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' †: The result is lower than the reported result in the original paper since they use a much larger development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ‡: We remove the noise-aware loss as well as graph-based regularization for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The effect of these two terms is presented in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Method AG News (Accuracy, ↑) Elec (Accuracy, ↑) 30 50 100 30 50 100 CEST 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 CEST w/ NRL 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8 NeST 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 NeST w/ NRL 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 Table 3: Performance comparison of CEST (Tsai, Lin, and Fu 2022) and NeST with noise-robust loss functions (NRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Instead, NeST does not lever- age any external resources but achieves better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For other self-training baselines, we observe that they cannot outperform our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' As we keep other components unchanged, the gain is mainly due to the pseudo-labeled data denosing benefit of NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We will il- lustrate this in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We also notice that the per- formance gain is more prominent on NYT and Chemprot datasets which have more classes, indicating NeST can be better adapted to tasks with fine-grained classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Incorporating Noise-robust Loss Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To demon- strate NeST can be combined with other loss functions, Ta- ble 3 further compares NeST with CEST (Tsai, Lin, and Fu 2022) which adds additional noise-aware loss (Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021) and graph-based regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The results show that these components can further improve the performance of NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Under both settings, NeST outperforms CEST, which justifies the efficacy of our proposed strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 Semi-supervised Learning on Graphs Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We choose molecular property prediction as the target task for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We conduct ex- periments on four widely used datasets from the Molecu- leNet (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2018), including BBBP (Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2012), BACE (Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2016), Esol (Delaney 2004) and Lipophilicity (Gaulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The statistics and evaluation metrics for each dataset are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' From the results in Table 4, we can see that NeST outperforms all the baselines on all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In particular, the performance gain compared to the best baseline is around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Compared to the Grover model using labeled data only, the gain is around 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Notice that the traditional self-training method (ST) sometimes performs even worse than Grover fine-tuned on labeled data only, because confidence-based selection introduces large label noise, which leads to many wrong predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' With proper control of noisy pseudo la- bels, UST generally outperforms other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' However, since they do not consider neighbor information, their per- formance is not as good as NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Adapting NeST to different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We use two datasets as an example to demonstrate that NeST can be adapted to different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Table 5 shows the results of training an AttentiveFP (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2019), another popu- lar GNN backbone based on graph attention networks for molecular property prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Unlike Grover, AttentiveFP is not pre-trained on massive molecules, but trained from scratch in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We can see that the perfor- mance of AttentiveFP is worse than Grover in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' A key reason is that pre-trained Grover has considerably more parameters than AttentiveFP, and incorporates rich do- main knowledge with self-supervised learning on unlabeled molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Nevertheless, NeST still outperforms all the baselines by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5% on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This indicates that NeST does not rely on any specific architecture, and it serves as an effective plug-in module for different GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 Parameter and Ablation Studies We study the effect of different parameters of NeST on NYT and Chemprot with 30 labels per class, shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Results on other datasets are in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The perfor- mance first improves as k increases, because larger k allows more labeled data in each neighborhood, introducing less randomness and regularizing divergence calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' When k > 7, the performance drops as the neighborhood includes labeled data far away, no longer serving as an appropriate regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Similarly, as c gets larger, the performance 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='563 Table 4: Performance on four datasets with various amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For classification datasets (BBBP, BACE), the higher value indicates better performance, while for regression datasets (Esol, Lipo), the lower value stands for better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Bold and underline indicate the best and second best results, respectively (Same as below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Method BBBP (ROC-AUC, ↑) BACE (ROC-AUC, ↑) 30 50 100 30 50 100 AttentiveFP (2019) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='9±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 ASGN (2020) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 84.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 InfoGraph (2020) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 ST (2005) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 NeST 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 Table 5: Performance on two datasets with various amounts of labeled data using AttentiveFP as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 1 3 5 7 9 54 55 56 57 58 83 84 85 86 87 Chemprot NYT (a) k 3 5 10 20 54 55 56 57 58 80 82 84 86 88 Chemprot NYT (b) c 1 2 3 4 5 48 50 52 54 56 Performance (in %) w/ Confidence (ST) w/ Uncertainty (UST) NeST w/o Agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' NeST (c) Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' over iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Figure 3: Effect of different components of NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' too small, the data are insufficient to train an effective stu- dent model, and when c is too large, unlabeled data tend to be noisier and can hurt the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We also inspect components of NeST, shown in Fig- ure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It is observed that both two strategies help to im- prove performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The aggregation module stabilizes the self-training as the fluctuation issue has been mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 Analysis We take a closer look at the performance of NeST and other self-training algorithms using four text datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For each dataset, we study the setting with 30 labels per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Error of Pseudo Labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To demonstrate how NeST re- duces the noise of pseudo labels, we compare the error rate of pseudo labels selected by ST, UST and NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' From Fig- ure 4 we can notice that ST and UST tend to have high er- ror rates due to their sole reliance on model prediction, and UST cannot stably improve the denoising ability of ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In contrast, NeST significantly reduces the pseudo labels error rate by 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8% on average compared to the best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' As a result, cleaner pseudo labels lead to performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2 3 4 5 Error Rate of Selected Pseudo Labels (in %) Amazon 1 3 5 7 NYT 17 19 21 23 Chemprot 2 4 6 8 AGNews Selected w/ Confidence (ST) Selected w/ Uncertainty (UST) NeST w/o Agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' NeST Figure 4: Error rates of pseudo labels selected by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' is the aggregation technique in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 0 500 1000 Running Time (in sec) Amazon 0 250 500 750 NYT 0 100 200 Chemprot 0 500 1000 1500 AGNews UST CEST NeST Figure 5: Running time of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Running Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We compare the running time for one self-training iteration of NeST with UST and CEST, which are the two strongest baselines using model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' As shown in Figure 5, the running time is reduced by 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The gain is even more significant on larger datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=', AGNews) where inference multiple times becomes the efficiency bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Instead, the KNN operation takes less than 2 seconds with FAISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' To sum up, NeST is more effi- cient and can be readily combined with self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 6 Conclusion In this paper, we propose NeST to improve sample selec- tion in self-training for robust label efficient learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We design a neighborhood-regularized approach to select more reliable samples based on representations for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Moreover, we propose to aggregate the predictions on differ- ent iterations to stabilize self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Experiments on four text datasets and four graph datasets show that NeST outper- forms the baselines by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='83% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='51% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' NeST also significantly reduce the noise in pseudo labels by 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8% and reduce the running time by 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3% when compared with the strongest baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For future works, we plan to extend NeST to other application domains and modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Acknowledgement We thank the anonymous reviewers for the valuable feed- backs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This research was partially supported by the in- ternal funds and GPU servers provided by the Computer Science Department of Emory University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In addition, YY and CZ were partly supported by NSF IIS-2008334, IIS- 2106961, and CAREER IIS-2144338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' JH was supported by NSF grants IIS-1838200 and IIS-2145411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' References Arazo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Ortego, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Albert, P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Structure-enhanced heterogeneous graph contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In SDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' and Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Graph contrastive learning with adaptive augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Dong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' and Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Detecting corrupted labels without training a model to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' In ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' A Dataset Information AGNews (Zhang, Zhao, and LeCun 2015): This dataset is a collection of more than one million news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It is constructed by Zhang, Zhao, and LeCun (2015) choosing the 4 largest topic classes from the original corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The total number of training samples is 120K and both vali- dation and testing are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Elec (McAuley and Leskovec 2013): This dataset is a subset of Amazon’s product reviews for binary senti- ment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It is constructed by McAuley and Leskovec (2013), including 25K training samples, and 25K testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' NYT (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2020): This dataset is collected using the New York Times API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It is constructed by Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2020), including 30K training samples, and 3K testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' ChemProt (Taboureau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This is a 10-class re- lation extraction dataset constructed by Taboureau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (2010), containing 12K training samples and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6K testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' BBBP (Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2012): This is a binary classifica- tion task for predicting whether a compound carries the permeability property of penetrating the blood-brain bar- rier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It contains 2039 molecules in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' BACE (Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2016): This is a binary clas- sification task for predicting compounds which could act as the inhibitors of human β-secretase 1 in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It contains 1513 molecules in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Esol (Delaney 2004) is a regression task for predicting the solubility of compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It contains 1128 molecules in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Lipophilicity (Gaulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 2012) is a regression task for predicting the property that affects the molecular membrane permeability and solubility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' It contains 4200 molecules in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' B Introduction of Confidence-based and Uncertainty-based Selection Criterion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Confidence-based Method For confidence-based strategy, we grant higher selection probability yi = p(xi) for sample xi with higher predictive confidence p(xi) ∝ max (yi) � xu∈Xu max (yu) (12) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 Uncertainty-based Method For uncertainty-based method, the overall goal is to select samples that model is less uncertain about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For each data 1 3 5 7 9 68 70 72 74 76 AUC 69 70 71 72 73 BBBP BACE (a) k 1 3 5 73 74 75 76 AUC 69 70 71 72 BBBP BACE (b) c Figure 6: Effect of different hyperparameters of NeST for two graph learning datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' sample xi ∈ Xu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' the information gain B with respect to its expected label yi as B (yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ | xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' DU) = H (y | x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xu) − Ep(θ|DU)[H(y | x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ)] (13) As the above equation is computationally intractable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' BALD (Gal and Ghahramani 2016) is used to approximate the model uncertainty as �B (yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' θ | xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xu) = − C � c=1 � 1 M M � m=1 ym c � log � 1 M M � m=1 ym c � + 1 M C � c=1 M � m=1 ym c log(ym c ) (14) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' the sample selection probability is calculated as p(xi) ∝ 1 − �B (yi | xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xu) � xu∈Xu 1 − �B (yu | xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Xu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' (15) Note that, in the above equations, M denotes the number of inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Usually M is set to a large value (M = 10 is used in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This causes the inefficiency issue for deep neural networks, especially for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' C Implementations C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Computing Infrastructure System: Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 LTS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' CPU: Intel(R) Core(TM) i7-5930K CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='50GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' GPU: NVIDIA A5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2 Hyperparameters Details of hyperparameters in text datasets is shown in Ta- ble 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Details of hyperparameters in graph datasets is shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' D Hyperparameter Study on Graph Datasets We use BBBP and BACE as an example to study the ef- fect of different hyperparameters on graph datasets, shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We find that for these two datasets, k = 5 and c = 3 lead to the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Hyper-parameter AGNews Elec NYT Chemprot Dropout Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Maximum Tokens 128 256 160 64 Batch Size for Labeled Data 16 8 16 32 Batch Size for Unlabeled Data 32 Learning Rate 2e-5 2e-5 1e-5 5e-5 Initialization Epochs 12 15 15 15 k 5 7 9 7 c 10 20 10 10 Table 6: Hyper-parameter configurations for semi-supervised text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Hyper-parameter BBBP BACE Esol Lipo Dropout Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1 Batch Size for Labeled Data 16 16 8 8 Batch Size for Unlabeled Data 16 Weight Decay 1e-4 Learning Rate 5e-5 Initialization Epochs 10 10 12 12 k 5 5 3 5 c 3 3 1 3 Table 7: Hyper-parameter configurations for semi-supervised graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' 10 12 14 16 Error Rate of Selected Pseudo Labels (in %) BBBP 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 BACE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='6 Esol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='5 Lipo Selected w/ Confidence (ST) Selected w/ Uncertainty (UST) NeST w/o Agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' NeST Figure 7: Error rates of pseudo labels selected by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' is the aggregation technique in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' E Error of Pseudo Labels and Running Time on Graph Datasets Figure 7 shows the error of pseudo labels for four graph learning datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Note that for BBBP and BACE, we show the error rate of pseudo labels, same as the text datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' For Esol and Lipo, since they are regression datasets, we show the average RMSE of the pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Note that the lower value indicates higher quality of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' The results indicate that our proposed two strategies also improve the quality of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Figure 8 shows the running time of the best baseline (UST) and NeST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' Note that CEST is designed for text data, and cannot be directly used for molecular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' From the re- sults, we find that NeST saves 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1%–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content='1% of the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' We point out that the improvement of running time is rather smaller when compared with the text datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' This is mainly because the size of datasets are much smaller, thus the additional time for multiple inferences in UST is not ex- cessive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' That being said, these results can well illustrate that NeST is more efficient and will serve as a complement to 0 50 100 150 Running Time (in sec) BBBP 0 50 100 150 BACE 0 50 100 Esol 0 100 200 300 Lipo UST NeST Figure 8: Running time of NeST and the best baseline (UST) on four graph datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'} +page_content=' existing self-training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfMgbP/content/2301.03726v1.pdf'}